程序代写代做代考 Java scheme chain Excel flex matlab cache Fortran database algorithm data structure gui Hive Preface
Preface
Numerical Computing with MATLAB is a textbook for an introductory course
in numerical methods, Matlab, and technical computing. The emphasis is on in-
formed use of mathematical software. We want you learn enough about the mathe-
matical functions in Matlab that you will be able to use them correctly, appreciate
their limitations, and modify them when necessary to suit your own needs. The
topics include
• introduction to Matlab,
• linear equations,
• interpolation,
• zero and roots,
• least squares,
• quadrature,
• ordinary differential equations,
• random numbers,
• Fourier analysis,
• eigenvalues and singular values,
• partial differential equations.
George Forsythe initiated a software-based numerical methods course at Stan-
ford University in the late 1960s. The textbooks by Forsythe, Malcolm, and Moler
[1] and Kahaner, Moler, and Nash [2] that evolved from the Stanford course were
based upon libraries of Fortran subroutines.
This textbook is based upon Matlab. NCM, a collection of over 70 M-
files, forms an essential part of the book. Many of the over 200 exercises involve
modifying and extending the programs in NCM. The book also makes extensive
use of computer graphics, including interactive graphical expositions of numerical
algorithms.
The prerequisites for the course, and the book, include
February 15, 2008
1
2 Preface
• calculus,
• some familiarity with ordinary differential equations,
• some familiarity with matrices,
• some computer programming experience.
If you’ve never used Matlab before, the first chapter will help you get started. If
you’re already familiar with Matlab, you can glance over most of the first chapter
quickly. Everyone should read the section in the first chapter about floating-point
arithmetic.
There is probably too much material here for a one-quarter or one-semester
course. Plan to cover the first several chapters and then choose the portions of the
last four chapters that interest you.
Make sure that the NCM collection is installed on your network or your per-
sonal computer as you read the book. The software is available from a Web site
devoted to the book [3]:
http://www.mathworks.com/moler
There are three types of NCM files:
• gui files: interactive graphical demonstrations;
• tx files: textbook implementations of built-in Matlab functions;
• others: miscellaneous files, primarily associated with exercises.
When you have NCM available,
ncmgui
produces the figure shown on the next page. Each thumbnail plot is actually a push
button that launches the corresponding gui.
This book would not have been possible without the people at The MathWorks
and at SIAM. Both groups are professional, creative, and delightful to work with.
They have been especially supportive of this book project. Out of the many friends
and colleagues who have made specific contributions, I want to mention five in
particular. Kathryn Ann Moler has used early drafts of the book several times in
courses at Stanford and has been my best critic. Tim Davis and Charlie Van Loan
wrote especially helpful reviews. Lisl Urban did an immaculate editing job. My wife
Patsy has lived with my work habits and my laptop and loves me anyway. Thanks,
everyone.
This second printing includes a revision of the section on Google PageRank
that improves the handling of web pages with no out links, a short new section
in the Random Numbers chapter, removal of material on inline and feval, and
correction of a few dozen minor typographical errors.
Cleve Moler
February 15, 2008
Preface 3
ncmgui.
4 Preface
Bibliography
[1] G. Forsythe, M. Malcolm, and C. Moler, Computer Methods for Math-
ematical Computations, Prentice–Hall, Englewood Cliffs, NJ, 1977.
[2] D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software,
Prentice–Hall, Englewood Cliffs, NJ, 1989.
[3] C. Moler, Numerical Computing with MATLAB,
Electronic edition: The MathWorks, Inc., Natick, MA, 2004.
http://www.mathworks.com/moler
Print edition: SIAM, Philadelphia, 2004.
http://www.ec-securehost.com/SIAM/ot87.html
5
Chapter 1
Introduction to MATLAB
This book is an introduction to two subjects: Matlab and numerical computing.
This first chapter introduces Matlab by presenting several programs that inves-
tigate elementary, but interesting, mathematical problems. If you already have
some experience programming in another language, we hope that you can see how
Matlab works by simply studying these programs.
If you want a more comprehensive introduction, an on-line manual from The
MathWorks is available. Select Help in the toolbar atop the Matlab command
window, then select MATLAB Help and Getting Started. A PDF version is
available under Printable versions. The document is also available from The
MathWorks Web site [10]. Many other manuals produced by The MathWorks are
available on line and from the Web site.
A list of over 600 Matlab-based books by other authors and publishers, in sev-
eral languages, is available at [11]. Three introductions to Matlab are of particular
interest here: a relatively short primer by Sigmon and Davis [8], a medium-sized,
mathematically oriented text by Higham and Higham [3], and a large, comprehen-
sive manual by Hanselman and Littlefield [2].
You should have a copy of Matlab close at hand so you can run our sample
programs as you read about them. All of the programs used in this book have been
collected in a directory (or folder) named
NCM
(The directory name is the initials of the book title.) You can either start Matlab
in this directory or use
pathtool
to add the directory to the Matlab path.
February 15, 2008
1
2 Chapter 1. Introduction to MATLAB
1.1 The Golden Ratio
What is the world’s most interesting number? Perhaps you like π, or e, or 17.
Some people might vote for φ, the golden ratio, computed here by our first Matlab
statement.
phi = (1 + sqrt(5))/2
This produces
phi =
1.6180
Let’s see more digits.
format long
phi
phi =
1.61803398874989
This didn’t recompute φ, it just displayed 15 significant digits instead of 5.
The golden ratio shows up in many places in mathematics; we’ll see several
in this book. The golden ratio gets its name from the golden rectangle, shown in
Figure 1.1. The golden rectangle has the property that removing a square leaves a
smaller rectangle with the same shape.
φ
φ − 1
1
1
Figure 1.1. The golden rectangle.
Equating the aspect ratios of the rectangles gives a defining equation for φ:
1
φ
=
φ− 1
1
.
This equation says that you can compute the reciprocal of φ by simply subtracting
one. How many numbers have that property?
Multiplying the aspect ratio equation by φ produces the polynomial equation
φ2 − φ− 1 = 0.
1.1. The Golden Ratio 3
The roots of this equation are given by the quadratic formula:
φ =
1±
√
5
2
.
The positive root is the golden ratio.
If you have forgotten the quadratic formula, you can ask Matlab to find
the roots of the polynomial. Matlab represents a polynomial by the vector of its
coefficients, in descending order. So the vector
p = [1 -1 -1]
represents the polynomial
p(x) = x2 − x− 1.
The roots are computed by the roots function.
r = roots(p)
produces
r =
-0.61803398874989
1.61803398874989
These two numbers are the only numbers whose reciprocal can be computed by
subtracting one.
You can use the Symbolic Toolbox, which connects Matlab to a computer
algebra system, to solve the aspect ratio equation without converting it to a polyno-
mial. The equation is represented by a character string. The solve function finds
two solutions.
r = solve(’1/x = x-1’)
produces
r =
[ 1/2*5^(1/2)+1/2]
[ 1/2-1/2*5^(1/2)]
The pretty function displays the results in a way that resembles typeset mathe-
matics.
pretty(r)
produces
[ 1/2 ]
[1/2 5 + 1/2]
[ ]
[ 1/2]
[1/2 – 1/2 5 ]
4 Chapter 1. Introduction to MATLAB
The variable r is a vector with two components, the symbolic forms of the two
solutions. You can pick off the first component with
phi = r(1)
which produces
phi =
1/2*5^(1/2)+1/2
This expression can be converted to a numerical value in two different ways. It can
be evaluated to any number of digits using variable-precision arithmetic with the
vpa function.
vpa(phi,50)
produces 50 digits.
1.6180339887498948482045868343656381177203091798058
It can also be converted to double-precision floating point, which is the principal
way that Matlab represents numbers, with the double function.
phi = double(phi)
produces
phi =
1.61803398874989
The aspect ratio equation is simple enough to have closed-form symbolic so-
lutions. More complicated equations have to be solved approximately. In Matlab
an anonymous function is a convenient way to define an object that can be used as
an argument to other functions. The statement
f = @(x) 1./x-(x-1)
defines f(x) = 1/x− (x− 1) and produces
f =
@(x) 1./x-(x-1)
The graph of f(x) over the interval 0 ≤ x ≤ 4 shown in Figure 1.2 is obtained
with
ezplot(f,0,4)
The name ezplot stands for “easy plot,” although some of the English-speaking
world would pronounce it “e-zed plot.” Even though f(x) becomes infinite as x → 0,
ezplot automatically picks a reasonable vertical scale.
The statement
phi = fzero(f,1)
1.1. The Golden Ratio 5
0 0.5 1 1.5 2 2.5 3 3.5 4
−3
−2
−1
0
1
2
3
4
5
6
7
x
1/x − (x−1)
Figure 1.2. f(φ) = 0.
looks for a zero of f(x) near x = 1. It produces an approximation to φ that is
accurate to almost full precision. The result can be inserted in Figure 1.2 with
hold on
plot(phi,0,’o’)
The following Matlab program produces the picture of the golden rectangle
shown in Figure 1.1. The program is contained in an M-file named goldrect.m, so
issuing the command
goldrect
runs the script and creates the picture.
% GOLDRECT Plot the golden rectangle
phi = (1+sqrt(5))/2;
x = [0 phi phi 0 0];
y = [0 0 1 1 0];
u = [1 1];
v = [0 1];
plot(x,y,’b’,u,v,’b–’)
text(phi/2,1.05,’phi’)
text((1+phi)/2,-.05,’phi – 1’)
text(-.05,.5,’1’)
text(.5,-.05,’1’)
axis equal
axis off
set(gcf,’color’,’white’)
6 Chapter 1. Introduction to MATLAB
The vectors x and y each contain five elements. Connecting consecutive
(xk, yk) pairs with straight lines produces the outside rectangle. The vectors u
and v each contain two elements. The line connecting (u1, v1) with (u2, v2) sepa-
rates the rectangle into the square and the smaller rectangle. The plot command
draws these lines—the x − y lines in solid blue and the u − v line in dashed blue.
The next four statements place text at various points; the string ’phi’ denotes the
Greek letter. The two axis statements cause the scaling in the x and y directions
to be equal and then turn off the display of the axes. The last statement sets the
background color of gcf, which stands for get current figure, to white.
A continued fraction is an infinite expression of the form
a0 +
1
a1 + 1a2+ 1a3+···
.
If all the ak’s are equal to 1, the continued fraction is another representation of the
golden ratio:
φ = 1 +
1
1 + 1
1+ 11+···
.
The following Matlab function generates and evaluates truncated continued frac-
tion approximations to φ. The code is stored in an M-file named goldfract.m.
function goldfract(n)
%GOLDFRACT Golden ratio continued fraction.
% GOLDFRACT(n) displays n terms.
p = ’1’;
for k = 1:n
p = [’1+1/(’ p ’)’];
end
p
p = 1;
q = 1;
for k = 1:n
s = p;
p = p + q;
q = s;
end
p = sprintf(’%d/%d’,p,q)
format long
p = eval(p)
format short
err = (1+sqrt(5))/2 – p
The statement
1.1. The Golden Ratio 7
goldfract(6)
produces
p =
1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1))))))
p =
21/13
p =
1.61538461538462
err =
0.0026
The three p’s are all different representations of the same approximation to φ.
The first p is the continued fraction truncated to six terms. There are six
right parentheses. This p is a string generated by starting with a single ‘1’ (that’s
goldfract(0)) and repeatedly inserting the string ‘1+1/(’ in front and the string ‘)’
in back. No matter how long this string becomes, it is a valid Matlab expression.
The second p is an “ordinary” fraction with a single integer numerator and
denominator obtained by collapsing the first p. The basis for the reformulation is
1 +
1
p
q
=
p + q
p
.
So the iteration starts with
1
1
and repeatedly replaces the fraction
p
q
with
p + q
p
.
The statement
p = sprintf(’%d/%d’,p,q)
prints the final fraction by formatting p and q as decimal integers and placing a ‘/’
between them.
The third p is the same number as the first two p’s, but is represented as
a conventional decimal expansion, obtained by having the Matlab eval function
actually do the division expressed in the second p.
The final quantity err is the difference between p and φ. With only 6 terms,
the approximation is accurate to less than 3 digits. How many terms does it take
to get 10 digits of accuracy?
8 Chapter 1. Introduction to MATLAB
As the number of terms n increases, the truncated continued fraction generated
by goldfract(n) theoretically approaches φ. But limitations on the size of the
integers in the numerator and denominator, as well as roundoff error in the actual
floating-point division, eventually intervene. Exercise 1.3 asks you to investigate
the limiting accuracy of goldfract(n).
1.2 Fibonacci Numbers
Leonardo Pisano Fibonacci was born around 1170 and died around 1250 in Pisa
in what is now Italy. He traveled extensively in Europe and Northern Africa. He
wrote several mathematical texts that, among other things, introduced Europe to
the Hindu-Arabic notation for numbers. Even though his books had to be tran-
scribed by hand, they were widely circulated. In his best known book, Liber Abaci,
published in 1202, he posed the following problem:
A man put a pair of rabbits in a place surrounded on all sides by a wall.
How many pairs of rabbits can be produced from that pair in a year if it
is supposed that every month each pair begets a new pair which from the
second month on becomes productive?
Today the solution to this problem is known as the Fibonacci sequence, or
Fibonacci numbers. There is a small mathematical industry based on Fibonacci
numbers. A search of the Internet for “Fibonacci” will find dozens of Web sites and
hundreds of pages of material. There is even a Fibonacci Association that publishes
a scholarly journal, the Fibonacci Quarterly.
If Fibonacci had not specified a month for the newborn pair to mature, he
would not have a sequence named after him. The number of pairs would simply
double each month. After n months there would be 2n pairs of rabbits. That’s a
lot of rabbits, but not distinctive mathematics.
Let fn denote the number of pairs of rabbits after n months. The key fact is
that the number of rabbits at the end of a month is the number at the beginning
of the month plus the number of births produced by the mature pairs:
fn = fn−1 + fn−2.
The initial conditions are that in the first month there is one pair of rabbits and in
the second there are two pairs:
f1 = 1, f2 = 2.
The following Matlab function, stored in the M-file fibonacci.m, produces
a vector containing the first n Fibonacci numbers.
function f = fibonacci(n)
% FIBONACCI Fibonacci sequence
% f = FIBONACCI(n) generates the first n Fibonacci numbers.
f = zeros(n,1);
1.2. Fibonacci Numbers 9
f(1) = 1;
f(2) = 2;
for k = 3:n
f(k) = f(k-1) + f(k-2);
end
With these initial conditions, the answer to Fibonacci’s original question about the
size of the rabbit population after one year is given by
fibonacci(12)
This produces
1
2
3
5
8
13
21
34
55
89
144
233
The answer is 233 pairs of rabbits. (It would be 4096 pairs if the number doubled
every month for 12 months.)
Let’s look carefully at fibonacci.m. It’s a good example of how to create a
Matlab function. The first line is
function f = fibonacci(n)
The first word on the first line says this is a function M-file, not a script. The
remainder of the first line says this particular function produces one output result,
f, and takes one input argument, n. The name of the function specified on the first
line is not actually used, because Matlab looks for the name of the M-file, but it
is common practice to have the two match. The next two lines are comments that
provide the text displayed when you ask for help.
help fibonacci
produces
FIBONACCI Fibonacci sequence
f = FIBONACCI(n) generates the first n Fibonacci numbers.
The name of the function is in uppercase because historically Matlab was case
insensitive and ran on terminals with only a single font. The use of capital letters
may be confusing to some first-time Matlab users, but the convention persists. It
10 Chapter 1. Introduction to MATLAB
is important to repeat the input and output arguments in these comments because
the first line is not displayed when you ask for help on the function.
The next line
f = zeros(n,1);
creates an n-by-1 matrix containing all zeros and assigns it to f. In Matlab, a
matrix with only one column is a column vector and a matrix with only one row is
a row vector.
The next two lines,
f(1) = 1;
f(2) = 2;
provide the initial conditions.
The last three lines are the for statement that does all the work.
for k = 3:n
f(k) = f(k-1) + f(k-2);
end
We like to use three spaces to indent the body of for and if statements, but other
people prefer two or four spaces, or a tab. You can also put the entire construction
on one line if you provide a comma after the first clause.
This particular function looks a lot like functions in other programming lan-
guages. It produces a vector, but it does not use any of the Matlab vector or
matrix operations. We will see some of these operations soon.
Here is another Fibonacci function, fibnum.m. Its output is simply the nth
Fibonacci number.
function f = fibnum(n)
% FIBNUM Fibonacci number.
% FIBNUM(n) generates the nth Fibonacci number.
if n <= 1
f = 1;
else
f = fibnum(n-1) + fibnum(n-2);
end
The statement
fibnum(12)
produces
ans =
233
The fibnum function is recursive. In fact, the term recursive is used in both a
mathematical and a computer science sense. The relationship fn = fn−1 + fn−2 is
known as a recursion relation and a function that calls itself is a recursive function.
A recursive program is elegant, but expensive. You can measure execution
time with tic and toc. Try
1.2. Fibonacci Numbers 11
tic, fibnum(24), toc
Do not try
tic, fibnum(50), toc
Now compare the results produced by goldfract(6) and fibonacci(7). The
first contains the fraction 21/13 while the second ends with 13 and 21. This is not
just a coincidence. The continued fraction is collapsed by repeating the statement
p = p + q;
while the Fibonacci numbers are generated by
f(k) = f(k-1) + f(k-2);
In fact, if we let φn denote the golden ratio continued fraction truncated at n terms,
then
fn+1
fn
= φn.
In the infinite limit, the ratio of successive Fibonacci numbers approaches the golden
ratio:
lim
n→∞
fn+1
fn
= φ.
To see this, compute 40 Fibonacci numbers.
n = 40;
f = fibonacci(n);
Then compute their ratios.
f(2:n)./f(1:n-1)
This takes the vector containing f(2) through f(n) and divides it, element by
element, by the vector containing f(1) through f(n-1). The output begins with
2.00000000000000
1.50000000000000
1.66666666666667
1.60000000000000
1.62500000000000
1.61538461538462
1.61904761904762
1.61764705882353
1.61818181818182
and ends with
1.61803398874990
1.61803398874989
1.61803398874990
1.61803398874989
1.61803398874989
12 Chapter 1. Introduction to MATLAB
Do you see why we chose n = 40? Use the up arrow key on your keyboard to bring
back the previous expression. Change it to
f(2:n)./f(1:n-1) - phi
and then press the Enter key. What is the value of the last element?
The population of Fibonacci’s rabbit pen doesn’t double every month; it is
multiplied by the golden ratio every month.
It is possible to find a closed-form solution to the Fibonacci number recurrence
relation. The key is to look for solutions of the form
fn = cρ
n
for some constants c and ρ. The recurrence relation
fn = fn−1 + fn−2
becomes
ρ2 = ρ + 1.
We’ve seen this equation before. There are two possible values of ρ, namely φ and
1− φ. The general solution to the recurrence is
fn = c1φ
n + c2(1− φ)n.
The constants c1 and c2 are determined by initial conditions, which are now
conveniently written
f0 = c1 + c2 = 1,
f1 = c1φ + c2(1− φ) = 1.
Exercise 1.4 asks you to use the Matlab backslash operator to solve this 2-by-2
system of simultaneous linear equations, but it is actually easier to solve the system
by hand:
c1 =
φ
2φ− 1 ,
c2 = −
(1− φ)
2φ− 1 .
Inserting these in the general solution gives
fn =
1
2φ− 1(φ
n+1 − (1− φ)n+1).
This is an amazing equation. The right-hand side involves powers and quo-
tients of irrational numbers, but the result is a sequence of integers. You can check
this with Matlab, displaying the results in scientific notation.
1.3. Fractal Fern 13
format long e
n = (1:40)’;
f = (phi.^(n+1) - (1-phi).^(n+1))/(2*phi-1)
The .^ operator is an element-by-element power operator. It is not necessary to
use ./ for the final division because (2*phi-1) is a scalar quantity. The computed
result starts with
f =
1.000000000000000e+000
2.000000000000000e+000
3.000000000000000e+000
5.000000000000001e+000
8.000000000000002e+000
1.300000000000000e+001
2.100000000000000e+001
3.400000000000001e+001
and ends with
5.702887000000007e+006
9.227465000000011e+006
1.493035200000002e+007
2.415781700000003e+007
3.908816900000005e+007
6.324598600000007e+007
1.023341550000001e+008
1.655801410000002e+008
Roundoff error prevents the results from being exact integers, but
f = round(f)
finishes the job.
1.3 Fractal Fern
The M-files fern.m and finitefern.m produce the “Fractal Fern” described by
Michael Barnsley in Fractals Everywhere [1]. They generate and plot a potentially
infinite sequence of random, but carefully choreographed, points in the plane. The
command
fern
runs forever, producing an increasingly dense plot. The command
finitefern(n)
generates n points and a plot like Figure 1.3. The command
14 Chapter 1. Introduction to MATLAB
Figure 1.3. Fractal fern.
finitefern(n,’s’)
shows the generation of the points one at a time. The command
F = finitefern(n);
generates, but does not plot, n points and returns an array of zeros and ones for
use with sparse matrix and image-processing functions.
The NCM collection also includes fern.png, a 768-by-1024 color image with
half a million points that you can view with a browser or a paint program. You can
also view the file with
F = imread(’fern.png’);
image(F)
If you like the image, you might even choose to make it your computer desktop
background. However, you should really run fern on your own computer to see the
dynamics of the emerging fern in high resolution.
The fern is generated by repeated transformations of a point in the plane. Let
x be a vector with two components, x1 and x2, representing the point. There are
1.3. Fractal Fern 15
four different transformations, all of them of the form
x → Ax + b,
with different matrices A and vectors b. These are known as affine transformations.
The most frequently used transformation has
A =
(
0.85 0.04
−0.04 0.85
)
, b =
(
0
1.6
)
.
This transformation shortens and rotates x a little bit, then adds 1.6 to its second
component. Repeated application of this transformation moves the point up and to
the right, heading toward the upper tip of the fern. Every once in a while, one of
the other three transformations is picked at random. These transformations move
the point into the lower subfern on the right, the lower subfern on the left, or the
stem.
Here is the complete fractal fern program.
function fern
%FERN MATLAB implementation of the Fractal Fern
%Michael Barnsley, Fractals Everywhere, Academic Press,1993
%This version runs forever, or until stop is toggled.
%See also: FINITEFERN.
shg
clf reset
set(gcf,’color’,’white’,’menubar’,’none’, ...
’numbertitle’,’off’,’name’,’Fractal Fern’)
x = [.5; .5];
h = plot(x(1),x(2),’.’);
darkgreen = [0 2/3 0];
set(h,’markersize’,1,’color’,darkgreen,’erasemode’,’none’);
axis([-3 3 0 10])
axis off
stop = uicontrol(’style’,’toggle’,’string’,’stop’, ...
’background’,’white’);
drawnow
p = [ .85 .92 .99 1.00];
A1 = [ .85 .04; -.04 .85]; b1 = [0; 1.6];
A2 = [ .20 -.26; .23 .22]; b2 = [0; 1.6];
A3 = [-.15 .28; .26 .24]; b3 = [0; .44];
A4 = [ 0 0 ; 0 .16];
cnt = 1;
tic
while ~get(stop,’value’)
16 Chapter 1. Introduction to MATLAB
r = rand;
if r < p(1)
x = A1*x + b1;
elseif r < p(2)
x = A2*x + b2;
elseif r < p(3)
x = A3*x + b3;
else
x = A4*x;
end
set(h,’xdata’,x(1),’ydata’,x(2));
cnt = cnt + 1;
drawnow
end
t = toc;
s = sprintf(’%8.0f points in %6.3f seconds’,cnt,t);
text(-1.5,-0.5,s,’fontweight’,’bold’);
set(stop,’style’,’pushbutton’,’string’,’close’, ...
’callback’,’close(gcf)’)
Let’s examine this program a few statements at a time.
shg
stands for “show graph window.” It brings an existing graphics window forward,
or creates a new one if necessary.
clf reset
resets most of the figure properties to their default values.
set(gcf,’color’,’white’,’menubar’,’none’, ...
’numbertitle’,’off’,’name’,’Fractal Fern’)
changes the background color of the figure window from the default gray to white
and provides a customized title for the window.
x = [.5; .5];
provides the initial coordinates of the point.
h = plot(x(1),x(2),’.’);
plots a single dot in the plane and saves a handle, h, so that we can later modify
the properties of the plot.
darkgreen = [0 2/3 0];
defines a color where the red and blue components are zero and the green component
is two-thirds of its full intensity.
1.3. Fractal Fern 17
set(h,’markersize’,1,’color’,darkgreen,’erasemode’,’none’);
makes the dot referenced by h smaller, changes its color, and specifies that the image
of the dot on the screen should not be erased when its coordinates are changed. A
record of these old points is kept by the computer’s graphics hardware (until the
figure is reset), but Matlab itself does not remember them.
axis([-3 3 0 10])
axis off
specifies that the plot should cover the region
−3 ≤ x1 ≤ 3, 0 ≤ x2 ≤ 10,
but that the axes should not be drawn.
stop = uicontrol(’style’,’toggle’,’string’,’stop’, ...
’background’,’white’);
creates a toggle user interface control, labeled ’stop’ and colored white, in the
default position near the lower left corner of the figure. The handle for the control
is saved in the variable stop.
drawnow
causes the initial figure, including the initial point, to actually be plotted on the
computer screen.
The statement
p = [ .85 .92 .99 1.00];
sets up a vector of probabilities. The statements
A1 = [ .85 .04; -.04 .85]; b1 = [0; 1.6];
A2 = [ .20 -.26; .23 .22]; b2 = [0; 1.6];
A3 = [-.15 .28; .26 .24]; b3 = [0; .44];
A4 = [ 0 0 ; 0 .16];
define the four affine transformations. The statement
cnt = 1;
initializes a counter that keeps track of the number of points plotted. The statement
tic
initializes a stopwatch timer. The statement
while ~get(stop,’value’)
begins a while loop that runs as long as the ’value’ property of the stop toggle is
equal to 0. Clicking the stop toggle changes the value from 0 to 1 and terminates
the loop.
18 Chapter 1. Introduction to MATLAB
r = rand;
generates a pseudorandom value between 0 and 1. The compound if statement
if r < p(1)
x = A1*x + b1;
elseif r < p(2)
x = A2*x + b2;
elseif r < p(3)
x = A3*x + b3;
else
x = A4*x;
end
picks one of the four affine transformations. Because p(1) is 0.85, the first trans-
formation is chosen 85% of the time. The other three transformations are chosen
relatively infrequently.
set(h,’xdata’,x(1),’ydata’,x(2));
changes the coordinates of the point h to the new (x1, x2) and plots this new point.
But get(h,’erasemode’) is ’none’, so the old point also remains on the screen.
cnt = cnt + 1;
counts one more point.
drawnow
tells Matlab to take the time to redraw the figure, showing the new point along
with all the old ones. Without this command, nothing would be plotted until stop
is toggled.
end
matches the while at the beginning of the loop. Finally,
t = toc;
reads the timer.
s = sprintf(’%8.0f points in %6.3f seconds’,cnt,t);
text(-1.5,-0.5,s,’fontweight’,’bold’);
displays the elapsed time since tic was called and the final count of the number of
points plotted. Finally,
set(stop,’style’,’pushbutton’,’string’,’close’, ...
’callback’,’close(gcf)’)
changes the control to a push button that closes the window.
1.4. Magic Squares 19
1.4 Magic Squares
Matlab stands for Matrix Laboratory. Over the years, Matlab has evolved into a
general-purpose technical computing environment, but operations involving vectors,
matrices, and linear algebra continue to be its most distinguishing feature.
Magic squares provide an interesting set of sample matrices. The command
help magic tells us the following:
MAGIC(N) is an N-by-N matrix constructed from the integers
1 through N^2 with equal row, column, and diagonal sums.
Produces valid magic squares for all N > 0 except N = 2.
Magic squares were known in China over 2,000 years before the birth of Christ.
The 3-by-3 magic square is known as Lo Shu. Legend has it that Lo Shu was
discovered on the shell of a turtle that crawled out of the Lo River in the 23rd
century b.c. Lo Shu provides a mathematical basis for feng shui, the ancient Chinese
philosophy of balance and harmony. Matlab can generate Lo Shu with
A = magic(3)
which produces
A =
8 1 6
3 5 7
4 9 2
The command
sum(A)
sums the elements in each column to produce
15 15 15
The command
sum(A’)’
transposes the matrix, sums the columns of the transpose, and then transposes the
results to produce the row sums
15
15
15
The command
sum(diag(A))
sums the main diagonal of A, which runs from upper left to lower right, to produce
15
20 Chapter 1. Introduction to MATLAB
The opposite diagonal, which runs from upper right to lower left, is less important
in linear algebra, so finding its sum is a little trickier. One way to do it makes use
of the function that “flips” a matrix “upside-down.”
sum(diag(flipud(A)))
produces
15
This verifies that A has equal row, column, and diagonal sums.
Why is the magic sum equal to 15? The command
sum(1:9)
tells us that the sum of the integers from 1 to 9 is 45. If these integers are allocated
to 3 columns with equal sums, that sum must be
sum(1:9)/3
which is 15.
There are eight possible ways to place a transparency on an overhead projec-
tor. Similarly, there are eight magic squares of order three that are rotations and
reflections of A. The statements
for k = 0:3
rot90(A,k)
rot90(A’,k)
end
display all eight of them.
8 1 6 8 3 4
3 5 7 1 5 9
4 9 2 6 7 2
6 7 2 4 9 2
1 5 9 3 5 7
8 3 4 8 1 6
2 9 4 2 7 6
7 5 3 9 5 1
6 1 8 4 3 8
4 3 8 6 1 8
9 5 1 7 5 3
2 7 6 2 9 4
These are all the magic squares of order three.
Now for some linear algebra. The determinant of our magic square,
1.4. Magic Squares 21
det(A)
is
-360
The inverse,
X = inv(A)
is
X =
0.1472 -0.1444 0.0639
-0.0611 0.0222 0.1056
-0.0194 0.1889 -0.1028
The inverse looks better if it is displayed with a rational format.
format rat
X
shows that the elements of X are fractions with det(A) in the denominator.
X =
53/360 -13/90 23/360
-11/180 1/45 19/180
-7/360 17/90 -37/360
The statement
format short
restores the output format to its default.
Three other important quantities in computational linear algebra are matrix
norms, eigenvalues, and singular values. The statements
r = norm(A)
e = eig(A)
s = svd(A)
produce
r =
15
e =
15.0000
4.8990
-4.8990
s =
15.0000
6.9282
3.4641
22 Chapter 1. Introduction to MATLAB
The magic sum occurs in all three because the vector of all ones is an eigenvector
and is also a left and right singular vector.
So far, all the computations in this section have been done using floating-point
arithmetic. This is the arithmetic used for almost all scientific and engineering
computation, especially for large matrices. But for a 3-by-3 matrix, it is easy to
repeat the computations using symbolic arithmetic and the Symbolic Toolbox. The
statement
A = sym(A)
changes the internal representation of A to a symbolic form that is displayed as
A =
[ 8, 1, 6]
[ 3, 5, 7]
[ 4, 9, 2]
Now commands like
sum(A), sum(A’)’, det(A), inv(A), eig(A), svd(A)
produce symbolic results. In particular, the eigenvalue problem for this matrix can
be solved exactly, and
e =
[ 15]
[ 2*6^(1/2)]
[ -2*6^(1/2)]
A 4-by-4 magic square is one of several mathematical objects on display in
Melancolia, a Renaissance etching by Albrecht Dürer. An electronic copy of the
etching is available in a Matlab data file.
load durer
whos
produces
X 648×509 2638656 double array
caption 2×28 112 char array
map 128×3 3072 double array
The elements of the matrix X are indices into the gray-scale color map named map.
The image is displayed with
image(X)
colormap(map)
axis image
Click the magnifying glass with a “+” in the toolbar and use the mouse to zoom
in on the magic square in the upper right-hand corner. The scanning resolution
becomes evident as you zoom in. The commands
1.4. Magic Squares 23
load detail
image(X)
colormap(map)
axis image
display a higher resolution scan of the area around the magic square.
The command
A = magic(4)
produces a 4-by-4 magic square.
A =
16 2 3 13
5 11 10 8
9 7 6 12
4 14 15 1
The commands
sum(A), sum(A’), sum(diag(A)), sum(diag(flipud(A)))
yield enough 34’s to verify that A is indeed a magic square.
The 4-by-4 magic square generated by Matlab is not the same as Dürer’s
magic square. We need to interchange the second and third columns.
A = A(:,[1 3 2 4])
changes A to
A =
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
Interchanging columns does not change the column sums or the row sums. It usually
changes the diagonal sums, but in this case both diagonal sums are still 34. So now
our magic square matches the one in Dürer’s etching. Dürer probably chose this
particular 4-by-4 square because the date he did the work, 1514, occurs in the
middle of the bottom row.
We have seen two different 4-by-4 magic squares. It turns out that there are
880 different magic squares of order 4 and 275305224 different magic squares of
order 5. Determining the number of different magic squares of order 6 or larger is
an unsolved mathematical problem.
The determinant of our 4-by-4 magic square, det(A), is 0. If we try to compute
its inverse
inv(A)
we get
24 Chapter 1. Introduction to MATLAB
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate.
So some magic squares represent singular matrices. Which ones? The rank of a
square matrix is the number of linearly independent rows or columns. An n-by-n
matrix is singular if and only if its rank is less than n.
The statements
for n = 1:24, r(n) = rank(magic(n)); end
[(1:24)’ r’]
produce a table of order versus rank.
1 1
2 2
3 3
4 3
5 5
6 5
7 7
8 3
9 9
10 7
11 11
12 3
13 13
14 9
15 15
16 3
17 17
18 11
19 19
20 3
21 21
22 13
23 23
24 3
Look carefully at this table. Ignore n = 2 because magic(2) is not really a magic
square. What patterns do you see? A bar graph makes the patterns easier to see.
bar(r)
title(’Rank of magic squares’)
produces Figure 1.4.
The rank considerations show that there are three different kinds of magic
squares:
1.4. Magic Squares 25
0 5 10 15 20 25
0
5
10
15
20
25
Rank of magic squares
Figure 1.4. Rank of magic squares.
• Odd order: n is odd.
• Singly even order: n is a multiple of 2, but not 4.
• Doubly even order: n is a multiple of 4.
Odd-ordered magic squares, n = 3, 5, 7, . . . , have full rank n. They are nonsingular
and have inverses. Doubly even magic squares, n = 4, 8, 12, . . . , have rank three no
matter how large n is. They might be called very singular. Singly even magic squares,
n = 6, 10, 14, . . . , have rank n/2 + 2. They are also singular, but have fewer row
and column dependencies than the doubly even squares.
If you have Matlab Version 6 or later, you can look at the M-file that gener-
ates magic squares with
edit magic.m
or
type magic.m
You will see the three different cases in the code.
The different kinds of magic squares also produce different three-dimensional
surface plots. Try the following for various values of n.
surf(magic(n))
axis off
set(gcf,’doublebuffer’,’on’)
cameratoolbar
Double buffering prevents flicker when you use the various camera tools to move
the viewpoint.
26 Chapter 1. Introduction to MATLAB
The following code produces Figure 1.5.
for n = 8:11
subplot(2,2,n-7)
surf(magic(n))
title(num2str(n))
axis off
view(30,45)
axis tight
end
8 9
10 11
Figure 1.5. Surface plots of magic squares.
1.5 Cryptography
This section uses a cryptography example to show how Matlab deals with text and
character strings. The cryptographic technique, which is known as a Hill cipher,
involves arithmetic in a finite field.
Almost all modern computers use the ASCII character set to store basic text.
ASCII stands for American Standard Code for Information Interchange. The char-
acter set uses 7 of the 8 bits in a byte to encode 128 characters. The first 32
characters are nonprinting control characters, such as tab, backspace, and end-of-
line. The 128th character is another nonprinting character that corresponds to the
Delete key on your keyboard. In between these control characters are 95 printable
1.5. Cryptography 27
characters, including a space, 10 digits, 26 lowercase letters, 26 uppercase letters,
and 32 punctuation marks.
Matlab can easily display all the printable characters in the order determined
by their ASCII encoding. Start with
x = reshape(32:127,32,3)’
This produces a 3-by-32 matrix.
x =
32 33 34 … 61 62 63
64 65 66 … 93 94 95
96 97 98 … 125 126 127
The char function converts numbers to characters. The statement
c = char(x)
produces
c =
!”#$%&’()*+,-./0123456789:;<=>?
@ABCDEFGHIJKLMNOPQRSTUVWXYZ[]^_
‘abcdefghijklmnopqrstuvwxyz{|}~
We have cheated a little bit because the last element of x is 127, which corresponds
to the nonprinting delete character, and we have not shown the last character in c.
You can try this on your computer and see what is actually displayed.
The first character in c is blank, indicating that
char(32)
is the same as
’ ’
The last printable character in c is the tilde, indicating that
char(126)
is the same as
’~’
The characters representing digits are in the first line of c. In fact,
d = char(48:57)
displays a 10-character string
d =
0123456789
28 Chapter 1. Introduction to MATLAB
This string can be converted to the corresponding numerical values with double or
real. The statement
double(d) – ’0’
produces
0 1 2 3 4 5 6 7 8 9
Comparing the second and third lines of c, we see that the ASCII encoding
of the lowercase letters is obtained by adding 32 to the ASCII encoding of the
uppercase letters. Understanding this encoding allows us to use vector and matrix
operations in Matlab to manipulate text.
The ASCII standard is often extended to make use of all eight bits in a byte,
but the characters that are displayed depend on the computer and operating system
you are using, the font you have chosen, and even the country you live in. Try
char(reshape(160:255,32,3)’)
and see what happens on your machine.
Our encryption technique involves modular arithmetic. All the quantities in-
volved are integers and the result of any arithmetic operation is reduced by tak-
ing the remainder or modulus with respect to a prime number p. The functions
rem(x,y) and mod(x,y) both compute the remainder if x is divided by y. They
produce the same result if x and y have the same sign; the result also has that sign.
But if x and y have opposite signs, then rem(x,y) has the same sign as x, while
mod(x,y) has the same sign as y. Here is a table:
x = [37 -37 37 -37]’;
y = [10 10 -10 -10]’;
r = [ x y rem(x,y) mod(x,y)]
produces
37 10 7 7
-37 10 -7 3
37 -10 7 -3
-37 -10 -7 -7
We have chosen to encrypt text that uses the entire ASCII character set, not
just the letters. There are 95 such characters. The next larger prime number is
p = 97, so we represent the p characters by the integers 0:p-1 and do arithmetic
mod p.
The characters are encoded two at a time. Each pair of characters is repre-
sented by a 2-vector, x. For example, suppose the text contains the pair of letters
’TV’. The ASCII values for this pair of letters are 84 and 86. Subtracting 32 to
make the representation start at 0 produces the column vector
x =
(
52
54
)
.
1.5. Cryptography 29
The encryption is done with a 2-by-2 matrix-vector multiplication over the
integers mod p. The symbol ≡ is used to indicate that two integers have the same
remainder, modulo the specified prime:
y ≡ Ax, mod p,
where A is the matrix
A =
(
71 2
2 26
)
.
For our example, the product Ax is
Ax =
(
3800
1508
)
.
If this is reduced mod p, the result is
y =
(
17
53
)
.
Converting this back to characters by adding 32 produces ’1U’.
Now comes the interesting part. Over the integers modulo p, the matrix A is
its own inverse. If
y ≡ Ax, mod p,
then
x ≡ Ay, mod p.
In other words, in arithmetic mod p, A2 is the identity matrix. You can check this
with Matlab.
p = 97;
A = [71 2; 2 26]
I = mod(A^2,p)
produces
A =
71 2
2 26
I =
1 0
0 1
This means that the encryption process is its own inverse. The same function can
be used to both encrypt and decrypt a message.
The M-file crypto.m begins with a preamble.
30 Chapter 1. Introduction to MATLAB
function y = crypto(x)
% CRYPTO Cryptography example.
% y = crypto(x) converts an ASCII text string into another
% coded string. The function is its own inverse, so
% crypto(crypto(x)) gives x back.
% See also: ENCRYPT.
A comment precedes the statement that assigns the prime p.
% Use a two-character Hill cipher with arithmetic
% modulo 97, a prime.
p = 97;
Choose two characters above ASCII 128 to expand the size of the character set from
95 to 97.
c1 = char(169);
c2 = char(174);
x(x==c1) = 127;
x(x==c2) = 128;
The conversion from characters to numerical values is done by
x = mod(real(x-32),p);
Prepare for the matrix-vector product by forming a matrix with two rows and lots
of columns.
n = 2*floor(length(x)/2);
X = reshape(x(1:n),2,n/2);
All this preparation has been so that we can do the actual finite field arithmetic
quickly and easily.
% Encode with matrix multiplication modulo p.
A = [71 2; 2 26];
Y = mod(A*X,p);
Reshape into a single row.
y = reshape(Y,1,n);
If length(x) is odd, encode the last character
if length(x) > n
y(n+1) = mod((p-1)*x(n+1),p);
end
Finally, convert the numbers back to characters.
y = char(y+32);
y(y==127) = c1;
y(y==128) = c2;
1.6. The 3n+1 Sequence 31
Let’s follow the computation of y = crypto(’Hello world’). We begin with
a character string.
x = ’Hello world’
This is converted to an integer vector.
x =
40 69 76 76 79 0 87 79 82 76 68
length(x) is odd, so the reshaping temporarily ignores the last element
X =
40 76 79 87 82
69 76 0 79 76
A conventional matrix-vector multiplication A*X produces an intermediate matrix.
2978 5548 5609 6335 5974
1874 2128 158 2228 2140
Then the mod(.,p) operation produces
Y =
68 19 80 30 57
31 91 61 94 6
This is rearranged to a row vector.
y =
68 31 19 91 80 61 30 94 57 6
Now the last element of x is encoded by itself and attached to the end of y.
y =
68 31 19 91 80 61 30 94 57 6 29
Finally, y is converted back to a character string to produce the encrypted result.
y = ’d?3{p]>~Y&=’
If we now compute crypto(y), we get back our original ’Hello world’.
1.6 The 3n+1 Sequence
This section describes a famous unsolved problem in number theory. Start with any
positive integer n. Repeat the following steps:
• If n = 1, stop.
• If n is even, replace it with n/2.
• If n is odd, replace it with 3n + 1.
32 Chapter 1. Introduction to MATLAB
For example, starting with n = 7 produces
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
The sequence terminates after 17 steps. Note that whenever n reaches a power of
2, the sequence terminates in log2 n more steps.
The unanswered question is, does the process always terminate? Or is there
some starting value that causes the process to go on forever, either because the
numbers get larger and larger, or because some periodic cycle is generated?
This problem is known as the 3n + 1 problem. It has been studied by many
eminent mathematicians, including Collatz, Ulam, and Kakatani, and is discussed
in a survey paper by Jeffrey Lagarias [5].
The following Matlab code fragment generates the sequence starting with
any specified n.
y = n;
while n > 1
if rem(n,2)==0
n = n/2;
else
n = 3*n+1;
end
y = [y n];
end
We don’t know ahead of time how long the resulting vector y is going to be. But
the statement
y = [y n];
automatically increases length(y) each time it is executed.
In principle, the unsolved mathematical problem is, Can this code fragment
run forever? In actual fact, floating-point roundoff error causes the calculation to
misbehave whenever 3n + 1 becomes greater than 253, but it is still interesting to
investigate modest values of n.
Let’s embed our code fragment in a GUI. The complete function is in the
M-file threenplus1.m. For example, the statement
threenplus1(7)
produces Figure 1.6.
The M-file begins with a preamble containing the function header and the
help information.
function threenplus1(n)
% ‘‘Three n plus 1’’.
% Study the 3n+1 sequence.
% threenplus1(n) plots the sequence starting with n.
% threenplus1 with no arguments starts with n = 1.
% uicontrols decrement or increment the starting n.
% Is it possible for this to run forever?
1.6. The 3n+1 Sequence 33
2 4 6 8 10 12 14 16
1
2
4
8
16
32
52
n = 7
Figure 1.6. threenplus1.
The next section of code brings the current graphics window forward and resets it.
Two push buttons, which are the default uicontrols, are positioned near the bot-
tom center of the figure at pixel coordinates [260,5] and [300,5]. Their size is 25
by 22 pixels and they are labeled with ’<’ and ’>’. If either button is subsequently
pushed, the ’callback’ string is executed, calling the function recursively with a
corresponding ’-1’ or ’+1’ string argument. The ’tag’ property of the current
figure, gcf, is set to a characteristic string that prevents this section of code from
being reexecuted on subsequent calls.
if ~isequal(get(gcf,’tag’),’3n+1’)
shg
clf reset
uicontrol( …
’position’,[260 5 25 22], …
’string’,’<’, ...
’callback’,’threenplus1(’’-1’’)’);
uicontrol( ...
’position’,[300 5 25 22], ...
’string’,’>’, …
’callback’,’threenplus1(’’+1’’)’);
set(gcf,’tag’,’3n+1’);
end
The next section of code sets n. If nargin, the number of input arguments, is 0,
then n is set to 1. If the input argument is either of the strings from the push
button callbacks, then n is retrieved from the ’userdata’ field of the figure and
decremented or incremented. If the input argument is not a string, then it is the
desired n. In all situations, n is saved in ’userdata’ for use on subsequent calls.
34 Chapter 1. Introduction to MATLAB
if nargin == 0
n = 1;
elseif isequal(n,’-1’)
n = get(gcf,’userdata’) – 1;
elseif isequal(n,’+1’)
n = get(gcf,’userdata’) + 1;
end
if n < 1, n = 1; end
set(gcf,’userdata’,n)
We’ve seen the next section of code before; it does the actual computation.
y = n;
while n > 1
if rem(n,2)==0
n = n/2;
else
n = 3*n+1;
end
y = [y n];
end
The final section of code plots the generated sequence with dots connected by
straight lines, using a logarithmic vertical scale and customized tick labels.
semilogy(y,’.-’)
axis tight
ymax = max(y);
ytick = [2.^(0:ceil(log2(ymax))-1) ymax];
if length(ytick) > 8, ytick(end-1) = []; end
set(gca,’ytick’,ytick)
title([’n = ’ num2str(y(1))]);
1.7 Floating-Point Arithmetic
Some people believe that
• numerical analysis is the study of floating-point arithmetic;
• floating-point arithmetic is unpredictable and hard to understand.
We intend to convince you that both of these assertions are false. Very little of
this book is actually about floating-point arithmetic. But when the subject does
arise, we hope you will find floating-point arithmetic is not only computationally
powerful, but also mathematically elegant.
If you look carefully at the definitions of fundamental arithmetic operations
like addition and multiplication, you soon encounter the mathematical abstraction
known as real numbers. But actual computation with real numbers is not very
1.7. Floating-Point Arithmetic 35
practical because it involves limits and infinities. Instead, Matlab and most other
technical computing environments use floating-point arithmetic, which involves a
finite set of numbers with finite precision. This leads to the phenomena of roundoff,
underflow, and overflow. Most of the time, it is possible to use Matlab effectively
without worrying about these details, but, every once in a while, it pays to know
something about the properties and limitations of floating-point numbers.
Twenty years ago, the situation was far more complicated than it is today.
Each computer had its own floating-point number system. Some were binary; some
were decimal. There was even a Russian computer that used trinary arithmetic.
Among the binary computers, some used 2 as the base; others used 8 or 16. And
everybody had a different precision. In 1985, the IEEE Standards Board and the
American National Standards Institute adopted the ANSI/IEEE Standard 754-1985
for Binary Floating-Point Arithmetic. This was the culmination of almost a decade
of work by a 92-person working group of mathematicians, computer scientists, and
engineers from universities, computer manufacturers, and microprocessor compa-
nies.
All computers designed since 1985 use IEEE floating-point arithmetic. This
doesn’t mean that they all get exactly the same results, because there is some
flexibility within the standard. But it does mean that we now have a machine-
independent model of how floating-point arithmetic behaves.
Matlab has traditionally used the IEEE double-precision format. There is
a single-precision format that saves space, but that isn’t much faster on modern
machines. Matlab 7 will have support for single-precision arithmetic, but we will
deal exclusively with double precision in this book. There is also an extended-
precision format, which is optional and therefore is one of the reasons for lack of
uniformity among different machines.
Most nonzero floating-point numbers are normalized. This means they can be
expressed as
x = ±(1 + f) · 2e.
The quantity f is the fraction or mantissa and e is the exponent. The fraction
satisfies
0 ≤ f < 1
and must be representable in binary using at most 52 bits. In other words, 252f is
an integer in the interval
0 ≤ 252f < 252.
The exponent e is an integer in the interval
−1022 ≤ e ≤ 1023.
The finiteness of f is a limitation on precision. The finiteness of e is a limitation
on range. Any numbers that don’t meet these limitations must be approximated
by ones that do.
Double-precision floating-point numbers are stored in a 64-bit word, with 52
bits for f , 11 bits for e, and 1 bit for the sign of the number. The sign of e is
accommodated by storing e + 1023, which is between 1 and 211− 2. The 2 extreme
36 Chapter 1. Introduction to MATLAB
values for the exponent field, 0 and 211−1, are reserved for exceptional floating-point
numbers that we will describe later.
The entire fractional part of a floating-point number is not f , but 1+f , which
has 53 bits. However, the leading 1 doesn’t need to be stored. In effect, the IEEE
format packs 65 bits of information into a 64-bit word.
The program floatgui shows the distribution of the positive numbers in a
model floating-point system with variable parameters. The parameter t specifies the
number of bits used to store f . In other words, 2tf is an integer. The parameters
emin and emax specify the range of the exponent, so emin ≤ e ≤ emax. Initially,
floatgui sets t = 3, emin = −4, and emax = 3 and produces the distribution shown
in Figure 1.7.
1/16 1/2 1 2 4 8−1/2
||||||||||||||||||||||||| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Figure 1.7. floatgui.
Within each binary interval 2e ≤ x ≤ 2e+1, the numbers are equally spaced
with an increment of 2e−t. If e = 0 and t = 3, for example, the spacing of the
numbers between 1 and 2 is 1/8. As e increases, the spacing increases.
It is also instructive to display the floating-point numbers with a logarithmic
scale. Figure 1.8 shows floatgui with logscale checked and t = 5, emin = −4,
and emax = 3. With this logarithmic scale, it is more apparent that the distribution
in each binary interval is the same.
A very important quantity associated with floating-point arithmetic is high-
lighted in red by floatgui. Matlab calls this quantity eps, which is short for
machine epsilon.
eps is the distance from 1 to the next larger floating-point number.
For the floatgui model floating-point system, eps = 2^(-t).
1/16 1/8 1/4 1/2 1 2 4 8 16−1/4
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Figure 1.8. floatgui(logscale).
Before the IEEE standard, different machines had different values of eps.
Now, for IEEE double-precision,
eps = 2^(-52).
1.7. Floating-Point Arithmetic 37
The approximate decimal value of eps is 2.2204 ·10−16. Either eps/2 or eps can be
called the roundoff level. The maximum relative error incurred when the result of
an arithmetic operation is rounded to the nearest floating-point number is eps/2.
The maximum relative spacing between numbers is eps. In either case, you can say
that the roundoff level is about 16 decimal digits.
A frequent instance of roundoff occurs with the simple Matlab statement
t = 0.1
The mathematical value t stored in t is not exactly 0.1 because expressing the
decimal fraction 1/10 in binary requires an infinite series. In fact,
1
10
=
1
24
+
1
25
+
0
26
+
0
27
+
1
28
+
1
29
+
0
210
+
0
211
+
1
212
+ · · · .
After the first term, the sequence of coefficients 1, 0, 0, 1 is repeated infinitely often.
Grouping the resulting terms together four at a time expresses 1/10 in a base 16,
or hexadecimal, series.
1
10
= 2−4 ·
(
1 +
9
16
+
9
162
+
9
163
+
9
164
+ · · ·
)
Floating-point numbers on either side of 1/10 are obtained by terminating the
fractional part of this series after 52 binary terms, or 13 hexadecimal terms, and
rounding the last term up or down. Thus
t1 < 1/10 < t2,
where
t1 = 2
−4 ·
(
1 +
9
16
+
9
162
+
9
163
+ · · ·+ 9
1612
+
9
1613
)
,
t2 = 2
−4 ·
(
1 +
9
16
+
9
162
+
9
163
+ · · ·+ 9
1612
+
10
1613
)
.
It turns out that 1/10 is closer to t2 than to t1, so t is equal to t2. In other words,
t = (1 + f) · 2e,
where
f =
9
16
+
9
162
+
9
163
+ · · ·+ 9
1612
+
10
1613
,
e = −4.
The Matlab command
format hex
causes t to be displayed as
3fb999999999999a
38 Chapter 1. Introduction to MATLAB
The characters a through f represent the hexadecimal “digits” 10 through 15. The
first three characters, 3fb, give the hexadecimal representation of decimal 1019,
which is the value of the biased exponent e+1023 if e is −4. The other 13 characters
are the hexadecimal representation of the fraction f .
In summary, the value stored in t is very close to, but not exactly equal to,
0.1. The distinction is occasionally important. For example, the quantity
0.3/0.1
is not exactly equal to 3 because the actual numerator is a little less than 0.3 and
the actual denominator is a little greater than 0.1.
Ten steps of length t are not precisely the same as one step of length 1.
Matlab is careful to arrange that the last element of the vector
0:0.1:1
is exactly equal to 1, but if you form this vector yourself by repeated additions of
0.1, you will miss hitting the final 1 exactly.
What does the floating-point approximation to the golden ratio look like?
format hex
phi = (1 + sqrt(5))/2
produces
phi =
3ff9e3779b97f4a8
The first hex digit, 3, is 0011 in binary. The first bit is the sign of the floating-
point number; 0 is positive, 1 is negative. So phi is positive. The remaining bits
of the first three hex digits contain e + 1023. In this example, 3ff in base 16 is
3 · 162 + 15 · 16 + 15 = 1023 in decimal. So
e = 0.
In fact, any floating-point number between 1.0 and 2.0 has e = 0, so its hex output
begins with 3ff. The other 13 hex digits contain f . In this example,
f =
9
16
+
14
162
+
3
163
+ · · ·+ 10
1612
+
8
1613
.
With these values of f and e,
(1 + f)2e ≈ φ.
Another example is provided by the following code segment.
format long
a = 4/3
b = a - 1
c = 3*b
e = 1 - c
1.7. Floating-Point Arithmetic 39
With exact computation, e would be 0. But with floating-point, the output pro-
duced is
a =
1.33333333333333
b =
0.33333333333333
c =
1.00000000000000
e =
2.220446049250313e-016
It turns out that the only roundoff occurs in the division in the first statement.
The quotient cannot be exactly 4/3, except on that Russian trinary computer.
Consequently the value stored in a is close to, but not exactly equal to, 4/3. The
subtraction b = a - 1 produces a b whose last bit is 0. This means that the
multiplication 3*b can be done without any roundoff. The value stored in c is
not exactly equal to 1, and so the value stored in e is not 0. Before the IEEE
standard, this code was used as a quick way to estimate the roundoff level on
various computers.
The roundoff level eps is sometimes called “floating-point zero,” but that’s a
misnomer. There are many floating-point numbers much smaller than eps. The
smallest positive normalized floating-point number has f = 0 and e = −1022. The
largest floating-point number has f a little less than 1 and e = 1023. Matlab
calls these numbers realmin and realmax. Together with eps, they characterize
the standard system.
Binary Decimal
eps 2^(-52) 2.2204e-16
realmin 2^(-1022) 2.2251e-308
realmax (2-eps)*2^1023 1.7977e+308
If any computation tries to produce a value larger than realmax, it is said to
overflow. The result is an exceptional floating-point value called infinity or Inf. It
is represented by taking e = 1024 and f = 0 and satisfies relations like 1/Inf = 0
and Inf+Inf = Inf.
If any computation tries to produce a value that is undefined even in the real
number system, the result is an exceptional value known as Not-a-Number, or NaN.
Examples include 0/0 and Inf-Inf. NaN is represented by taking e = 1024 and f
nonzero.
If any computation tries to produce a value smaller than realmin, it is said to
underflow. This involves one of the optional, and controversial, aspects of the IEEE
standard. Many, but not all, machines allow exceptional denormal or subnormal
floating-point numbers in the interval between realmin and eps*realmin. The
smallest positive subnormal number is about 0.494e-323. Any results smaller than
this are set to 0. On machines without subnormals, any results less than realmin
are set to 0. The subnormal numbers fill in the gap you can see in the floatgui
model system between 0 and the smallest positive number. They do provide an
40 Chapter 1. Introduction to MATLAB
elegant way to handle underflow, but their practical importance for Matlab-style
computation is very rare. Denormal numbers are represented by taking e = −1023,
so the biased exponent e + 1023 is 0.
Matlab uses the floating-point system to handle integers. Mathematically,
the numbers 3 and 3.0 are the same, but many programming languages would use
different representations for the two. Matlab does not distinguish between them.
We sometimes use the term flint to describe a floating-point number whose value is
an integer. Floating-point operations on flints do not introduce any roundoff error,
as long as the results are not too large. Addition, subtraction, and multiplication of
flints produce the exact flint result if it is not larger than 253. Division and square
root involving flints also produce a flint if the result is an integer. For example,
sqrt(363/3) produces 11, with no roundoff.
Two Matlab functions that take apart and put together floating-point num-
bers are log2 and pow2.
help log2
help pow2
produces
[F,E] = LOG2(X) for a real array X, returns an array F
of real numbers, usually in the range 0.5 <= abs(F) < 1,
and an array E of integers, so that X = F .* 2.^E.
Any zeros in X produce F = 0 and E = 0.
X = POW2(F,E) for a real array F and an integer array E
computes X = F .* (2 .^ E). The result is computed quickly
by simply adding E to the floating-point exponent of F.
The quantities F and E used by log2 and pow2 predate the IEEE floating-point
standard and so are slightly different from the f and e we are using in this section.
In fact, f = 2*F-1 and e = E-1.
[F,E] = log2(phi)
produces
F =
0.80901699437495
E =
1
Then
phi = pow2(F,E)
gives back
phi =
1.61803398874989
1.7. Floating-Point Arithmetic 41
As an example of how roundoff error affects matrix computations, consider
the 2-by-2 set of linear equations
17x1 + 5x2 = 22,
1.7x1 + 0.5x2 = 2.2.
The obvious solution is x1 = 1, x2 = 1. But the Matlab statements
A = [17 5; 1.7 0.5]
b = [22; 2.2]
x = A
produce
x =
-1.0588
8.0000
Where did this come from? Well, the equations are singular, but consistent. The
second equation is just 0.1 times the first. The computed x is one of infinitely
many possible solutions. But the floating-point representation of the matrix A is
not exactly singular because A(2,1) is not exactly 17/10.
The solution process subtracts a multiple of the first equation from the second.
The multiplier is mu = 1.7/17, which turns out to be the floating-point number
obtained by truncating, rather than rounding, the binary expansion of 1/10. The
matrix A and the right-hand side b are modified by
A(2,:) = A(2,:) - mu*A(1,:)
b(2) = b(2) - mu*b(1)
With exact computation, both A(2,2) and b(2) would become zero, but with
floating-point arithmetic, they both become nonzero multiples of eps.
A(2,2) = (1/4)*eps
= 5.5511e-17
b(2) = 2*eps
= 4.4408e-16
Matlab notices the tiny value of the new A(2,2) and displays a message
warning that the matrix is close to singular. It then computes the solution of the
modified second equation by dividing one roundoff error by another.
x(2) = b(2)/A(2,2)
= 8
This value is substituted back into the first equation to give
x(1) = (22 - 5*x(2))/17
= -1.0588
42 Chapter 1. Introduction to MATLAB
The details of the roundoff error lead Matlab to pick out one particular solution
from among the infinitely many possible solutions to the singular system.
Our final example plots a seventh-degree polynomial.
x = 0.988:.0001:1.012;
y = x.^7-7*x.^6+21*x.^5-35*x.^4+35*x.^3-21*x.^2+7*x-1;
plot(x,y)
The resulting plot in Figure 1.9 doesn’t look anything like a polynomial. It
isn’t smooth. You are seeing roundoff error in action. The y-axis scale factor is tiny,
10−14. The tiny values of y are being computed by taking sums and differences of
0.985 0.99 0.995 1 1.005 1.01 1.015
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10
−14
Figure 1.9. Is this a polynomial?
numbers as large as 35 · 1.0124. There is severe subtractive cancellation. The
example was contrived by using the Symbolic Toolbox to expand (x − 1)7 and
carefully choosing the range for the x-axis to be near x = 1. If the values of y are
computed instead by
y = (x-1).^7;
then a smooth (but very flat) plot results.
1.8 Further Reading
Additional information about floating-point arithmetic and roundoff error can be
found in Higham [4] and Overton [6].
Exercises 43
Exercises
1.1. Which of these familiar rectangles is closest to a golden rectangle? Use Mat-
lab to do the calculations with an element-by-element vector division, w./h.
• 3-by-5 inch index card,
• 8.5-by-11 inch U.S. letter paper,
• 8.5-by-14 inch U.S. legal paper,
• 9-by-12 foot rug,
• 9:16 “letterbox” TV picture,
• 768-by-1024 pixel computer monitor.
1.2. ISO standard A4 paper is commonly used throughout most of the world,
except in the United States and Canada. Its dimensions are 210 by 297 mm.
This is not a golden rectangle, but the aspect ratio is close to another familiar
irrational mathematical quantity. What is that quantity? If you fold a piece
of A4 paper in half, what is the aspect ratio of each of the halves? Modify
the M-file goldrect.m to illustrate this property.
1.3. How many terms in the truncated continued fraction does it take to approx-
imate φ with an error less than 10−10? As the number of terms increases
beyond this roundoff, error eventually intervenes. What is the best accuracy
you can hope to achieve with double-precision floating-point arithmetic and
how many terms does it take?
1.4. Use the Matlab backslash operator to solve the 2-by-2 system of simultane-
ous linear equations
c1 + c2 = 1,
c1φ + c2(1− φ) = 1
for c1 and c2. You can find out about the backslash operator by taking a
peek at the next chapter of this book, or with the commands
help
help slash
1.5. The statement
semilogy(fibonacci(18),’-o’)
makes a logarithmic plot of Fibonacci numbers versus their index. The graph
is close to a straight line. What is the slope of this line?
1.6. How does the execution time of fibnum(n) depend on the execution time
for fibnum(n-1) and fibnum(n-2)? Use this relationship to obtain an ap-
proximate formula for the execution time of fibnum(n) as a function of n.
Estimate how long it would take your computer to compute fibnum(50).
Warning: You probably do not want to actually run fibnum(50).
44 Chapter 1. Introduction to MATLAB
1.7. What is the index of the largest Fibonacci number that can be represented
exactly as a Matlab double-precision quantity without roundoff error? What
is the index of the largest Fibonacci number that can be represented approx-
imately as a Matlab double-precision quantity without overflowing?
1.8. Enter the statements
A = [1 1; 1 0]
X = [1 0; 0 1]
Then enter the statement
X = A*X
Now repeatedly press the up arrow key, followed by the Enter key. What
happens? Do you recognize the matrix elements being generated? How many
times would you have to repeat this iteration before X overflows?
1.9. Change the fern color scheme to use pink on a black background. Don’t forget
the stop button.
1.10. (a) What happens if you resize the figure window while the fern is being
generated? Why?
(b) The M-file finitefern.m can be used to produce printed output of the
fern. Explain why printing is possible with finitefern.m but not with
fern.m.
1.11. Flip the fern by interchanging its x- and y-coordinates.
1.12. What happens to the fern if you change the only nonzero element in the
matrix A4?
1.13. What are the coordinates of the lower end of the fern’s stem?
1.14. The coordinates of the point at the upper tip end of the fern can be computed
by solving a certain 2-by-2 system of simultaneous linear equations. What is
that system and what are the coordinates of the tip?
1.15. The fern algorithm involves repeated random choices from four different for-
mulas for advancing the point. If the kth formula is used repeatedly by itself,
without random choices, it defines a deterministic trajectory in the (x, y)
plane. Modify finitefern.m so that plots of each of these four trajectories
are superimposed on the plot of the fern. Start each trajectory at the point
(−1, 5). Plot o’s connected with straight lines for the steps along each trajec-
tory. Take as many steps as are needed to show each trajectory’s limit point.
You can superimpose several plots with
plot(...)
hold on
plot(...)
plot(...)
hold off
1.16. Use the following code to make your own Portable Network Graphics file from
the fern. Then compare your image with one obtained from ncm/fern.png.
Exercises 45
bg = [0 0 85]; % Dark blue background
fg = [255 255 255]; % White dots
sz = get(0,’screensize’);
rand(’state’,0)
X = finitefern(500000,sz(4),sz(3));
d = fg - bg;
R = uint8(bg(1) + d(1)*X);
G = uint8(bg(2) + d(2)*X);
B = uint8(bg(3) + d(3)*X);
F = cat(3,R,G,B);
imwrite(F,’myfern.png’,’png’,’bitdepth’,8)
1.17. Modify fern.m or finitefern.m so that it produces Sierpinski’s triangle.
Start at
x =
(
0
0
)
.
At each iterative step, the current point x is replaced with Ax+ b, where the
matrix A is always
A =
(
1/2 0
0 1/2
)
and the vector b is chosen at random with equal probability from among the
three vectors
b =
(
0
0
)
, b =
(
1/2
0
)
, and b =
(
1/4√
3/4
)
.
1.18. greetings(phi) generates a seasonal holiday fractal that depends upon the
parameter phi. The default value of phi is the golden ratio. What hap-
pens for other values of phi? Try both simple fractions and floating-point
approximations to irrational values.
1.19. A = magic(4) is singular. Its columns are linearly dependent. What do
null(A), null(A,’r’), null(sym(A)), and rref(A) tell you about that de-
pendence?
1.20. Let A = magic(n) for n = 3, 4, or 5. What does
p = randperm(n); q = randperm(n); A = A(p,q);
do to
sum(A)
sum(A’)’
sum(diag(A))
sum(diag(flipud(A)))
rank(A)
1.21. The character char(7) is a control character. What does it do?
1.22. What does char([169 174]) display on your computer?
46 Chapter 1. Introduction to MATLAB
1.23. What fundamental physical law is hidden in this string?
s = ’/b_t3{$H~MO6JTQI>v~#3GieW*l(p,nF’
1.24. Find the two files encrypt.m and gettysburg.txt. Use encrypt to encrypt
gettysburg.txt. Then decrypt the result. Use encrypt to encrypt itself.
1.25. With the NCM directory on your path, you can read the text of Lincoln’s
Gettysburg Address with
fp = fopen(’gettysburg.txt’);
G = char(fread(fp))’
fclose(fp);
(a) How many characters are in the text?
(b) Use the unique function to find the unique characters in the text.
(c) How many blanks are in the text? What punctuation characters, and how
many of each, are there?
(d) Remove the blanks and the punctuation and convert the text to all upper-
or lowercase. Use the histc function to count the number of letters. What
is the most frequent letter? What letters are missing?
(e) Use the bar function as described in help histc to plot a histogram of
the letter frequencies.
(f) Use get(gca,’xtick’) and get(gca,’xticklabel’) to see how the x-
axis of the histogram is labeled. Then use
set(gca,’xtick’,…,’xticklabel’,…)
to relabel the x-axis with the letters in the text.
1.26. If x is the character string consisting of just two blanks,
x = ’ ’
then crypto(x) is actually equal to x. Why does this happen? Are there
any other two-character strings that crypto does not change?
1.27. Find another 2-by-2 integer matrix A for which
mod(A*A,97)
is the identity matrix. Replace the matrix in crypto.m with your matrix and
verify that the function still works correctly.
1.28. The function crypto works with 97 characters instead of 95. It can produce
output, and correctly handle input, that contains two characters with ASCII
values greater than 127. What are these characters? Why are they necessary?
What happens to other characters with ASCII values greater than 127?
1.29. Create a new crypto function that works with just 29 characters: the 26
lowercase letters, plus blank, period, and comma. You will need to find a
2-by-2 integer matrix A for which mod(A*A,29) is the identity matrix.
1.30. The graph of the 3n + 1 sequence has a particular characteristic shape if the
starting n is 5, 10, 20, 40, . . . , that is, n is five times a power of 2. What is
this shape and why does it happen?
Exercises 47
1.31. The graphs of the 3n+1 sequences starting at n = 108, 109, and 110 are very
similar to each other. Why?
1.32. Let L(n) be the number of terms in the 3n + 1 sequence that starts with n.
Write a Matlab function that computes L(n) without using any vectors or
unpredictable amounts of storage. Plot L(n) for 1 ≤ n ≤ 1000. What is the
maximum value of L(n) for n in this range, and for what value of n does it
occur? Use threenplus1 to plot the sequence that starts with this particular
value of n.
1.33. Modify floatgui.m by changing its last line from a comment to an executable
statement and changing the question mark to a simple expression that counts
the number of floating-point numbers in the model system.
1.34. Explain the output produced by
t = 0.1
n = 1:10
e = n/10 – n*t
1.35. What does each of these programs do? How many lines of output does each
program produce? What are the last two values of x printed?
x = 1; while 1+x > 1, x = x/2, pause(.02), end
x = 1; while x+x > x, x = 2*x, pause(.02), end
x = 1; while x+x > x, x = x/2, pause(.02), end
1.36. Which familiar real numbers are approximated by floating-point numbers
that display the following values with format hex?
4059000000000000
3f847ae147ae147b
3fe921fb54442d18
1.37. Let F be the set of all IEEE double-precision floating-point numbers, except
NaNs and Infs, which have biased exponent 7ff (hex), and denormals, which
have biased exponent 000 (hex).
(a) How many elements are there in F?
(b) What fraction of the elements of F are in the interval 1 ≤ x < 2?
(c) What fraction of the elements of F are in the interval 1/64 ≤ x < 1/32?
(d) Determine by random sampling approximately what fraction of the ele-
ments x of F satisfy the Matlab logical relation
x*(1/x) == 1
1.38. The classic quadratic formula says that the two roots of the quadratic equa-
tion
ax2 + bx + c = 0
are
x1, x2 =
−b±
√
b2 − 4ac
2a
.
48 Chapter 1. Introduction to MATLAB
Use this formula in Matlab to compute both roots for
a = 1, b = −100000000, c = 1.
Compare your computed results with
roots([a b c])
What happens if you try to compute the roots by hand or with a hand
calculator?
You should find that the classic formula is good for computing one root, but
not the other. So use it to compute one root accurately and then use the fact
that
x1x2 =
c
a
to compute the other.
1.39. The power series for sin x is
sin x = x− x
3
3!
+
x5
5!
− x
7
7!
+ · · · .
This Matlab function uses the series to compute sin x.
function s = powersin(x)
% POWERSIN. Power series for sin(x).
% POWERSIN(x) tries to compute sin(x)
% from a power series
s = 0;
t = x;
n = 1;
while s+t ~= s;
s = s + t;
t = -x.^2/((n+1)*(n+2)).*t;
n = n + 2;
end
What causes the while loop to terminate?
Answer the following questions for x = π/2, 11π/2, 21π/2, and 31π/2:
How accurate is the computed result?
How many terms are required?
What is the largest term in the series?
What do you conclude about the use of floating-point arithmetic and power
series to evaluate functions?
1.40. Steganography is the technique of hiding messages or other images in the
low-order bits of the data for an image. The Matlab image function has a
hidden image that contains other hidden images. To see the top-level image,
just execute the single command
Exercises 49
image
Then, to improve its appearance,
colormap(gray(32))
truesize
axis ij
axis image
axis off
But that’s just the beginning. The NCM program stegano helps you continue
the investigation.
(a) How many images are hidden in the cdata for the default image?
(b) What does this have to do with the structure of floating-point numbers?
1.41. Prime spirals. A Ulam prime spiral is a plot of the location of the prime
numbers using a numbering scheme that spirals outward from the center of
a grid. Our NCM file primespiral(n,c) generates an n-by-n prime spiral
starting with the number c in the center. The default is c = 1. Figure 1.10
is primespiral(7) and Figure 1.11 is primespiral(250).
43 44 45 46 47 48 49
42 21 22 23 24 25 26
41 20 7 8 9 10 27
40 19 6 1 2 11 28
39 18 5 4 3 12 29
38 17 16 15 14 13 30
37 36 35 34 33 32 31
Figure 1.10. primespiral(7).
The concentration of primes on some diagonal segments is remarkable, and
not completely understood. The value of the element at position (i, j) is a
piecewise quadratic function of i and j, so each diagonal segment represents
a mini-theorem about the distribution of primes. The phenomenon was dis-
covered by Stanislaw Ulam in 1963 and appeared on the cover of Scientific
American in 1964. There are a number of interesting Web pages devoted to
prime spirals. Start with [7] and [9].
(a) The Matlab demos directory contains an M-file spiral.m. The integers
from 1 to n2 are arranged in a spiral pattern, starting in the center of the
matrix. The code in demos/spiral.m is not very elegant. Here is a better
version.
50 Chapter 1. Introduction to MATLAB
0 50 100 150 200 250
0
50
100
150
200
250
nz = 6275
Figure 1.11. primespiral(250).
function S = spiral(n)
%SPIRAL SPIRAL(n) is an n-by-n matrix with elements
% 1:n^2 arranged in a rectangular spiral pattern.
S = [];
for m = 1:n
S = rot90(S,2);
S(m,m) = 0;
p = ???
v = (m-1:-1:0);
S(:,m) = p-v’;
S(m,:) = p+v;
end
if mod(n,2)==1
S = rot90(S,2);
end
What value should be assigned to p each time through the loop so that this
function generates the same matrices as spiral.m in the demos directory?
(b) Why do half of the diagonals of spiral(n) contain no primes?
(c) Let S = spiral(2*n) and let r1 and r2 be rows that go nearly halfway
across the middle of the matrix:
Exercises 51
r1 = S(n+1,1:n-2)
r2 = S(n-1,n+2:end)
Why do these rows contain no primes?
(d) There is something particularly remarkable about
primespiral(17,17)
primespiral(41,41)
What is it?
(e) Find values of n and c, both less than 50, and not equal to 17 or 41, so
that
[S,P] = primespiral(n,c)
contains a diagonal segment with 8 or more primes.
1.42. Triangular numbers are integers of the form n(n + 1)/2. The term comes
from the fact that a triangular grid with n points on a side has a total
of n(n + 1)/2 points. Write a function trinums(m) that generates all the
triangular numbers less than or equal to m. Modify primespiral to use your
trinums and become trinumspiral.
1.43. Here is a puzzle that does not have much to do with this chapter, but you
might find it interesting nevertheless. What familiar property of the integers
is represented by the following plot?
0 10 20 30 40 50 60 70 80 90 100
0
2
4
6
8
1.44. In the Gregorian calendar, a year y is a leap year if and only if
(mod(y,4) == 0) & (mod(y,100) ~= 0) | (mod(y,400) == 0)
Thus 2000 was a leap year, but 2100 will not be a leap year. This rule implies
that the Gregorian calendar repeats itself every 400 years. In that 400-year
period, there are 97 leap years, 4800 months, 20871 weeks, and 146097 days.
The Matlab functions datenum, datevec, datestr, and weekday use these
facts to facilitate computations involving calendar dates. For example, either
of the statements
[d,w] = weekday(’Aug. 17, 2003’)
and
[d,w] = weekday(datenum([2003 8 17]))
tells me that my birthday was on a Sunday in 2003.
Use Matlab to answer the following questions.
(a) On which day of the week were you born?
52 Chapter 1. Introduction to MATLAB
(b) In a 400-year Gregorian calendar cycle, which weekday is the most likely
for your birthday?
(c) What is the probability that the 13th of any month falls on a Friday?
The answer is close to, but not exactly equal to, 1/7.
1.45. Biorhythms were very popular in the 1960s. You can still find many Web sites
today that offer to prepare personalized biorhythms, or that sell software to
compute them. Biorhythms are based on the notion that three sinusoidal
cycles influence our lives. The physical cycle has a period of 23 days, the
emotional cycle has a period of 28 days, and the intellectual cycle has a
period of 33 days. For any individual, the cycles are initialized at birth.
Figure 1.12 is my biorhythm, which begins on August 17, 1939, plotted for
an eight-week period centered around the date this is being written, October
19, 2003. It shows that my intellectual power reached a peak yesterday, that
my physical strength and emotional wellbeing will reach their peaks within
6 h of each other on the same day next week, and that all three cycles will
be at their low point within a few days of each other early in November.
09/21 09/28 10/05 10/12 10/19 10/26 11/02 11/09 11/16
−100
−50
0
50
100
10/19/03
birthday: 08/17/39
Physical
Emotional
Intellectual
Figure 1.12. My biorhythm.
The date and graphics functions in Matlab make the computation and dis-
play of biorhythms particularly convenient. Dates are represented by their
date number, which is the number of days since the zeroth day of a theoretical
calendar year zero. The function datenum returns the date number for any
given date and time. For example, datenum(’Oct. 19, 2003’) is 731873.
The expression fix(now) returns the date number of the current date.
The following code segment is part of a program that plots a biorhythm for
an eight-week period centered on the current date.
t0 = datenum(mybirthday);
t1 = fix(now);
t = (t1-28):1:(t1+28);
y = 100*[sin(2*pi*(t-t0)/23)
sin(2*pi*(t-t0)/28)
sin(2*pi*(t-t0)/33)];
plot(t,y)
Exercises 53
(a) Complete this program, using your own birthday, and the line, datetick,
title, datestr, and legend functions. Your program should produce some-
thing like Figure 1.12.
(b) All three cycles started at zero when you were born. How long does it
take until all three simultaneously return to that initial condition? How old
were you, or will you be, on that date? Plot your biorhythm near that date.
You should find the lcm function helpful.
(c) Is it possible for all three cycles to reach their maximum or minimum at
exactly the same time?
54 Chapter 1. Introduction to MATLAB
Bibliography
[1] M. Barnsley, Fractals Everywhere, Academic Press, Boston, 1993.
[2] D. C. Hanselman and B. Littlefield, Mastering MATLAB 6, A Compre-
hensive Tutorial and Reference, Prentice–Hall, Upper Saddle River, NJ, 2000.
[3] D. J. Higham and N. J. Higham, MATLAB Guide, SIAM, Philadelphia,
2000.
[4] N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM,
Philadelphia, 2002.
[5] J. Lagarias, The 3x+1 problem and its generalizations, American Mathemat-
ical Monthly, 92 (1985), pp. 3–23.
http://www.cecm.sfu.ca/organics/papers/lagarias
[6] M. Overton, Numerical Computing with IEEE Floating Point Arithmetic,
SIAM, Philadelphia, 2001.
[7] I. Peterson, Prime Spirals, Science News Online, 161 (2002).
http://www.sciencenews.org/20020504/mathtrek.asp
[8] K. Sigmon and T. A. Davis, MATLAB Primer, Sixth Edition, Chapman and
Hall/CRC, Boca Raton, FL, 2002.
[9] E. Weisstein, World of Mathematics, Prime Spiral,
http://mathworld.wolfram.com/PrimeSpiral.html
[10] The MathWorks, Inc., Getting Started with MATLAB.
http://www.mathworks.com/access/helpdesk/help/techdoc
/learn_matlab/learn_matlab.shtml
[11] The MathWorks, Inc., List of Matlab-based books.
http://www.mathworks.com/support/books/index.jsp
55
Chapter 2
Linear Equations
One of the problems encountered most frequently in scientific computation is the
solution of systems of simultaneous linear equations. This chapter covers the solu-
tion of linear systems by Gaussian elimination and the sensitivity of the solution to
errors in the data and roundoff errors in the computation.
2.1 Solving Linear Systems
With matrix notation, a system of simultaneous linear equations is written
Ax = b.
In the most frequent case, when there are as many equations as unknowns, A is a
given square matrix of order n, b is a given column vector of n components, and x
is an unknown column vector of n components.
Students of linear algebra learn that the solution to Ax = b can be written
x = A−1b, where A−1 is the inverse of A. However, in the vast majority of practical
computational problems, it is unnecessary and inadvisable to actually compute
A−1. As an extreme but illustrative example, consider a system consisting of just
one equation, such as
7x = 21.
The best way to solve such a system is by division:
x =
21
7
= 3.
Use of the matrix inverse would lead to
x = 7−1 × 21 = 0.142857× 21 = 2.99997.
The inverse requires more arithmetic—a division and a multiplication instead of
just a division—and produces a less accurate answer. Similar considerations apply
February 15, 2008
1
2 Chapter 2. Linear Equations
to systems of more than one equation. This is even true in the common situation
where there are several systems of equations with the same matrix A but different
right-hand sides b. Consequently, we shall concentrate on the direct solution of
systems of equations rather than the computation of the inverse.
2.2 The MATLAB Backslash Operator
To emphasize the distinction between solving linear equations and computing in-
verses, Matlab has introduced nonstandard notation using backward slash and
forward slash operators, “” and “/”.
If A is a matrix of any size and shape and B is a matrix with as many rows
as A, then the solution to the system of simultaneous equations
AX = B
is denoted by
X = AB.
Think of this as dividing both sides of the equation by the coefficient matrix A.
Because matrix multiplication is not commutative and A occurs on the left in the
original equation, this is left division.
Similarly, the solution to a system with A on the right and B with as many
columns as A,
XA = B,
is obtained by right division,
X = B/A.
This notation applies even if A is not square, so that the number of equations is not
the same as the number of unknowns. However, in this chapter, we limit ourselves
to systems with square coefficient matrices.
2.3 A 3-by-3 Example
To illustrate the general linear equation solution algorithm, consider an example of
order three:
10 −7 0
−3 2 6
5 −1 5
x1
x2
x3
=
7
4
6
.
This, of course, represents the three simultaneous equations
10x1 − 7x2 = 7,
−3x1 + 2x2 + 6x3 = 4,
5x1 − x2 + 5x3 = 6.
The first step of the solution algorithm uses the first equation to eliminate x1 from
the other equations. This is accomplished by adding 0.3 times the first equation
2.3. A 3-by-3 Example 3
to the second equation and subtracting 0.5 times the first equation from the third
equation. The coefficient 10 of x1 in the first equation is called the first pivot
and the quantities −0.3 and 0.5, obtained by dividing the coefficients of x1 in the
other equations by the pivot, are called the multipliers. The first step changes the
equations to
10 −7 0
0 −0.1 6
0 2.5 5
x1
x2
x3
=
7
6.1
2.5
.
The second step might use the second equation to eliminate x2 from the third
equation. However, the second pivot, which is the coefficient of x2 in the second
equation, would be −0.1, which is smaller than the other coefficients. Consequently,
the last two equations are interchanged. This is called pivoting. It is not actually
necessary in this example because there are no roundoff errors, but it is crucial in
general:
10 −7 0
0 2.5 5
0 −0.1 6
x1
x2
x3
=
7
2.5
6.1
.
Now the second pivot is 2.5 and the second equation can be used to eliminate x2 from
the third equation. This is accomplished by adding 0.04 times the second equation
to the third equation. (What would the multiplier have been if the equations had
not been interchanged?)
10 −7 0
0 2.5 5
0 0 6.2
x1
x2
x3
=
7
2.5
6.2
.
The last equation is now
6.2x3 = 6.2.
This can be solved to give x3 = 1. This value is substituted into the second equation:
2.5x2 + (5)(1) = 2.5.
Hence x2 = −1. Finally, the values of x2 and x3 are substituted into the first
equation:
10x1 + (−7)(−1) = 7.
Hence x1 = 0. The solution is
x =
0
−1
1
.
This solution can be easily checked using the original equations:
10 −7 0
−3 2 6
5 −1 5
0
−1
1
=
7
4
6
.
4 Chapter 2. Linear Equations
The entire algorithm can be compactly expressed in matrix notation. For this
example, let
L =
1 0 0
0.5 1 0
−0.3 −0.04 1
, U =
10 −7 0
0 2.5 5
0 0 6.2
, P =
1 0 0
0 0 1
0 1 0
.
The matrix L contains the multipliers used during the elimination, the matrix U
is the final coefficient matrix, and the matrix P describes the pivoting. With these
three matrices, we have
LU = PA.
In other words, the original coefficient matrix can be expressed in terms of products
involving matrices with simpler structure.
2.4 Permutation and Triangular Matrices
A permutation matrix is an identity matrix with the rows and columns interchanged.
It has exactly one 1 in each row and column; all the other elements are 0. For
example,
P =
0 0 0 1
1 0 0 0
0 0 1 0
0 1 0 0
.
Multiplying a matrix A on the left by a permutation matrix to give PA permutes
the rows of A. Multiplying on the right, AP , permutes the columns of A.
Matlab can also use a permutation vector as a row or column index to rear-
range the rows or columns of a matrix. Continuing with the P above, let p be the
vector
p = [4 1 3 2]
Then P*A and A(p,:) are equal. The resulting matrix has the fourth row of A as
its first row, the first row of A as its second row, and so on. Similarly, A*P and
A(:,p) both produce the same permutation of the columns of A. The P*A notation
is closer to traditional mathematics, PA, while the A(p,:) notation is faster and
uses less memory.
Linear equations involving permutation matrices are trivial to solve. The
solution to
Px = b
is simply a rearrangement of the components of b:
x = PT b.
An upper triangular matrix has all its nonzero elements above or on the main
diagonal. A unit lower triangular matrix has ones on the main diagonal and all the
2.5. LU Factorization 5
rest of its nonzero elements below the main diagonal. For example,
U =
1 2 3 4
0 5 6 7
0 0 8 9
0 0 0 10
is upper triangular, and
L =
1 0 0 0
2 1 0 0
3 5 1 0
4 6 7 1
is unit lower triangular.
Linear equations involving triangular matrices are also easily solved. There are
two variants of the algorithm for solving an n-by-n upper triangular system Ux = b.
Both begin by solving the last equation for the last variable, then the next-to-last
equation for the next-to-last variable, and so on. One subtracts multiples of the
columns of U from b.
x = zeros(n,1);
for k = n:-1:1
x(k) = b(k)/U(k,k);
i = (1:k-1)’;
b(i) = b(i) - x(k)*U(i,k);
end
The other uses inner products between the rows of U and portions of the
emerging solution x.
x = zeros(n,1);
for k = n:-1:1
j = k+1:n;
x(k) = (b(k) - U(k,j)*x(j))/U(k,k);
end
2.5 LU Factorization
The algorithm that is almost universally used to solve square systems of simultane-
ous linear equations is one of the oldest numerical methods, the systematic elimi-
nation method, generally named after C. F. Gauss. Research in the period 1955 to
1965 revealed the importance of two aspects of Gaussian elimination that were not
emphasized in earlier work: the search for pivots and the proper interpretation of
the effect of rounding errors.
In general, Gaussian elimination has two stages, the forward elimination and
the back substitution. The forward elimination consists of n − 1 steps. At the kth
step, multiples of the kth equation are subtracted from the remaining equations
to eliminate the kth variable. If the coefficient of xk is “small,” it is advisable to
6 Chapter 2. Linear Equations
interchange equations before this is done. The elimination steps can be simultane-
ously applied to the right-hand side, or the interchanges and multipliers saved and
applied to the right-hand side later. The back substitution consists of solving the
last equation for xn, then the next-to-last equation for xn−1, and so on, until x1 is
computed from the first equation.
Let Pk, k = 1, . . . , n − 1, denote the permutation matrix obtained by in-
terchanging the rows of the identity matrix in the same way the rows of A are
interchanged at the kth step of the elimination. Let Mk denote the unit lower tri-
angular matrix obtained by inserting the negatives of the multipliers used at the
kth step below the diagonal in the kth column of the identity matrix. Let U be the
final upper triangular matrix obtained after the n− 1 steps. The entire process can
be described by one matrix equation,
U = Mn−1Pn−1 · · ·M2P2M1P1A.
It turns out that this equation can be rewritten
L1L2 · · ·Ln−1U = Pn−1 · · ·P2P1A,
where Lk is obtained from Mk by permuting and changing the signs of the multi-
pliers below the diagonal. So, if we let
L = L1L2 · · ·Ln−1,
P = Pn−1 · · ·P2P1,
then we have
LU = PA.
The unit lower triangular matrix L contains all the multipliers used during the
elimination and the permutation matrix P accounts for all the interchanges.
For our example
A =
10 −7 0
−3 2 6
5 −1 5
,
the matrices defined during the elimination are
P1 =
1 0 0
0 1 0
0 0 1
, M1 =
1 0 0
0.3 1 0
−0.5 0 1
,
P2 =
1 0 0
0 0 1
0 1 0
, M2 =
1 0 0
0 1 0
0 0.04 1
.
The corresponding L’s are
L1 =
1 0 0
0.5 1 0
−0.3 0 1
, L2 =
1 0 0
0 1 0
0 −0.04 1
.
2.6. Why Is Pivoting Necessary? 7
The relation LU = PA is called the LU factorization or the triangular de-
composition of A. It should be emphasized that nothing new has been introduced.
Computationally, elimination is done by row operations on the coefficient matrix,
not by actual matrix multiplication. LU factorization is simply Gaussian elimina-
tion expressed in matrix notation.
With this factorization, a general system of equations
Ax = b
becomes a pair of triangular systems
Ly = Pb,
Ux = y.
2.6 Why Is Pivoting Necessary?
The diagonal elements of U are called pivots. The kth pivot is the coefficient of the
kth variable in the kth equation at the kth step of the elimination. In our 3-by-3
example, the pivots are 10, 2.5, and 6.2. Both the computation of the multipliers and
the back substitution require divisions by the pivots. Consequently, the algorithm
cannot be carried out if any of the pivots are zero. Intuition should tell us that it
is a bad idea to complete the computation if any of the pivots are nearly zero. To
see this, let us change our example slightly to
10 −7 0
−3 2.099 6
5 −1 5
x1
x2
x3
=
7
3.901
6
.
The (2, 2) element of the matrix has been changed from 2.000 to 2.099, and the
right-hand side has also been changed so that the exact answer is still (0,−1, 1)T .
Let us assume that the solution is to be computed on a hypothetical machine that
does decimal floating-point arithmetic with five significant digits.
The first step of the elimination produces
10 −7 0
0 −0.001 6
0 2.5 5
x1
x2
x3
=
7
6.001
2.5
.
The (2, 2) element is now quite small compared with the other elements in the ma-
trix. Nevertheless, let us complete the elimination without using any interchanges.
The next step requires adding 2.5 · 103 times the second equation to the third:
(5 + (2.5 · 103)(6))x3 = (2.5 + (2.5 · 103)(6.001)).
On the right-hand side, this involves multiplying 6.001 by 2.5 · 103. The result is
1.50025 ·104, which cannot be exactly represented in our hypothetical floating-point
number system. It must be rounded to 1.5002 · 104. The result is then added to 2.5
and rounded again. In other words, both of the 5’s shown in italics in
(5 + 1.5000 · 104)x3 = (2.5 + 1.50025 · 104)
8 Chapter 2. Linear Equations
are lost in roundoff errors. On this hypothetical machine, the last equation becomes
1.5005 · 104x3 = 1.5004 · 104.
The back substitution begins with
x3 =
1.5004 · 104
1.5005 · 104 = 0.99993.
Because the exact answer is x3 = 1, it does not appear that the error is too serious.
Unfortunately, x2 must be determined from the equation
−0.001x2 + (6)(0.99993) = 6.001,
which gives
x2 =
1.5 · 10−3
−1.0 · 10−3 = −1.5.
Finally, x1 is determined from the first equation,
10x1 + (−7)(−1.5) = 7,
which gives
x1 = −0.35.
Instead of (0,−1, 1)T , we have obtained (−0.35,−1.5, 0.99993)T .
Where did things go wrong? There was no “accumulation of rounding error”
caused by doing thousands of arithmetic operations. The matrix is not close to
singular. The difficulty comes from choosing a small pivot at the second step of the
elimination. As a result, the multiplier is 2.5 · 103, and the final equation involves
coefficients that are 103 times as large as those in the original problem. Roundoff
errors that are small if compared to these large coefficients are unacceptable in
terms of the original matrix and the actual solution.
We leave it to the reader to verify that if the second and third equations are
interchanged, then no large multipliers are necessary and the final result is accurate.
This turns out to be true in general: If the multipliers are all less than or equal
to one in magnitude, then the computed solution can be proved to be satisfactory.
Keeping the multipliers less than one in absolute value can be ensured by a process
known as partial pivoting. At the kth step of the forward elimination, the pivot is
taken to be the largest (in absolute value) element in the unreduced part of the kth
column. The row containing this pivot is interchanged with the kth row to bring
the pivot element into the (k, k) position. The same interchanges must be done
with the elements of the right-hand side b. The unknowns in x are not reordered
because the columns of A are not interchanged.
2.7 lutx, bslashtx, lugui
We have three functions implementing the algorithms discussed in this chapter.
The first function, lutx, is a readable version of the built-in Matlab function lu.
2.7. lutx, bslashtx, lugui 9
There is one outer for loop on k that counts the elimination steps. The inner loops
on i and j are implemented with vector and matrix operations, so that the overall
function is reasonably efficient.
function [L,U,p] = lutx(A)
%LU Triangular factorization
% [L,U,p] = lutx(A) produces a unit lower triangular
% matrix L, an upper triangular matrix U, and a
% permutation vector p, so that L*U = A(p,:).
[n,n] = size(A);
p = (1:n)’
for k = 1:n-1
% Find largest element below diagonal in k-th column
[r,m] = max(abs(A(k:n,k)));
m = m+k-1;
% Skip elimination if column is zero
if (A(m,k) ~= 0)
% Swap pivot row
if (m ~= k)
A([k m],:) = A([m k],:);
p([k m]) = p([m k]);
end
% Compute multipliers
i = k+1:n;
A(i,k) = A(i,k)/A(k,k);
% Update the remainder of the matrix
j = k+1:n;
A(i,j) = A(i,j) - A(i,k)*A(k,j);
end
end
% Separate result
L = tril(A,-1) + eye(n,n);
U = triu(A);
Study this function carefully. Almost all the execution time is spent in the
statement
A(i,j) = A(i,j) - A(i,k)*A(k,j);
10 Chapter 2. Linear Equations
At the kth step of the elimination, i and j are index vectors of length n-k. The
operation A(i,k)*A(k,j) multiplies a column vector by a row vector to produce
a square, rank one matrix of order n-k. This matrix is then subtracted from the
submatrix of the same size in the bottom right corner of A. In a programming
language without vector and matrix operations, this update of a portion of A would
be done with doubly nested loops on i and j.
The second function, bslashtx, is a simplified version of the built-in Matlab
backslash operator. It begins by checking for three important special cases: lower
triangular, upper triangular, and symmetric positive definite. Linear systems with
these properties can be solved in less time than a general system.
function x = bslashtx(A,b)
% BSLASHTX Solve linear system (backslash)
% x = bslashtx(A,b) solves A*x = b
[n,n] = size(A);
if isequal(triu(A,1),zeros(n,n))
% Lower triangular
x = forward(A,b);
return
elseif isequal(tril(A,-1),zeros(n,n))
% Upper triangular
x = backsubs(A,b);
return
elseif isequal(A,A’)
[R,fail] = chol(A);
if ~fail
% Positive definite
y = forward(R’,b);
x = backsubs(R,y);
return
end
end
If none of the special cases is detected, bslashtx calls lutx to permute and fac-
tor the coefficient matrix, then uses the permutation and factors to complete the
solution of a linear system.
% Triangular factorization
[L,U,p] = lutx(A);
% Permutation and forward elimination
y = forward(L,b(p));
% Back substitution
x = backsubs(U,y);
2.8. Effect of Roundoff Errors 11
The bslashtx function employs subfunctions to carry out the solution of lower and
upper triangular systems.
function x = forward(L,x)
% FORWARD. Forward elimination.
% For lower triangular L, x = forward(L,b) solves L*x = b.
[n,n] = size(L);
for k = 1:n
j = 1:k-1;
x(k) = (x(k) - L(k,j)*x(j))/L(k,k);
end
function x = backsubs(U,x)
% BACKSUBS. Back substitution.
% For upper triangular U, x = backsubs(U,b) solves U*x = b.
[n,n] = size(U);
for k = n:-1:1
j = k+1:n;
x(k) = (x(k) - U(k,j)*x(j))/U(k,k);
end
A third function, lugui, shows the steps in LU decomposition by Gaussian
elimination. It is a version of lutx that allows you to experiment with various pivot
selection strategies. At the kth step of the elimination, the largest element in the
unreduced portion of the kth column is shown in magenta. This is the element that
partial pivoting would ordinarily select as the pivot. You can then choose among
four different pivoting strategies:
• Pick a pivot. Use the mouse to pick the magenta element, or any other
element, as pivot.
• Diagonal pivoting. Use the diagonal element as the pivot.
• Partial pivoting. Same strategy as lu and lutx.
• Complete pivoting. Use the largest element in the unfactored submatrix as
the pivot.
The chosen pivot is shown in red and the resulting elimination step is taken. As the
process proceeds, the emerging columns of L are shown in green and the emerging
rows of U in blue.
2.8 Effect of Roundoff Errors
The rounding errors introduced during the solution of a linear system of equations
almost always cause the computed solution—which we now denote by x∗—to differ
somewhat from the theoretical solution x = A−1b. In fact, if the elements of x
12 Chapter 2. Linear Equations
are not floating-point numbers, then x∗ cannot equal x. There are two common
measures of the discrepancy in x∗: the error,
e = x− x∗,
and the residual,
r = b−Ax∗.
Matrix theory tells us that, because A is nonsingular, if one of these is zero, the
other must also be zero. But they are not necessarily both “small” at the same
time. Consider the following example:
(
0.780 0.563
0.913 0.659
) (
x1
x2
)
=
(
0.217
0.254
)
.
What happens if we carry out Gaussian elimination with partial pivoting on a
hypothetical three-digit decimal computer? First, the two rows (equations) are
interchanged so that 0.913 becomes the pivot. Then the multiplier
0.780
0.913
= 0.854 (to three places)
is computed. Next, 0.854 times the new first row is subtracted from the new second
row to produce the system
(
0.913 0.659
0 0.001
) (
x1
x2
)
=
(
0.254
0.001
)
.
Finally, the back substitution is carried out:
x2 =
0.001
0.001
= 1.00 (exactly),
x1 =
0.254− 0.659x2
0.913
= −0.443 (to three places).
Thus the computed solution is
x∗ =
(
−0.443
1.000
)
.
To assess the accuracy without knowing the exact answer, we compute the residuals
(exactly):
r = b−Ax∗ =
(
0.217− ((0.780)(−0.443) + (0.563)(1.00))
0.254− ((0.913)(−0.443) + (0.659)(1.00))
)
=
(
−0.000460
−0.000541
)
.
2.8. Effect of Roundoff Errors 13
The residuals are less than 10−3. We could hardly expect better on a three-digit
machine. However, it is easy to see that the exact solution to this system is
x =
(
1.000
−1.000
)
.
So the components of our computed solution actually have the wrong signs; the
error is larger than the solution itself.
Were the small residuals just a lucky fluke? You should realize that this
example is highly contrived. The matrix is very close to being singular and is not
typical of most problems encountered in practice. Nevertheless, let us track down
the reason for the small residuals.
If Gaussian elimination with partial pivoting is carried out for this example on
a computer with six or more digits, the forward elimination will produce a system
something like
(
0.913000 0.659000
0 −0.000001
)(
x1
x2
)
=
(
0.254000
0.000001
)
.
Notice that the sign of U2,2 differs from that obtained with three-digit computation.
Now the back substitution produces
x2 =
0.000001
−0.000001 = −1.00000,
x1 =
0.254− 0.659x2
0.913
= 1.00000,
the exact answer. On our three-digit machine, x2 was computed by dividing two
quantities, both of which were on the order of rounding errors and one of which did
not even have the correct sign. Hence x2 can turn out to be almost anything. Then
this arbitrary value of x2 was substituted into the first equation to obtain x1.
We can reasonably expect the residual from the first equation to be small—
x1 was computed in such a way as to make this certain. Now comes a subtle but
crucial point. We can also expect the residual from the second equation to be small,
precisely because the matrix is so close to being singular. The two equations are very
nearly multiples of one another, so any pair (x1, x2) that nearly satisfies the first
equation will also nearly satisfy the second. If the matrix were known to be exactly
singular, we would not need the second equation at all—any solution of the first
would automatically satisfy the second.
In Figure 2.1, the exact solution is marked with a circle and the computed
solution with an asterisk. Even though the computed solution is far from the exact
intersection, it is close to both lines because they are nearly parallel.
Although this example is contrived and atypical, the conclusion we reached
is not. It is probably the single most important fact that we have learned about
matrix computation since the invention of the digital computer:
Gaussian elimination with partial pivoting is guaranteed to produce small
residuals.
14 Chapter 2. Linear Equations
−1.5 −1 −0.5 0 0.5 1 1.5 2
−1.5
−1
−0.5
0
0.5
1
1.5
Figure 2.1. The computed solution, marked by an asterisk, shows a large
error, but a small residual.
Now that we have stated it so strongly, we must make a couple of qualifying
remarks. By “guaranteed” we mean it is possible to prove a precise theorem that
assumes certain technical details about how the floating-point arithmetic system
works and that establishes certain inequalities that the components of the residual
must satisfy. If the arithmetic units work some other way or if there is a bug in
the particular program, then the “guarantee” is void. Furthermore, by “small” we
mean on the order of roundoff error relative to three quantities: the size of the
elements of the original coefficient matrix, the size of the elements of the coefficient
matrix at intermediate steps of the elimination process, and the size of the elements
of the computed solution. If any of these are “large,” then the residual will not
necessarily be small in an absolute sense. Finally, even if the residual is small, we
have made no claims that the error will be small. The relationship between the size
of the residual and the size of the error is determined in part by a quantity known
as the condition number of the matrix, which is the subject of the next section.
2.9 Norms and Condition Numbers
The coefficients in the matrix and right-hand side of a system of simultaneous linear
equations are rarely known exactly. Some systems arise from experiments, and so
the coefficients are subject to observational errors. Other systems have coefficients
given by formulas that involve roundoff error in their evaluation. Even if the system
can be stored exactly in the computer, it is almost inevitabe that roundoff errors will
be introduced during its solution. It can be shown that roundoff errors in Gaussian
elimination have the same effect on the answer as errors in the original coefficients.
2.9. Norms and Condition Numbers 15
Consequently, we are led to a fundamental question. If perturbations are made
in the coefficients of a system of linear equations, how much is the solution altered?
In other words, if Ax = b, how can we measure the sensitivity of x to changes in A
and b?
The answer to this question lies in making the idea of nearly singular precise.
If A is a singular matrix, then for some b’s a solution x will not exist, while for
others it will not be unique. So if A is nearly singular, we can expect small changes
in A and b to cause very large changes in x. On the other hand, if A is the identity
matrix, then b and x are the same vector. So if A is nearly the identity, small
changes in A and b should result in correspondingly small changes in x.
At first glance, it might appear that there is some connection between the size
of the pivots encountered in Gaussian elimination with partial pivoting and nearness
to singularity, because if the arithmetic could be done exactly, all the pivots would
be nonzero if and only if the matrix is nonsingular. To some extent, it is also true
that if the pivots are small, then the matrix is close to singular. However, when
roundoff errors are encountered, the converse is no longer true—a matrix might be
close to singular even though none of the pivots are small.
To get a more precise, and reliable, measure of nearness to singularity than
the size of the pivots, we need to introduce the concept of a norm of a vector. This
is a single number that measures the general size of the elements of the vector.
The family of vector norms known as lp depends on a parameter p in the range
1 ≤ p ≤ ∞:
‖x‖p =
(
n∑
i=1
|xi|p
)1/p
.
We almost always use p = 1, p = 2, or lim p →∞:
‖x‖1 =
n∑
i=1
|xi|,
‖x‖2 =
(
n∑
i=1
|xi|2
)1/2
,
‖x‖∞ = max
i
|xi|.
The l1-norm is also known as the Manhattan norm because it corresponds to the
distance traveled on a grid of city streets. The l2-norm is the familiar Euclidean
distance. The l∞-norm is also known as the Chebyshev norm.
The particular value of p is often unimportant and we simply use ‖x‖. All
vector norms have the following basic properties associated with the notion of dis-
tance:
‖x‖ > 0 if x 6= 0,
‖0‖ = 0,
‖cx‖ = |c|‖x‖ for all scalars c,
‖x + y‖ ≤ ‖x‖+ ‖y‖ (the triangle inequality).
16 Chapter 2. Linear Equations
In Matlab, ‖x‖p is computed by norm(x,p), and norm(x) is the same as
norm(x,2). For example,
x = (1:4)/5
norm1 = norm(x,1)
norm2 = norm(x)
norminf = norm(x,inf)
produces
x =
0.2000 0.4000 0.6000 0.8000
norm1 =
2.0000
norm2 =
1.0954
norminf =
0.8000
Multiplication of a vector x by a matrix A results in a new vector Ax that can
have a very different norm from x. This change in norm is directly related to the
sensitivity we want to measure. The range of the possible change can be expressed
by two numbers:
M = max
‖Ax‖
‖x‖ ,
m = min
‖Ax‖
‖x‖ .
The max and min are taken over all nonzero vectors x. Note that if A is singular,
then m = 0. The ratio M/m is called the condition number of A:
κ(A) =
max ‖Ax‖‖x‖
min ‖Ax‖‖x‖
.
The actual numerical value of κ(A) depends on the vector norm being used,
but we are usually only interested in order of magnitude estimates of the condition
number, so the particular norm is usually not very important.
Consider a system of equations
Ax = b
and a second system obtained by altering the right-hand side:
A(x + δx) = b + δb.
2.9. Norms and Condition Numbers 17
We think of δb as being the error in b and δx as being the resulting error in x,
although we need not make any assumptions that the errors are small. Because
A(δx) = δb, the definitions of M and m immediately lead to
‖b‖ ≤ M‖x‖
and
‖δb‖ ≥ m‖δx‖.
Consequently, if m 6= 0,
‖δx‖
‖x‖ ≤ κ(A)
‖δb‖
‖b‖ .
The quantity ‖δb‖/‖b‖ is the relative change in the right-hand side, and the quantity
‖δx‖/‖x‖ is the relative error caused by this change. The advantage of using relative
changes is that they are dimensionless, that is, they are not affected by overall scale
factors.
This shows that the condition number is a relative error magnification factor.
Changes in the right-hand side can cause changes κ(A) times as large in the solution.
It turns out that the same is true of changes in the coefficient matrix itself.
The condition number is also a measure of nearness to singularity. Although
we have not yet developed the mathematical tools necessary to make the idea pre-
cise, the condition number can be thought of as the reciprocal of the relative distance
from the matrix to the set of singular matrices. So, if κ(A) is large, A is close to
singular.
Some of the basic properties of the condition number are easily derived.
Clearly, M ≥ m, and so
κ(A) ≥ 1.
If P is a permutation matrix, then the components of Px are simply a rearrangement
of the components of x. It follows that ‖Px‖ = ‖x‖ for all x, and so
κ(P ) = 1.
In particular, κ(I) = 1. If A is multiplied by a scalar c, then M and m are both
multiplied by the same scalar, and so
κ(cA) = κ(A).
If D is a diagonal matrix, then
κ(D) =
max |dii|
min |dii|
.
These last two properties are two of the reasons that κ(A) is a better measure of
nearness to singularity than the determinant of A. As an extreme example, consider
a 100-by-100 diagonal matrix with 0.1 on the diagonal. Then det(A) = 10−100,
which is usually regarded as a small number. But κ(A) = 1, and the components of
Ax are simply 0.1 times the corresponding components of x. For linear systems of
equations, such a matrix behaves more like the identity than like a singular matrix.
18 Chapter 2. Linear Equations
The following example uses the l1-norm:
A =
(
4.1 2.8
9.7 6.6
)
,
b =
(
4.1
9.7
)
,
x =
(
1
0
)
.
Clearly, Ax = b, and
‖b‖ = 13.8, ‖x‖ = 1.
If the right-hand side is changed to
b̃ =
(
4.11
9.70
)
,
the solution becomes
x̃ =
(
0.34
0.97
)
.
Let δb = b− b̃ and δx = x− x̃. Then
‖δb‖ = 0.01,
‖δx‖ = 1.63.
We have made a fairly small perturbation in b that completely changes x. In fact,
the relative changes are
‖δb‖
‖b‖ = 0.0007246,
‖δx‖
‖x‖ = 1.63.
Because κ(A) is the maximum magnification factor,
κ(A) ≥ 1.63
0.0007246
= 2249.4.
We have actually chosen the b and δb that give the maximum, and so, for this
example with the l1-norm,
κ(A) = 2249.4.
It is important to realize that this example is concerned with the exact so-
lutions to two slightly different systems of equations and that the method used to
obtain the solutions is irrelevant. The example is constructed to have a fairly large
condition number so that the effect of changes in b is quite pronounced, but similar
behavior can be expected in any problem with a large condition number.
The condition number also plays a fundamental role in the analysis of the
roundoff errors introduced during the solution by Gaussian elimination. Let us
2.9. Norms and Condition Numbers 19
assume that A and b have elements that are exact floating-point numbers, and let
x∗ be the vector of floating-point numbers obtained from a linear equation solver
such as the function we shall present in the next section. We also assume that exact
singularity is not detected and that there are no underflows or overflows. Then it
is possible to establish the following inequalities:
‖b−Ax∗‖
‖A‖‖x∗‖
≤ ρ²,
‖x− x∗‖
‖x∗‖
≤ ρκ(A)².
Here ² is the relative machine precision eps and ρ is defined more carefully later,
but it usually has a value no larger than about 10.
The first inequality says that the relative residual can usually be expected to
be about the size of roundoff error, no matter how badly conditioned the matrix is.
This was illustrated by the example in the previous section. The second inequality
requires that A be nonsingular and involves the exact solution x. It follows directly
from the first inequality and the definition of κ(A) and says that the relative error
will also be small if κ(A) is small but might be quite large if the matrix is nearly
singular. In the extreme case where A is singular but the singularity is not detected,
the first inequality still holds but the second has no meaning.
To be more precise about the quantity ρ, it is necessary to introduce the idea
of a matrix norm and establish some further inequalities. Readers who are not
interested in such details can skip the remainder of this section. The quantity M
defined earlier is known as the norm of the matrix. The notation for the matrix
norm is the same as for the vector norm:
‖A‖ = max ‖Ax‖‖x‖ .
It is not hard to see that ‖A−1‖ = 1/m, so an equivalent definition of the condition
number is
κ(A) = ‖A‖‖A−1‖.
Again, the actual numerical values of the matrix norm and condition number
depend on the underlying vector norm. It is easy to compute the matrix norms
corresponding to the l1 and l∞ vector norms. In fact, it is not hard to show that
‖A‖1 = max
j
∑
i
|ai,j |,
‖A‖∞ = max
i
∑
j
|ai,j |.
Computing the matrix norm corresponding to the l2 vector norm involves the sin-
gular value decomposition (SVD), which is discussed in a later chapter. Matlab
computes matrix norms with norm(A,p) for p = 1, 2, or inf.
The basic result in the study of roundoff error in Gaussian elimination is due
to J. H. Wilkinson. He proved that the computed solution x∗ exactly satisfies
(A + E)x∗ = b,
20 Chapter 2. Linear Equations
where E is a matrix whose elements are about the size of roundoff errors in the
elements of A. There are some rare situations where the intermediate matrices
obtained during Gaussian elimination have elements that are larger than those of
A, and there is some effect from accumulation of rounding errors in large matrices,
but it can be expected that if ρ is defined by
‖E‖
‖A‖ = ρ²,
then ρ will rarely be bigger than about 10.
From this basic result, we can immediately derive inequalities involving the
residual and the error in the computed solution. The residual is given by
b−Ax∗ = Ex∗,
and hence
‖b−Ax∗‖ = ‖Ex∗‖ ≤ ‖E‖‖x∗‖.
The residual involves the product Ax∗, so it is appropriate to consider the relative
residual, which compares the norm of b− Ax to the norms of A and x∗. It follows
directly from the above inequalities that
‖b−Ax∗‖
‖A‖‖x∗‖
≤ ρ².
If A is nonsingular, the error can be expressed using the inverse of A by
x− x∗ = A−1(b−Ax∗),
and so
‖x− x∗‖ ≤ ‖A−1‖‖E‖‖x∗‖.
It is simplest to compare the norm of the error with the norm of the computed
solution. Thus the relative error satisfies
‖x− x∗‖
‖x∗‖
≤ ρ‖A‖‖A−1‖².
Hence
‖x− x∗‖
‖x∗‖
≤ ρκ(A)².
The actual computation of κ(A) requires knowing ‖A−1‖. But computing A−1
requires roughly three times as much work as solving a single linear system. Com-
puting the l2 condition number requires the SVD and even more work. Fortunately,
the exact value of κ(A) is rarely required. Any reasonably good estimate of it is
satisfactory.
Matlab has several functions for computing or estimating condition numbers.
• cond(A) or cond(A,2) computes κ2(A). Uses svd(A). Suitable for smaller
matrices where the geometric properties of the l2-norm are important.
2.10. Sparse Matrices and Band Matrices 21
• cond(A,1) computes κ1(A). Uses inv(A). Less work than cond(A,2).
• cond(A,inf) computes κ∞(A). Uses inv(A). Same as cond(A’,1).
• condest(A) estimates κ1(A). Uses lu(A) and a recent algorithm of Higham
and Tisseur [9]. Especially suitable for large, sparse matrices.
• rcond(A) estimates 1/κ1(A). Uses lu(A) and an older algorithm developed
by the LINPACK and LAPACK projects. Primarily of historical interest.
2.10 Sparse Matrices and Band Matrices
Sparse matrices and band matrices occur frequently in technical computing. The
sparsity of a matrix is the fraction of its elements that are zero. The Matlab
function nnz counts the number of nonzeros in a matrix, so the sparsity of A is
given by
density = nnz(A)/prod(size(A))
sparsity = 1 – density
A sparse matrix is a matrix whose sparsity is nearly equal to 1.
The bandwidth of a matrix is the maximum distance of the nonzero elements
from the main diagonal.
[i,j] = find(A)
bandwidth = max(abs(i-j))
A band matrix is a matrix whose bandwidth is small.
As you can see, both sparsity and bandwidth are matters of degree. An n-by-n
diagonal matrix with no zeros on the diagonal has sparsity 1− 1/n and bandwidth
0, so it is an extreme example of both a sparse matrix and a band matrix. On the
other hand, an n-by-n matrix with no zero elements, such as the one created by
rand(n,n), has sparsity equal to zero and bandwidth equal to n− 1, and so is far
from qualifying for either category.
The Matlab sparse data structure stores the nonzero elements together with
information about their indices. The sparse data structure also provides efficient
handling of band matrices, so Matlab does not have a separate band matrix storage
class. The statement
S = sparse(A)
converts a matrix to its sparse representation. The statement
A = full(S)
reverses the process. However, most sparse matrices have orders so large that it is
impractical to store the full representation. More frequently, sparse matrices are
created by
S = sparse(i,j,x,m,n)
22 Chapter 2. Linear Equations
This produces a matrix S with
[i,j,x] = find(S)
[m,n] = size(S)
Most Matlab matrix operations and functions can be applied to both full and
sparse matrices. The dominant factor in determining the execution time and mem-
ory requirements for sparse matrix operations is the number of nonzeros, nnz(S),
in the various matrices involved.
A matrix with bandwidth equal to 1 is known as a tridiagonal matrix. It is
worthwhile to have a specialized function for one particular band matrix operation,
the solution of a tridiagonal system of simultaneous linear equations:
b1 c1
a1 b2 c2
a2 b3 c3
. . .
. . .
. . .
an−2 bn−1 cn−1
an−1 bn
x1
x2
x3
…
xn−1
xn
=
d1
d2
d3
…
dn−1
dn
.
The function tridisolve is included in the NCM directory. The statement
x = tridisolve(a,b,c,d)
solves the tridiagonal system with subdiagonal a, diagonal b, superdiagonal c, and
right-hand side d. We have already seen the algorithm that tridisolve uses; it
is Gaussian elimination. In many situations involving tridiagonal matrices, the
diagonal elements dominate the off-diagonal elements, so pivoting is unnecessary.
Furthermore, the right-hand side is processed at the same time as the matrix itself.
In this context, Gaussian elimination without pivoting is also known as the Thomas
algorithm.
The body of tridisolve begins by copying the right-hand side to a vector
that will become the solution.
x = d;
n = length(x);
The forward elimination step is a simple for loop.
for j = 1:n-1
mu = a(j)/b(j);
b(j+1) = b(j+1) – mu*c(j);
x(j+1) = x(j+1) – mu*x(j);
end
The mu’s would be the multipliers on the subdiagonal of L if we were saving the LU
factorization. Instead, the right-hand side is processed in the same loop. The back
substitution step is another simple loop.
2.11. PageRank and Markov Chains 23
x(n) = x(n)/b(n);
for j = n-1:-1:1
x(j) = (x(j)-c(j)*x(j+1))/b(j);
end
Because tridisolve does not use pivoting, the results might be inaccurate if abs(b)
is much smaller than abs(a)+abs(c). More robust, but slower, alternatives that
do use pivoting include generating a full matrix with diag:
T = diag(a,-1) + diag(b,0) + diag(c,1);
x = Td
or generating a sparse matrix with spdiags:
S = spdiags([a b c],[-1 0 1],n,n);
x = Sd
2.11 PageRank and Markov Chains
One of the reasons why GoogleTM is such an effective search engine is the PageRankTM
algorithm developed by Google’s founders, Larry Page and Sergey Brin, when they
were graduate students at Stanford University. PageRank is determined entirely by
the link structure of the World Wide Web. It is recomputed about once a month
and does not involve the actual content of any Web pages or individual queries.
Then, for any particular query, Google finds the pages on the Web that match that
query and lists those pages in the order of their PageRank.
Imagine surfing the Web, going from page to page by randomly choosing an
outgoing link from one page to get to the next. This can lead to dead ends at
pages with no outgoing links, or cycles around cliques of interconnected pages.
So, a certain fraction of the time, simply choose a random page from the Web.
This theoretical random walk is known as a Markov chain or Markov process. The
limiting probability that an infinitely dedicated random surfer visits any particular
page is its PageRank. A page has high rank if other pages with high rank link to
it.
Let W be the set of Web pages that can be reached by following a chain of
hyperlinks starting at some root page, and let n be the number of pages in W . For
Google, the set W actually varies with time, In June 2004, their value of n was over
4 billion. Today, Google does not reveal how many pages they reach. Let G be the
n-by-n connectivity matrix of a portion of the Web, that is, gij = 1 if there is a
hyperlink to page i from page j and gij = 0 otherwise. The matrix G can be huge,
but it is very sparse. Its jth column shows the links on the jth page. The number
of nonzeros in G is the total number of hyperlinks in W .
Let ri and cj be the row and column sums of G:
ri =
∑
j
gij , cj =
∑
i
gij .
The quantities rj and cj are the in-degree and out-degree of the jth page. Let p be
the probability that the random walk follows a link. A typical value is p = 0.85.
24 Chapter 2. Linear Equations
Then 1− p is the probability that some arbitrary page is chosen and δ = (1− p)/n
is the probability that a particular random page is chosen. Let A be the n-by-n
matrix whose elements are
aij =
{
pgij/cj + δ : cj 6= 0
1/n : cj = 0.
Notice that A comes from scaling the connectivity matrix by its column sums. The
jth column is the probability of jumping from the jth page to the other pages on
the Web. If the jth page is a dead end, that is has no out-links, then we assign a
uniform probability of 1/n to all the elements in its column. Most of the elements
of A are equal to δ, the probability of jumping from one page to another without
following a link. If n = 4 · 109 and p = 0.85, then δ = 3.75 · 10−11.
The matrix A is the transition probability matrix of the Markov chain. Its
elements are all strictly between zero and one and its column sums are all equal to
one. An important result in matrix theory known as the Perron–Frobenius theorem
applies to such matrices. It concludes that a nonzero solution of the equation
x = Ax
exists and is unique to within a scaling factor. If this scaling factor is chosen so
that ∑
i
xi = 1,
then x is the state vector of the Markov chain and is Google’s PageRank. The
elements of x are all positive and less than one.
The vector x is the solution to the singular, homogeneous linear system
(I −A)x = 0.
For modest n, an easy way to compute x in Matlab is to start with some approx-
imate solution, such as the PageRanks from the previous month, or
x = ones(n,1)/n
Then simply repeat the assignment statement
x = A*x
until successive vectors agree to within a specified tolerance. This is known as the
power method and is about the only possible approach for very large n.
In practice, the matrices G and A are never actually formed. One step of the
power method would be done by one pass over a database of Web pages, updating
weighted reference counts generated by the hyperlinks between pages.
The best way to compute PageRank in Matlab is to take advantage of the
particular structure of the Markov matrix. Here is an approach that preserves the
sparsity of G. The transition matrix can be written
A = pGD + ezT
2.11. PageRank and Markov Chains 25
where D is the diagonal matrix formed from the reciprocals of the outdegrees,
djj =
{
1/cj : cj 6= 0
0 : cj = 0,
e is the n-vector of all ones, and z is the vector with components
zj =
{
δ : cj 6= 0
1/n : cj = 0.
The rank-one matrix ezT accounts for the random choices of Web pages that do
not follow links. The equation
x = Ax
can be written
(I − pGD)x = γe
where
γ = zT x.
We do not know the value of γ because it depends upon the unknown vector x, but
we can temporarily take γ = 1. As long as p is strictly less than one, the coefficient
matrix I − pGD is nonsingular and the equation
(I − pGD)x = e
can be solved for x. Then the resulting x can be rescaled so that
∑
i
xi = 1.
Notice that the vector z is not actually involved in this calculation.
The following Matlab statements implement this approach
c = sum(G,1);
k = find(c~=0);
D = sparse(k,k,1./c(k),n,n);
e = ones(n,1);
I = speye(n,n);
x = (I – p*G*D)e;
x = x/sum(x);
The power method can also be implemented in a way that does not actually
form the Markov matrix and so preserves sparsity. Compute
G = p*G*D;
z = ((1-p)*(c~=0) + (c==0))/n;
Start with
x = e/n
26 Chapter 2. Linear Equations
Then repeat the statement
x = G*x + e*(z*x)
until x settles down to several decimal places.
It is also possible to use an algorithm known as inverse iteration.
A = p*G*D + delta
x = (I – A)e
x = x/sum(x)
At first glance, this appears to be a very dangerous idea. Because I − A is the-
oretically singular, with exact computation some diagonal element of the upper
triangular factor of I − A should be zero and this computation should fail. But
with roundoff error, the computed matrix I – A is probably not exactly singular.
Even if it is singular, roundoff during Gaussian elimination will most likely pre-
vent any exact zero diagonal elements. We know that Gaussian elimination with
partial pivoting always produces a solution with a small residual, relative to the
computed solution, even if the matrix is badly conditioned. The vector obtained
with the backslash operation, (I – A)e, usually has very large components. If it
is rescaled by its sum, the residual is scaled by the same factor and becomes very
small. Consequently, the two vectors x and A*x equal each other to within roundoff
error. In this setting, solving the singular system with Gaussian elimination blows
up, but it blows up in exactly the right direction.
alpha
beta
gamma
delta
sigma rho
Figure 2.2. A tiny Web.
Figure 2.2 is the graph for a tiny example, with n = 6 instead of n = 4 · 109.
Pages on the Web are identified by strings known as uniform resource locators,
or URLs. Most URLs begin with http because they use the hypertext transfer
protocol. In Matlab , we can store the URLs as an array of strings in a cell array.
This example involves a 6-by-1 cell array.
2.11. PageRank and Markov Chains 27
U = {’http://www.alpha.com’
’http://www.beta.com’
’http://www.gamma.com’
’http://www.delta.com’
’http://www.rho.com’
’http://www.sigma.com’}
Two different kinds of indexing into cell arrays are possible. Parentheses denote
subarrays, including individual cells, and curly braces denote the contents of the
cells. If k is a scalar, then U(k) is a 1-by-1 cell array consisting of the kth cell in U,
while U{k} is the string in that cell. Thus U(1) is a single cell and U{1} is the string
’http://www.alpha.com’. Think of mail boxes with addresses on a city street.
B(502) is the box at number 502, while B{502} is the mail in that box.
We can generate the connectivity matrix by specifying the pairs of indices
(i,j) of the nonzero elements. Because there is a link to beta.com from alpha.com,
the (2,1) element of G is nonzero. The nine connections are described by
i = [ 2 6 3 4 4 5 6 1 1]
j = [ 1 1 2 2 3 3 3 4 6]
A sparse matrix is stored in a data structure that requires memory only for the
nonzero elements and their indices. This is hardly necessary for a 6-by-6 matrix
with only 27 zero entries, but it becomes crucially important for larger problems.
The statements
n = 6
G = sparse(i,j,1,n,n);
full(G)
generate the sparse representation of an n-by-n matrix with ones in the positions
specified by the vectors i and j and display its full representation.
0 0 0 1 0 1
1 0 0 0 0 0
0 1 0 0 0 0
0 1 1 0 0 0
0 0 1 0 0 0
1 0 1 0 0 0
The statement
c = full(sum(G))
computes the column sums
c =
2 2 3 1 0 1
Notice that c(5) = 0 because the 5th page, labeled rho, has no out-links.
The statements
28 Chapter 2. Linear Equations
x = (I – p*G*D)e
x = x/sum(x)
solve the sparse linear system to produce
x =
0.3210
0.1705
0.1066
0.1368
0.0643
0.2007
1 2 3 4 5 6
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Page Rank
Figure 2.3. Page Rank for the tiny Web
The bar graph of x is shown in figure 2.3. If the URLs are sorted in PageRank
order and listed along with their in- and out-degrees, the result is
page-rank in out url
1 0.3210 2 2 http://www.alpha.com
6 0.2007 2 1 http://www.sigma.com
2 0.1705 1 2 http://www.beta.com
4 0.1368 2 1 http://www.delta.com
3 0.1066 1 3 http://www.gamma.com
5 0.0643 1 0 http://www.rho.com
We see that alpha has a higher PageRank than delta or sigma, even though they
all have the same number of in-links. A random surfer will visit alpha over 32% of
the time and rho only about 6% of the time.
For this tiny example with p = .85, the smallest element of the Markov tran-
sition matrix is δ = .15/6 = .0250.
2.11. PageRank and Markov Chains 29
A =
0.0250 0.0250 0.0250 0.8750 0.1667 0.8750
0.4500 0.0250 0.0250 0.0250 0.1667 0.0250
0.0250 0.4500 0.0250 0.0250 0.1667 0.0250
0.0250 0.4500 0.3083 0.0250 0.1667 0.0250
0.0250 0.0250 0.3083 0.0250 0.1667 0.0250
0.4500 0.0250 0.3083 0.0250 0.1667 0.0250
Notice that the column sums of A are all equal to one.
Our collection of NCM programs includes surfer.m. A statement like
[U,G] = surfer(’http://www.xxx.zzz’,n)
starts at a specified URL and tries to surf the Web until it has visited n pages. If
successful, it returns an n-by-1 cell array of URLs and an n-by-n sparse connectivity
matrix. The function uses urlread, which was introduced in Matlab 6.5, along
with underlying Java utilities to access the Web. Surfing the Web automatically
is a dangerous undertaking and this function must be used with care. Some URLs
contain typographical errors and illegal characters. There is a list of URLs to
avoid that includes .gif files and Web sites known to cause difficulties. Most
importantly, surfer can get completely bogged down trying to read a page from
a site that appears to be responding, but that never delivers the complete page.
When this happens, it may be necessary to have the computer’s operating system
ruthlessly terminate Matlab. With these precautions in mind, you can use surfer
to generate your own PageRank examples.
0 100 200 300 400 500
0
50
100
150
200
250
300
350
400
450
500
nz = 2636
Figure 2.4. Spy plot of the harvard500 graph.
30 Chapter 2. Linear Equations
The statement
[U,G] = surfer(’http://www.harvard.edu’,500)
accesses the home page of Harvard University and generates a 500-by-500 test case.
The graph generated in August 2003 is available in the NCM directory. The state-
ments
load harvard500
spy(G)
produce a spy plot (Figure 2.4) that shows the nonzero structure of the connectivity
matrix. The statement
pagerank(U,G)
computes page ranks, produces a bar graph (Figure 2.5) of the ranks, and prints
the most highly ranked URLs in PageRank order.
For the harvard500 data, the dozen most highly ranked pages are
page-rank in out url
1 0.0843 195 26 http://www.harvard.edu
10 0.0167 21 18 http://www.hbs.edu
42 0.0166 42 0 http://search.harvard.edu:8765/
custom/query.html
130 0.0163 24 12 http://www.med.harvard.edu
18 0.0139 45 46 http://www.gse.harvard.edu
15 0.0131 16 49 http://www.hms.harvard.edu
9 0.0114 21 27 http://www.ksg.harvard.edu
17 0.0111 13 6 http://www.hsph.harvard.edu
46 0.0100 18 21 http://www.gocrimson.com
13 0.0086 9 1 http://www.hsdm.med.harvard.edu
260 0.0086 26 1 http://search.harvard.edu:8765/
query.html
19 0.0084 23 21 http://www.radcliffe.edu
The URL where the search began, www.harvard.edu, dominates. Like most uni-
versities, Harvard is organized into various colleges and institutes, including the
Kennedy School of Government, the Harvard Medical School, the Harvard Busi-
ness School, and the Radcliffe Institute. You can see that the home pages of these
schools have high PageRank. With a different sample, such as the one generated
by Google itself, the ranks would be different.
2.12 Further Reading
Further reading on matrix computation includes books by Demmel [2], Golub and
Van Loan [3], Stewart [4, 5], and Trefethen and Bau [6]. The definitive references
on Fortran matrix computation software are the LAPACK Users’ Guide and Web
site [1]. The Matlab sparse matrix data structure and operations are described
Exercises 31
0 50 100 150 200 250 300 350 400 450 500
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Page Rank
Figure 2.5. PageRank of the harvard500 graph.
in [8]. Information available on Web sites about PageRank includes a brief expla-
nation at Google [7], a technical report by Page, Brin, and colleagues [11], and a
comprehensive survey by Langville and Meyer [10].
Exercises
2.1. Alice buys three apples, a dozen bananas, and one cantaloupe for $2.36. Bob
buys a dozen apples and two cantaloupes for $5.26. Carol buys two bananas
and three cantaloupes for $2.77. How much do single pieces of each fruit
cost? (You might want to set format bank.)
2.2. What Matlab function computes the reduced row echelon form of a ma-
trix? What Matlab function generates magic square matrices? What is the
reduced row echelon form of the magic square of order six?
2.3. Figure 2.6 depicts a plane truss having 13 members (the numbered lines)
connecting 8 joints (the numbered circles). The indicated loads, in tons, are
applied at joints 2, 5, and 6, and we want to determine the resulting force on
each member of the truss.
For the truss to be in static equilibrium, there must be no net force, hor-
izontally or vertically, at any joint. Thus, we can determine the member
forces by equating the horizontal forces to the left and right at each joint,
and similarly equating the vertical forces upward and downward at each joint.
For the eight joints, this would give 16 equations, which is more than the 13
unknown factors to be determined. For the truss to be statically determi-
nate, that is, for there to be a unique solution, we assume that joint 1 is
rigidly fixed both horizontally and vertically and that joint 8 is fixed verti-
32 Chapter 2. Linear Equations
1 2 5 6 8
3 4 7
1 3 5 7 9 11 12
2 6 10 13
4 8
10 15 20
Figure 2.6. A plane truss.
cally. Resolving the member forces into horizontal and vertical components
and defining α = 1/
√
2, we obtain the following system of equations for the
member forces fi:
Joint 2: f2 = f6,
f3 = 10;
Joint 3: αf1 = f4 + αf5,
αf1 + f3 + αf5 = 0;
Joint 4: f4 = f8,
f7 = 0;
Joint 5: αf5 + f6 = αf9 + f10,
αf5 + f7 + αf9 = 15;
Joint 6: f10 = f13,
f11 = 20;
Joint 7: f8 + αf9 = αf12,
αf9 + f11 + αf12 = 0;
Joint 8: f13 + αf12 = 0.
Solve this system of equations to find the vector f of member forces.
2.4. Figure 2.7 is the circuit diagram for a small network of resistors.
There are five nodes, eight resistors, and one constant voltage source. We
want to compute the voltage drops between the nodes and the currents around
each of the loops.
Several different linear systems of equations can be formed to describe this
circuit. Let vk, k = 1, . . . , 4, denote the voltage difference between each of
the first four nodes and node number 5 and let ik, k = 1, . . . , 4, denote the
clockwise current around each of the loops in the diagram. Ohm’s law says
that the voltage drop across a resistor is the resistance times the current. For
Exercises 33
1
2
3
4
5
r
23
r
34
r
45
r
25
r
13
r
12
r
14
r
35
v
s
i
1
i
2
i
3
i
4
Figure 2.7. A resistor network.
example, the branch between nodes 1 and 2 gives
v1 − v2 = r12(i2 − i1).
Using the conductance, which is the reciprocal of the resistance, gkj = 1/rkj ,
Ohm’s law becomes
i2 − i1 = g12(v1 − v2).
The voltage source is included in the equation
v3 − vs = r35i4.
Kirchhoff’s voltage law says that the sum of the voltage differences around
each loop is zero. For example, around loop 1,
(v1 − v4) + (v4 − v5) + (v5 − v2) + (v2 − v1) = 0.
Combining the voltage law with Ohm’s law leads to the loop equations for
the currents:
Ri = b.
Here i is the current vector,
i =
i1
i2
i3
i4
,
b is the source voltage vector,
b =
0
0
0
vs
,
34 Chapter 2. Linear Equations
and R is the resistance matrix,
r25 + r12 + r14 + r45 −r12 −r14 −r45
−r12 r23 + r12 + r13 −r13 0
−r14 −r13 r14 + r13 + r34 −r34
−r45 0 −r34 r35 + r45 + r34
.
Kirchhoff’s current law says that the sum of the currents at each node is zero.
For example, at node 1,
(i1 − i2) + (i2 − i3) + (i3 − i1) = 0.
Combining the current law with the conductance version of Ohm’s law leads
to the nodal equations for the voltages:
Gv = c.
Here v is the voltage vector,
v =
v1
v2
v3
v4
,
c is the source current vector,
c =
0
0
g35vs
0
,
and G is the conductance matrix,
g12 + g13 + g14 −g12 −g13 −g14
−g12 g12 + g23 + g25 −g23 0
−g13 −g23 g13 + g23 + g34 + g35 −g34
−g14 0 −g34 g14 + g34 + g45
.
You can solve the linear system obtained from the loop equations to compute
the currents and then use Ohm’s law to recover the voltages. Or you can solve
the linear system obtained from the node equations to compute the voltages
and then use Ohm’s law to recover the currents. Your assignment is to verify
that these two approaches produce the same results for this circuit. You can
choose your own numerical values for the resistances and the voltage source.
2.5. The Cholesky algorithm factors an important class of matrices known as
positive definite matrices. Andre-Louis Cholesky (1875–1918) was a French
military officer involved in geodesy and surveying in Crete and North Africa
just before World War I. He developed the method now named after him to
compute solutions to the normal equations for some least squares data-fitting
Exercises 35
problems arising in geodesy. His work was posthumously published on his
behalf in 1924 by a fellow officer, Benoit, in the Bulletin Geodesique.
A real symmetric matrix A = AT is positive definite if any of the following
equivalent conditions hold:
• The quadratic form
xT Ax
is positive for all nonzero vectors x.
• All determinants formed from symmetric submatrices of any order cen-
tered on the diagonal of A are positive.
• All eigenvalues λ(A) are positive.
• There is a real matrix R such that
A = RT R.
These conditions are difficult or expensive to use as the basis for checking if
a particular matrix is positive definite. In Matlab , the best way to check
positive definiteness is with the chol function. See
help chol
(a) Which of the following families of matrices are positive definite?
M = magic(n)
H = hilb(n)
P = pascal(n)
I = eye(n,n)
R = randn(n,n)
R = randn(n,n); A = R’ * R
R = randn(n,n); A = R’ + R
R = randn(n,n); I = eye(n,n); A = R’ + R + n*I
(b) If the matrix R is upper triangular, then equating individual elements in
the equation A = RT R gives
akj =
k∑
i=1
rikrij , k ≤ j.
Using these equations in different orders yields different variants of the Cholesky
algorithm for computing the elements of R. What is one such algorithm?
2.6. This example shows that a badly conditioned matrix does not necessarily
lead to small pivots in Gaussian elimination. The matrix is the n-by-n upper
triangular matrix A with elements
aij =
−1, i < j,
1, i = j,
0, i > j.
36 Chapter 2. Linear Equations
Show how to generate this matrix in Matlab with eye, ones, and triu.
Show that
κ1(A) = n2
n−1.
For what n does κ1(A) exceed 1/eps?
This matrix is not singular, so Ax cannot be zero unless x is zero. However,
there are vectors x for which ‖Ax‖ is much smaller than ‖x‖. Find one such
x.
Because this matrix is already upper triangular, Gaussian elimination with
partial pivoting has no work to do. What are the pivots?
Use lugui to design a pivot strategy that will produce smaller pivots than
partial pivoting. (Even these pivots do not completely reveal the large con-
dition number.)
2.7. The matrix factorization
LU = PA
can be used to compute the determinant of A. We have
det(L)det(U) = det(P )det(A).
Because L is triangular with ones on the diagonal, det(L) = 1. Because U is
triangular, det(U) = u11u22 · · ·unn. Because P is a permutation, det(P ) =
+1 if the number of interchanges is even and −1 if it is odd. So
det(A) = ±u11u22 · · ·unn.
Modify the lutx function so that it returns four outputs.
function [L,U,p,sig] = lutx(A)
%LU Triangular factorization
% [L,U,p,sig] = lutx(A) computes a unit lower triangular
% matrix L, an upper triangular matrix U, a permutation
% vector p, and a scalar sig, so that L*U = A(p,:) and
% sig = +1 or -1 if p is an even or odd permutation.
Write a function mydet(A) that uses your modified lutx to compute the
determinant of A. In Matlab, the product u11u22 · · ·unn can be computed
by the expression prod(diag(U)).
2.8. Modify the lutx function so that it uses explicit for loops instead of Matlab
vector notation. For example, one section of your modified program will read
% Compute the multipliers
for i = k+1:n
A(i,k) = A(i,k)/A(k,k);
end
Compare the execution time of your modified lutx program with the origi-
nal lutx program and with the built-in lu function by finding the order of
the matrix for which each of the three programs takes about 10 s on your
computer.
Exercises 37
2.9. Let
A =
1 2 3
4 5 6
7 8 9
, b =
1
3
5
.
(a) Show that the set of linear equations Ax = b has infinitely many solutions.
Describe the set of possible solutions.
(b) Suppose Gaussian elimination is used to solve Ax = b using exact arith-
metic. Because there are infinitely many solutions, it is unreasonable to
expect one particular solution to be computed. What does happen?
(c) Use bslashtx to solve Ax = b on an actual computer with floating-point
arithmetic. What solution is obtained? Why? In what sense is it a “good”
solution? In what sense is it a “bad” solution?
(d) Explain why the built-in backslash operator x = A gives a different
solution from x = bslashtx(A,b).
2.10. Section 2.4 gives two algorithms for solving triangular systems. One subtracts
columns of the triangular matrix from the right-hand side; the other uses
inner products between the rows of the triangular matrix and the emerging
solution.
(a) Which of these two algorithms does bslashtx use?
(b) Write another function, bslashtx2, that uses the other algorithm.
2.11. The inverse of a matrix A can be defined as the matrix X whose columns xj
solve the equations
Axj = ej ,
where ej is the jth column of the identity matrix.
(a) Starting with the function bslashtx, write a Matlab function
X = myinv(A)
that computes the inverse of A. Your function should call lutx only once and
should not use the built-in Matlab backslash operator or inv function.
(b) Test your function by comparing the inverses it computes with the inverses
obtained from the built-in inv(A) on a few test matrices.
2.12. If the built-in Matlab lu function is called with only two output arguments
[L,U] = lu(A)
the permutations are incorporated into the output matrix L. The help entry
for lu describes L as “psychologically lower triangular.” Modify lutx so that
it does the same thing. You can use
if nargout == 2, …
to test the number of output arguments.
2.13. (a) Why is
M = magic(8)
lugui(M)
38 Chapter 2. Linear Equations
an interesting example?
(b) How is the behavior of lugui(M) related to rank(M)?
(c) Can you pick a sequence of pivots so that no roundoff error occurs in
lugui(M)?
2.14. The pivot selection strategy known as complete pivoting is one of the options
available in lugui. It has some slight numerical advantages over partial
pivoting. At each stage of the elimination, the element of largest magnitude
in the entire unreduced matrix is selected as the pivot. This involves both
row and column interchanges and produces two permutation vectors p and q
so that
L*U = A(p,q)
Modify lutx and bslashtx so that they use complete pivoting.
2.15. The function golub in the NCM directory is named after Stanford professor
Gene Golub. The function generates test matrices with random integer en-
tries. The matrices are very badly conditioned, but Gaussian elimination
without pivoting fails to produce the small pivots that would reveal the large
condition number.
(a) How does condest(golub(n)) grow with increasing order n? Because
these are random matrices you can’t be very precise here, but you can give
some qualitative description.
(b) What atypical behavior do you observe with the diagonal pivoting option
in lugui(golub(n))?
(c) What is det(golub(n))? Why?
2.16. The function pascal generates symmetric test matrices based on Pascal’s
triangle.
(a) How are the elements of pascal(n+1) related to the binomial coefficients
generated by nchoosek(n,k)?
(b) How is chol(pascal(n)) related to pascal(n)?
(c) How does condest(pascal(n)) grow with increasing order n?
(d) What is det(pascal(n))? Why?
(e) Let Q be the matrix generated by
Q = pascal(n);
Q(n,n) = Q(n,n) – 1;
How is chol(Q) related to chol(pascal(n))? Why?
(f) What is det(Q)? Why?
2.17. Play “Pivot Pickin’ Golf” with pivotgolf. The goal is to use lugui to
compute the LU decompositions of nine matrices with as little roundoff error
as possible. The score for each hole is
‖R‖∞ + ‖L²‖∞ + ‖U²‖∞,
where R = LU−PAQ is the residual and ‖L²‖∞ and ‖U²‖∞ are the nonzeros
that should be zero in L and U .
Exercises 39
(a) Can you beat the scores obtained by partial pivoting on any of the courses?
(b) Can you get a perfect score of zero on any of the courses?
2.18. The object of this exercise is to investigate how the condition numbers of
random matrices grow with their order. Let Rn denote an n-by-n matrix with
normally distributed random elements. You should observe experimentally
that there is an exponent p so that
κ1(Rn) = O(n
p).
In other words, there are constants c1 and c2 so that most values of κ1(Rn)
satisfy
c1n
p ≤ κ1(Rn) ≤ c2np.
Your job is to find p, c1, and c2.
The NCM M-file randncond.m is the starting point for your experiments.
This program generates random matrices with normally distributed elements
and plots their l1 condition numbers versus their order on a loglog scale.
The program also plots two lines that are intended to enclose most of the
observations. (On a loglog scale, power laws like κ = cnp produce straight
lines.)
(a) Modify randncond.m so that the two lines have the same slope and enclose
most of the observations.
(b) Based on this experiment, what is your guess for the exponent p in
κ(Rn) = O(np)? How confident are you?
(c) The program uses (’erasemode’,’none’), so you cannot print the re-
sults. What would you have to change to make printing possible?
2.19. For n = 100, solve this tridiagonal system of equations three different ways:
2×1 − x2 =1,
−xj−1 + 2xj − xj+1 =j, j = 2, . . . , n− 1,
−xn−1 + 2xn =n.
(a) Use diag three times to form the coefficient matrix and then use lutx
and bslashtx to solve the system.
(b) Use spdiags once to form a sparse representation of the coefficient matrix
and then use the backslash operator to solve the system.
(c) Use tridisolve to solve the system.
(d) Use condest to estimate the condition of the coefficient matrix.
2.20. Use surfer and pagerank to compute PageRanks for some subset of the Web
that you choose. Do you see any interesting structure in the results?
2.21. Suppose that U and G are the URL cell array and the connectivity matrix
produced by surfer and that k is an integer. Explain what
U{k}, U(k), G(k,:), G(:,k), U(G(k,:)), U(G(:,k))
are.
40 Chapter 2. Linear Equations
2.22. The connectivity matrix for the harvard500 data set has four small, almost
entirely nonzero, submatrices that produce dense patches near the diagonal
of the spy plot. You can use the zoom button to find their indices. The
first submatrix has indices around 170 and the other three have indices in
the 200s and 300s. Mathematically, a graph with every node connected to
every other node is known as a clique. Identify the organizations within the
Harvard community that are responsible for these near cliques.
2.23. A Web connectivity matrix G has gij = 1 if it is possible to get to page i
from page j with one click. If you multiply the matrix by itself, the entries
of the matrix G2 count the number of different paths of length two to page i
from page j. The matrix power Gp shows the number of paths of length p.
(a) For the harvard500 data set, find the power p where the number of
nonzeros stops increasing. In other words, for any q greater than p, nnz(G^q)
is equal to nnz(G^p).
(b) What fraction of the entries in Gp are nonzero?
(c) Use subplot and spy to show the nonzeros in the successive powers.
(d) Is there a set of interconnected pages that do not link to the other pages?
2.24. The function surfer uses a subfunction, hashfun, to speed up the search for a
possibly new URL in the list of URLs that have already been processed. Find
two different URLs on The MathWorks home page http://www.mathworks.com
that have the same hashfun value.
2.25. Figure 2.8 is the graph of another six-node subset of the Web. In this example,
there are two disjoint subgraphs.
alpha
beta
gamma
delta
sigma rho
Figure 2.8. Another tiny Web.
(a) What is the connectivity matrix G?
(b) What are the PageRanks if the hyperlink transition probability p is the
default value 0.85?
(c) Describe what happens with this example to both the definition of PageR-
ank and the computation done by pagerank in the limit p → 1.
Exercises 41
2.26. The function pagerank(U,G) computes PageRanks by solving a sparse linear
system. It then plots a bar graph and prints the dominant URLs.
(a) Create pagerank1(G) by modifying pagerank so that it just computes
the PageRanks, but does not do any plotting or printing.
(b) Create pagerank2(G) by modifying pagerank1 to use inverse iteration
instead of solving the sparse linear system. The key statements are
x = (I – A)e
x = x/sum(x)
What should be done in the unlikely event that the backslash operation in-
volves a division by zero?
(c) Create pagerank3(G) by modifying pagerank1 to use the power method
instead of solving the sparse linear system. The key statements are
G = p*G*D
z = ((1-p)*(c~=0) + (c==0))/n;
while termination_test
x = G*x + e*(z*x)
end
What is an appropriate test for terminating the power iteration?
(d) Use your functions to compute the PageRanks of the six-node example
discussed in the text. Make sure you get the correct result from each of your
three functions.
2.27. Here is yet another function for computing PageRank. This version uses
the power method, but does not do any matrix operations. Only the link
structure of the connectivity matrix is involved.
function [x,cnt] = pagerankpow(G)
% PAGERANKPOW PageRank by power method.
% x = pagerankpow(G) is the PageRank of the graph G.
% [x,cnt] = pagerankpow(G)
% counts the number of iterations.
% Link structure
[n,n] = size(G);
for j = 1:n
L{j} = find(G(:,j));
c(j) = length(L{j});
end
% Power method
p = .85;
delta = (1-p)/n;
x = ones(n,1)/n;
42 Chapter 2. Linear Equations
z = zeros(n,1);
cnt = 0;
while max(abs(x-z)) > .0001
z = x;
x = zeros(n,1);
for j = 1:n
if c(j) == 0
x = x + z(j)/n;
else
x(L{j}) = x(L{j}) + z(j)/c(j);
end
end
x = p*x + delta;
cnt = cnt+1;
end
(a) How do the storage requirements and execution time of this function
compare with the three pagerank functions from the previous exercise?
(b) Use this function as a template to write a function that computes PageR-
ank in some other programming language.
Bibliography
[1] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J.
Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney,
and D. Sorensen, LAPACK Users’ Guide, Third Edition, SIAM, Philadelphia,
1999.
http://www.netlib.org/lapack
[2] J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
[3] G. H. Golub and C. F. Van Loan, Matrix Computations, Third Edition,
The Johns Hopkins University Press, Baltimore, 1996.
[4] G. W. Stewart, Introduction to Matrix Computations, Academic Press, New
York, 1973.
[5] G. W. Stewart, Matrix Algorithms: Basic Decompositions, SIAM, Philadel-
phia, 1998.
[6] L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM,
Philadelphia, 1997.
[7] Google, Google Technology.
http://www.google.com/technology/index.html
[8] J. R. Gilbert, C. Moler, and R. Schreiber, Sparse matrices in MATLAB:
Design and implementation, SIAM Journal on Matrix Analysis and Applica-
tions, 13 (1992), pp. 333–356.
[9] N. J. Higham, and F. Tisseur, A block algorithm for matrix 1-norm esti-
mation, with an application to 1-norm pseudospectra, SIAM Journal on Matrix
Analysis and Applications, 21 (2000), pp. 1185–1201.
[10] A. Langville, and C. Meyer, Deeper Inside PageRank,
[11] L. Page, S. Brin, R. Motwani, and T. Winograd, The PageRank Cita-
tion Ranking: Bringing Order to the Web.
http://dbpubs.stanford.edu:8090/pub/1999-66
43
Chapter 3
Interpolation
Interpolation is the process of defining a function that takes on specified values at
specified points. This chapter concentrates on two closely related interpolants: the
piecewise cubic spline and the shape-preserving piecewise cubic named “pchip.”
3.1 The Interpolating Polynomial
We all know that two points determine a straight line. More precisely, any two
points in the plane, (x1, y1) and (x2, y2), with x1 6= x2, determine a unique first-
degree polynomial in x whose graph passes through the two points. There are
many different formulas for the polynomial, but they all lead to the same straight
line graph.
This generalizes to more than two points. Given n points in the plane,
(xk, yk), k = 1, . . . , n, with distinct xk’s, there is a unique polynomial in x of degree
less than n whose graph passes through the points. It is easiest to remember that n,
the number of data points, is also the number of coefficients, although some of the
leading coefficients might be zero, so the degree might actually be less than n− 1.
Again, there are many different formulas for the polynomial, but they all define the
same function.
This polynomial is called the interpolating polynomial because it exactly re-
produces the given data:
P (xk) = yk, k = 1, . . . , n.
Later, we examine other polynomials, of lower degree, that only approximate the
data. They are not interpolating polynomials.
The most compact representation of the interpolating polynomial is the La-
grange form
P (x) =
∑
k
∏
j 6=k
x− xj
xk − xj
yk.
February 15, 2008
1
2 Chapter 3. Interpolation
There are n terms in the sum and n − 1 terms in each product, so this expression
defines a polynomial of degree at most n− 1. If P (x) is evaluated at x = xk, all the
products except the kth are zero. Furthermore, the kth product is equal to one, so
the sum is equal to yk and the interpolation conditions are satisfied.
For example, consider the following data set.
x = 0:3;
y = [-5 -6 -1 16];
The command
disp([x; y])
displays
0 1 2 3
-5 -6 -1 16
The Lagrangian form of the polynomial interpolating these data is
P (x) =
(x− 1)(x− 2)(x− 3)
(−6) (−5) +
x(x− 2)(x− 3)
(2)
(−6)
+
x(x− 1)(x− 3)
(−2) (−1) +
x(x− 1)(x− 2)
(6)
(16).
We can see that each term is of degree three, so the entire sum has degree at
most three. Because the leading term does not vanish, the degree is actually three.
Moreover, if we plug in x = 0, 1, 2, or 3, three of the terms vanish and the fourth
produces the corresponding value from the data set.
Polynomials are not usually represented in their Lagrangian form. More fre-
quently, they are written as something like
x3 − 2x− 5.
The simple powers of x are called monomials, and this form of a polynomial is said
to be using the power form.
The coefficients of an interpolating polynomial using its power form,
P (x) = c1x
n−1 + c2x
n−2 + · · ·+ cn−1x + cn,
can, in principle, be computed by solving a system of simultaneous linear equations
xn−11 x
n−2
1 · · · x1 1
xn−12 x
n−2
2 · · · x2 1
· · · · · · · · · · · · 1
xn−1n x
n−2
n · · · xn 1
c1
c2
…
cn
=
y1
y2
…
yn
.
The matrix V of this linear system is known as a Vandermonde matrix. Its
elements are
vk,j = x
n−j
k .
3.1. The Interpolating Polynomial 3
The columns of a Vandermonde matrix are sometimes written in the opposite order,
but polynomial coefficient vectors in Matlab always have the highest power first.
The Matlab function vander generates Vandermonde matrices. For our ex-
ample data set,
V = vander(x)
generates
V =
0 0 0 1
1 1 1 1
8 4 2 1
27 9 3 1
Then
c = Vy’
computes the coefficients.
c =
1.0000
0.0000
-2.0000
-5.0000
In fact, the example data were generated from the polynomial x3 − 2x− 5.
Exercise 3.6 asks you to show that Vandermonde matrices are nonsingular if
the points xk are distinct. But Exercise 3.18 asks you to show that a Vandermonde
matrix can be very badly conditioned. Consequently, using the power form and
the Vandermonde matrix is a satisfactory technique for problems involving a few
well-spaced and well-scaled data points. But as a general-purpose approach, it is
dangerous.
In this chapter, we describe several Matlab functions that implement various
interpolation algorithms. All of them have the calling sequence
v = interp(x,y,u)
The first two input arguments, x and y, are vectors of the same length that define
the interpolating points. The third input argument, u, is a vector of points where
the function is to be evaluated. The output v is the same length as u and has
elements
v(k)=interp(x,y,u(k))
Our first such interpolation function, polyinterp, is based on the Lagrange
form. The code uses Matlab array operations to evaluate the polynomial at all
the components of u simultaneously.
4 Chapter 3. Interpolation
function v = polyinterp(x,y,u)
n = length(x);
v = zeros(size(u));
for k = 1:n
w = ones(size(u));
for j = [1:k-1 k+1:n]
w = (u-x(j))./(x(k)-x(j)).*w;
end
v = v + w*y(k);
end
To illustrate polyinterp, create a vector of densely spaced evaluation points.
u = -.25:.01:3.25;
Then
v = polyinterp(x,y,u);
plot(x,y,’o’,u,v,’-’)
creates Figure 3.1.
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
−10
−5
0
5
10
15
20
25
Figure 3.1. polyinterp.
The polyinterp function also works correctly with symbolic variables. For
example, create
symx = sym(’x’)
Then evaluate and display the symbolic form of the interpolating polynomial with
3.2. Piecewise Linear Interpolation 5
P = polyinterp(x,y,symx)
pretty(P)
which produces
-5 (-1/3 x + 1)(-1/2 x + 1)(-x + 1) – 6 (-1/2 x + 3/2)(-x + 2)x
-1/2 (-x + 3)(x – 1)x + 16/3 (x – 2)(1/2 x – 1/2)x
This expression is a rearrangement of the Lagrange form of the interpolating poly-
nomial. Simplifying the Lagrange form with
P = simplify(P)
changes P to the power form
P =
x^3-2*x-5
Here is another example, with a data set that is used by the other methods
in this chapter.
x = 1:6;
y = [16 18 21 17 15 12];
disp([x; y])
u = .75:.05:6.25;
v = polyinterp(x,y,u);
plot(x,y,’o’,u,v,’-’);
produces
1 2 3 4 5 6
16 18 21 17 15 12
and Figure 3.2.
Already in this example, with only six nicely spaced points, we can begin to
see the primary difficulty with full-degree polynomial interpolation. In between
the data points, especially in the first and last subintervals, the function shows
excessive variation. It overshoots the changes in the data values. As a result, full-
degree polynomial interpolation is hardly ever used for data and curve fitting. Its
primary application is in the derivation of other numerical methods.
3.2 Piecewise Linear Interpolation
You can create a simple picture of the data set from the last section by plotting
the data twice, once with circles at the data points and once with straight lines
connecting the points. The following statements produce Figure 3.3.
x = 1:6;
y = [16 18 21 17 15 12];
plot(x,y,’o’,x,y,’-’);
6 Chapter 3. Interpolation
1 2 3 4 5 6
10
12
14
16
18
20
22
Full degree polynomial interpolation
Figure 3.2. Full-degree polynomial interpolation.
To generate the lines, the Matlab graphics routines use piecewise linear in-
terpolation. The algorithm sets the stage for more sophisticated algorithms. Three
quantities are involved. The interval index k must be determined so that
xk ≤ x < xk+1.
The local variable, s, is given by
s = x− xk.
The first divided difference is
δk =
yk+1 − yk
xk+1 − xk
.
With these quantities in hand, the interpolant is
L(x) = yk + (x− xk)
yk+1 − yk
xk+1 − xk
= yk + sδk.
This is clearly a linear function that passes through (xk, yk) and (xk+1, yk+1).
The points xk are sometimes called breakpoints or breaks. The piecewise linear
interpolant L(x) is a continuous function of x, but its first derivative, L′(x), is not
continuous. The derivative has a constant value, δk, on each subinterval and jumps
at the breakpoints.
Piecewise linear interpolation is implemented in piecelin.m. The input u
can be a vector of points where the interpolant is to be evaluated, so the index k is
actually a vector of indices. Read this code carefully to see how k is computed.
3.2. Piecewise Linear Interpolation 7
1 2 3 4 5 6
10
12
14
16
18
20
22
Piecewise linear interpolation
Figure 3.3. Piecewise linear interpolation.
function v = piecelin(x,y,u)
%PIECELIN Piecewise linear interpolation.
% v = piecelin(x,y,u) finds the piecewise linear L(x)
% with L(x(j)) = y(j) and returns v(k) = L(u(k)).
% First divided difference
delta = diff(y)./diff(x);
% Find subinterval indices k so that x(k) <= u < x(k+1)
n = length(x);
k = ones(size(u));
for j = 2:n-1
k(x(j) <= u) = j;
end
% Evaluate interpolant
s = u - x(k);
v = y(k) + s.*delta(k);
8 Chapter 3. Interpolation
3.3 Piecewise Cubic Hermite Interpolation
Many of the most effective interpolation techniques are based on piecewise cubic
polynomials. Let hk denote the length of the kth subinterval:
hk = xk+1 − xk.
Then the first divided difference, δk, is given by
δk =
yk+1 − yk
hk
.
Let dk denote the slope of the interpolant at xk:
dk = P
′(xk).
For the piecewise linear interpolant, dk = δk−1 or δk, but this is not necessarily true
for higher order interpolants.
Consider the following function on the interval xk ≤ x ≤ xk+1, expressed in
terms of local variables s = x− xk and h = hk:
P (x) =
3hs2 − 2s3
h3
yk+1 +
h3 − 3hs2 + 2s3
h3
yk
+
s2(s− h)
h2
dk+1 +
s(s− h)2
h2
dk.
This is a cubic polynomial in s, and hence in x, that satisfies four interpolation
conditions, two on function values and two on the possibly unknown derivative
values:
P (xk) = yk, P (xk+1) = yk+1,
P ′(xk) = dk, P
′(xk+1) = dk+1.
Functions that satisfy interpolation conditions on derivatives are known as Hermite
or osculatory interpolants, because of the higher order contact at the interpolation
sites. (“Osculari” means “to kiss” in Latin.)
If we happen to know both function values and first derivative values at a
set of data points, then piecewise cubic Hermite interpolation can reproduce those
data. But if we are not given the derivative values, we need to define the slopes
dk somehow. Of the many possible ways to do this, we will describe two, which
Matlab calls pchip and spline.
3.4 Shape-Preserving Piecewise Cubic
The acronym pchip abbreviates “piecewise cubic Hermite interpolating polyno-
mial.” Although it is fun to say, the name does not specify which of the many
possible interpolants is actually being used. In fact, spline interpolants are also
piecewise cubic Hermite interpolating polynomials, but with different slopes. Our
3.4. Shape-Preserving Piecewise Cubic 9
1 2 3 4 5 6
10
12
14
16
18
20
22
Shape−preserving Hermite interpolation
Figure 3.4. Shape-preserving piecewise cubic Hermite interpolation.
particular pchip is a shape-preserving, “visually pleasing” interpolant that was in-
troduced into Matlab fairly recently. It is based on an old Fortran program by
Fritsch and Carlson [2] that is described by Kahaner, Moler, and Nash [3]. Figure
3.4 shows how pchip interpolates our sample data.
The key idea is to determine the slopes dk so that the function values do not
overshoot the data values, at least locally. If δk and δk−1 have opposite signs or if
either of them is zero, then xk is a discrete local minimum or maximum, so we set
dk = 0.
This is illustrated in the first half of Figure 3.5. The lower solid line is the piecewise
linear interpolant. Its slopes on either side of the breakpoint in the center have
opposite signs. Consequently, the dashed line has slope zero. The curved line is
the shape-preserving interpolant, formed from two different cubics. The two cubics
interpolate the center value and their derivatives are both zero there. But there is
a jump in the second derivative at the breakpoint.
If δk and δk−1 have the same sign and the two intervals have the same length,
then dk is taken to be the harmonic mean of the two discrete slopes:
1
dk
=
1
2
(
1
δk−1
+
1
δk
)
.
In other words, at the breakpoint, the reciprocal slope of the Hermite interpolant
is the average of the reciprocal slopes of the piecewise linear interpolant on either
10 Chapter 3. Interpolation
side. This is shown in the other half of Figure 3.5. At the breakpoint, the reciprocal
slope of the piecewise linear interpolant changes from 1 to 5. The reciprocal slope
of the dashed line is 3, the average of 1 and 5. The shape-preserving interpolant is
formed from the 2 cubics that interpolate the center value and that have slope equal
to 1/3 there. Again, there is a jump in the second derivative at the breakpoint.
Figure 3.5. Slopes for pchip.
If δk and δk−1 have the same sign, but the two intervals have different lengths,
then dk is a weighted harmonic mean, with weights determined by the lengths of
the two intervals:
w1 + w2
dk
=
w1
δk−1
+
w2
δk
,
where
w1 = 2hk + hk−1, w2 = hk + 2hk−1.
This defines the pchip slopes at interior breakpoints, but the slopes d1 and dn
at either end of the data interval are determined by a slightly different, one-sided
analysis. The details are in pchiptx.m.
3.5 Cubic Spline
Our other piecewise cubic interpolating function is a cubic spline. The term “spline”
refers to an instrument used in drafting. It is a thin, flexible wooden or plastic tool
that is passed through given data points and defines a smooth curve in between. The
physical spline minimizes potential energy subject to the interpolation constraints.
The corresponding mathematical spline must have a continuous second derivative
and satisfy the same interpolation constraints. The breakpoints of a spline are also
referred to as its knots.
The world of splines extends far beyond the basic one-dimensional, cubic,
interpolatory spline we are describing here. There are multidimensional, high-order,
variable knot, and approximating splines. A valuable expository and reference text
for both the mathematics and the software is A Practical Guide to Splines by Carl
3.5. Cubic Spline 11
de Boor [1]. De Boor is also the author of the spline function and the Spline
Toolbox for Matlab.
Figure 3.6 shows how spline interpolates our sample data.
1 2 3 4 5 6
10
12
14
16
18
20
22
Spline interpolation
Figure 3.6. Cubic spline interpolation.
The first derivative P ′(x) of our piecewise cubic function is defined by different
formulas on either side of a knot xk. Both formulas yield the same value dk at the
knots, so P ′(x) is continuous.
On the kth subinterval, the second derivative is a linear function of s = x−xk:
P ′′(x) =
(6h− 12s)δk + (6s− 2h)dk+1 + (6s− 4h)dk
h2
.
If x = xk, s = 0 and
P ′′(xk+) =
6δk − 2dk+1 − 4dk
hk
.
The plus sign in xk+ indicates that this is a one-sided derivative. If x = xk+1,
s = hk and
P ′′(xk+1−) =
−6δk + 4dk+1 + 2dk
hk
.
On the (k − 1)st interval, P ′′(x) is given by a similar formula involving δk−1, dk,
and dk−1. At the knot xk,
P ′′(xk−) =
−6δk−1 + 4dk + 2dk−1
hk−1
.
Requiring P ′′(x) to be continuous at x = xk means that
P ′′(xk+) = P
′′(xk−).
12 Chapter 3. Interpolation
This leads to the condition
hkdk−1 + 2(hk−1 + hk)dk + hk−1dk+1 = 3(hkδk−1 + hk−1δk).
If the knots are equally spaced, so that hk does not depend on k, this becomes
dk−1 + 4dk + dk+1 = 3δk−1 + 3δk.
Like our other interpolants, the slopes dk of a spline are closely related to the
differences δk. In the spline case, they are a kind of running average of the δk’s.
The preceding approach can be applied at each interior knot xk, k = 2, . . . , n−
1, to give n− 2 equations involving the n unknowns dk. As with pchip, a different
approach must be used near the ends of the interval. One effective strategy is known
as “not-a-knot.” The idea is to use a single cubic on the first two subintervals,
x1 ≤ x ≤ x3, and on the last two subintervals, xn−2 ≤ x ≤ xn. In effect, x2 and
xn−1 are not knots. If the knots are equally spaced with all hk = 1, this leads to
d1 + 2d2 =
5
2
δ1 +
1
2
δ2
and
2dn−1 + dn =
1
2
δn−2 +
5
2
δn−1.
The details if the spacing is not equal to one are in splinetx.m.
With the two end conditions included, we have n linear equations in n un-
knowns:
Ad = r.
The vector of unknown slopes is
d =
d1
d2
...
dn
.
The coefficient matrix A is tridiagonal:
A =
h2 h2 + h1
h2 2(h1 + h2) h1
h3 2(h2 + h3) h2
. . .
. . .
. . .
hn−1 2(hn−2 + hn−1) hn−2
hn−1 + hn−2 hn−2
.
The right-hand side is
r = 3
r1
h2δ1 + h1δ2
h3δ2 + h2δ3
...
hn−1δn−2 + hn−2δn−1
rn
.
3.6. pchiptx, splinetx 13
The two values r1 and rn are associated with the end conditions.
If the knots are equally spaced with all hk = 1, the coefficient matrix is quite
simple:
A =
1 2
1 4 1
1 4 1
. . .
. . .
. . .
1 4 1
2 1
.
The right-hand side is
r = 3
5
6
δ1 + 16δ2
δ1 + δ2
δ2 + δ3
...
δn−2 + δn−1
1
6
δn−2 + 56δn−1
.
In our textbook function, splinetx, the linear equations defining the slopes
are solved with the tridisolve function introduced in Chapter 2, Linear Equations.
In the spline functions distributed with Matlab and the Spline Toolbox, the slopes
are computed by the Matlab backslash operator
d = A
Because most of the elements of A are zero, it is appropriate to store A in a sparse
data structure. The backslash operator can then take advantage of the tridiagonal
structure and solve the linear equations in time and storage proportional to n, the
number of data points.
Figure 3.7 compares the spline interpolant, s(x), with the pchip interpolant,
p(x). The difference between the functions themselves is barely noticeable. The
first derivative of spline, s′(x), is smooth, while the first derivative of pchip, p′(x),
is continuous, but shows “kinks.” The spline second derivative s′′(x) is continuous,
while the pchip second derivative p′′(x) has jumps at the knots. Because both func-
tions are piecewise cubics, their third derivatives, s′′′(x) and p′′′(x), are piecewise
constant. The fact that s′′′(x) takes on the same values in the first two intervals
and the last two intervals reflects the “not-a-knot” spline end conditions.
3.6 pchiptx, splinetx
The M-files pchiptx and splinetx are both based on piecewise cubic Hermite
interpolation. On the kth interval, this is
P (x) =
3hs2 − 2s3
h3
yk+1 +
h3 − 3hs2 + 2s3
h3
yk
+
s2(s− h)
h2
dk+1 +
s(s− h)2
h2
dk,
14 Chapter 3. Interpolation
2 4 6 8
12
15
18
21
spline
2 4 6 8
12
15
18
21
pchip
2 4 6 8
−5
0
5
2 4 6 8
−5
0
5
2 4 6 8
−20
0
20
2 4 6 8
−20
0
20
2 4 6 8
−30
0
30
2 4 6 8
−30
0
30
Figure 3.7. The spline and pchip interpolants, and their first three derivatives.
where s = x−xk and h = hk. The two functions differ in the way they compute the
slopes, dk. Once the slopes have been computed, the interpolant can be efficiently
evaluated using the power form with the local variable s:
P (x) = yk + sdk + s
2ck + s
3bk,
where the coefficients of the quadratic and cubic terms are
ck =
3δk − 2dk − dk+1
h
,
bk =
dk − 2δk + dk+1
h2
.
Here is the first portion of code for pchiptx. It calls an internal subfunction to
compute the slopes, then computes the other coefficients, finds a vector of interval
indices, and evaluates the interpolant. After the preamble, this part of the code for
splinetx is the same.
function v = pchiptx(x,y,u)
%PCHIPTX Textbook piecewise cubic Hermite interpolation.
% v = pchiptx(x,y,u) finds the shape-preserving piecewise
% P(x), with P(x(j)) = y(j), and returns v(k) = P(u(k)).
%
% See PCHIP, SPLINETX.
3.6. pchiptx, splinetx 15
% First derivatives
h = diff(x);
delta = diff(y)./h;
d = pchipslopes(h,delta);
% Piecewise polynomial coefficients
n = length(x);
c = (3*delta - 2*d(1:n-1) - d(2:n))./h;
b = (d(1:n-1) - 2*delta + d(2:n))./h.^2;
% Find subinterval indices k so that x(k) <= u < x(k+1)
k = ones(size(u));
for j = 2:n-1
k(x(j) <= u) = j;
end
% Evaluate interpolant
s = u - x(k);
v = y(k) + s.*(d(k) + s.*(c(k) + s.*b(k)));
The code for computing the pchip slopes uses the weighted harmonic mean
at interior breakpoints and a one-sided formula at the endpoints.
function d = pchipslopes(h,delta)
% PCHIPSLOPES Slopes for shape-preserving Hermite cubic
% pchipslopes(h,delta) computes d(k) = P’(x(k)).
% Slopes at interior points
% delta = diff(y)./diff(x).
% d(k) = 0 if delta(k-1) and delta(k) have opposites
% signs or either is zero.
% d(k) = weighted harmonic mean of delta(k-1) and
% delta(k) if they have the same sign.
n = length(h)+1;
d = zeros(size(h));
k = find(sign(delta(1:n-2)).*sign(delta(2:n-1))>0)+1;
w1 = 2*h(k)+h(k-1);
w2 = h(k)+2*h(k-1);
d(k) = (w1+w2)./(w1./delta(k-1) + w2./delta(k));
% Slopes at endpoints
16 Chapter 3. Interpolation
d(1) = pchipend(h(1),h(2),delta(1),delta(2));
d(n) = pchipend(h(n-1),h(n-2),delta(n-1),delta(n-2));
function d = pchipend(h1,h2,del1,del2)
% Noncentered, shape-preserving, three-point formula.
d = ((2*h1+h2)*del1 – h1*del2)/(h1+h2);
if sign(d) ~= sign(del1)
d = 0;
elseif (sign(del1)~=sign(del2))&(abs(d)>abs(3*del1))
d = 3*del1;
end
The splinetx M-file computes the slopes by setting up and solving a tridiag-
onal system of simultaneous linear equations.
function d = splineslopes(h,delta);
% SPLINESLOPES Slopes for cubic spline interpolation.
% splineslopes(h,delta) computes d(k) = S’(x(k)).
% Uses not-a-knot end conditions.
% Diagonals of tridiagonal system
n = length(h)+1;
a = zeros(size(h)); b = a; c = a; r = a;
a(1:n-2) = h(2:n-1);
a(n-1) = h(n-2)+h(n-1);
b(1) = h(2);
b(2:n-1) = 2*(h(2:n-1)+h(1:n-2));
b(n) = h(n-2);
c(1) = h(1)+h(2);
c(2:n-1) = h(1:n-2);
% Right-hand side
r(1) = ((h(1)+2*c(1))*h(2)*delta(1)+ …
h(1)^2*delta(2))/c(1);
r(2:n-1) = 3*(h(2:n-1).*delta(1:n-2)+ …
h(1:n-2).*delta(2:n-1));
r(n) = (h(n-1)^2*delta(n-2)+ …
(2*a(n-1)+h(n-1))*h(n-2)*delta(n-1))/a(n-1);
% Solve tridiagonal linear system
d = tridisolve(a,b,c,r);
3.7. interpgui 17
Piecewise linear interpolation Full degree polynomial interpolation
Shape−preserving Hermite interpolation Spline interpolation
Figure 3.8. Four interpolants.
3.7 interpgui
Figure 3.8 illustrates the tradeoff between smoothness and a somewhat subjective
property that we might call local monotonicity or shape preservation.
The piecewise linear interpolant is at one extreme. It has hardly any smooth-
ness. It is continuous, but there are jumps in its first derivative. On the other hand,
it preserves the local monotonicity of the data. It never overshoots the data and it
is increasing, decreasing, or constant on the same intervals as the data.
The full-degree polynomial interpolant is at the other extreme. It is infinitely
differentiable. But it often fails to preserve shape, particularly near the ends of the
interval.
The pchip and spline interpolants are in between these two extremes. The
spline is smoother than pchip. The spline has two continuous derivatives, while
pchip has only one. A discontinuous second derivative implies discontinuous cur-
vature. The human eye can detect large jumps in curvature in graphs and in me-
chanical parts made by numerically controlled machine tools. On the other hand,
pchip is guaranteed to preserve shape, but the spline might not.
The M-file interpgui allows you to experiment with the four interpolants
discussed in this chapter:
• piecewise linear interpolant,
• full-degree interpolating polynomial,
18 Chapter 3. Interpolation
0 1 2 3 4 5 6
8
10
12
14
16
18
20
22
24
Interpolation
Figure 3.9. interpgui.
• piecewise cubic spline,
• shape-preserving piecewise cubic.
The program can be initialized in several different ways:
• With no arguments, interpgui starts with 8 zeros.
• With a scalar argument, interpgui(n) starts with n zeros.
• With one vector argument, interpgui(y) starts with equally spaced x’s.
• With two arguments, interpgui(x,y) starts with a plot of y versus x.
After initialization, the interpolation points can be varied with the mouse. If x
has been specified, it remains fixed. Figure 3.9 is the initial plot generated by our
example data.
Exercises
3.1. Reproduce Figure 3.8, with four subplots showing the four interpolants dis-
cussed in this chapter.
3.2. Tom and Ben are twin boys born on October 27, 2001. Here is a table of
their weights, in pounds and ounces, over their first few months.
Exercises 19
% Date Tom Ben
W = [10 27 2001 5 10 4 8
11 19 2001 7 4 5 11
12 03 2001 8 12 6 4
12 20 2001 10 14 8 7
01 09 2002 12 13 10 3
01 23 2002 14 8 12 0
03 06 2002 16 10 13 10];
You can use datenum to convert the date in the first three columns to a serial
date number measuring time in days.
t = datenum(W(:,[3 1 2]));
Make a plot of their weights versus time, with circles at the data points and
the pchip interpolating curve in between. Use datetick to relabel the time
axis. Include a title and a legend. The result should look something like
Figure 3.10.
Nov01 Dec01 Jan02 Feb02 Mar02
4
6
8
10
12
14
16
18
Twins’ weights
P
o
u
n
d
s
Tom
Ben
Figure 3.10. Twins’ weights.
3.3. (a) Interpolate these data by each of the four interpolants discussed in this
chapter: piecelin, polyinterp, splinetx, and pchiptx. Plot the results
for −1 ≤ x ≤ 1.
20 Chapter 3. Interpolation
x y
-1.00 -1.0000
-0.96 -0.1512
-0.65 0.3860
0.10 0.4802
0.40 0.8838
1.00 1.0000
(b) What are the values of each of the four interpolants at x = −0.3? Which
of these values do you prefer? Why?
(c) The data were actually generated from a low-degree polynomial with
integer coefficients. What is that polynomial?
3.4. Make a plot of your hand. Start with
figure(’position’,get(0,’screensize’))
axes(’position’,[0 0 1 1])
[x,y] = ginput;
Place your hand on the computer screen. Use the mouse to select a few
dozen points outlining your hand. Terminate the ginput with a carriage
return. You might find it easier to trace your hand on a piece of paper and
then put the paper on the computer screen. You should be able to see the
ginput cursor through the paper. (Save these data. We will refer to them in
other exercises later in this book.)
Figure 3.11. A hand.
Exercises 21
Now think of x and y as two functions of an independent variable that goes
from one to the number of points you collected. You can interpolate both
functions on a finer grid and plot the result with
n = length(x);
s = (1:n)’;
t = (1:.05:n)’;
u = splinetx(s,x,t);
v = splinetx(s,y,t);
clf reset
plot(x,y,’.’,u,v,’-’);
Do the same thing with pchiptx. Which do you prefer?
Figure 3.11 is the plot of my hand. Can you tell if it was done with splinetx
or pchiptx?
3.5. The previous exercise uses the data index number as the independent variable
for two-dimensional parametric interpolation. This exercise uses, instead, the
angle θ from polar coordinates. In order to do this, the data must be centered
so that they lie on a curve that is starlike with respect to the origin, that is,
every ray emanating from the origin meets the data only once. This means
that you must be able to find values x0 and y0 so that the Matlab statements
x = x – x0
y = y – y0
theta = atan2(y,x)
r = sqrt(x.^2 + y.^2)
plot(theta,r)
produce a set of points that can be interpolated with a single-valued function,
r = r(θ). For the data obtained by sampling the outline of your hand, the
point (x0, y0) is located near the base of your palm. See the small circle in
Figure 3.11. Furthermore, in order to use splinetx and pchiptx, it is also
necessary to order the data so that theta is monotonically increasing.
Choose a subsampling increment, delta, and let
t = (theta(1):delta:theta(end))’;
p = pchiptx(theta,r,t);
s = splinetx(theta,r,t);
Examine two plots:
plot(theta,r,’o’,t,[p s],’-’)
and
plot(x,y,’o’,p.*cos(t),p.*sin(t),’-’,…
s.*cos(t),s.*sin(t),’-’)
Compare this approach with the one used in the previous exercise. Which do
you prefer? Why?
22 Chapter 3. Interpolation
3.6. This exercise requires the Symbolic Toolbox.
(a) What does vandal(n) compute and how does it compute it?
(b) Under what conditions on x is the matrix vander(x) nonsingular?
3.7. Prove that the interpolating polynomial is unique. That is, if P (x) and Q(x)
are two polynomials with degree less than n that agree at n distinct points,
then they agree at all points.
3.8. Give convincing arguments that each of the following descriptions defines
the same polynomial, the Chebyshev polynomial of degree five, T5(x). Your
arguments can involve analytic proofs, symbolic computation, numeric com-
putation, or all three. Two of the representations involve the golden ratio,
φ =
1 +
√
5
2
.
(a) Power form basis:
T5(x) = 16x
5 − 20×3 + 5x.
(b) Relation to trigonometric functions:
T5(x) = cos (5 cos
−1 x).
(c) Horner representation:
T5(x) = ((((16x + 0)x− 20)x + 0)x + 5)x + 0.
(d) Lagrange form:
x1, x6 = ±1,
x2, x5 = ±φ/2,
x3, x4 = ±(φ− 1)/2,
yk = (−1)k, k = 1, . . . , 6,
T5(x) =
∑
k
∏
j 6=k
x− xj
xk − xj
yk.
(e) Factored representation:
z1, z5 = ±
√
(2 + φ)/4,
z2, z4 = ±
√
(3− φ)/4,
z3 = 0,
T5(x) = 16
5∏
1
(x− zk).
(f) Three-term recurrence:
T0(x) = 1,
T1(x) = x,
Tn(x) = 2xTn−1(x)− Tn−2(x) for n = 2, . . . , 5.
Exercises 23
3.9. The M-file rungeinterp.m provides an experiment with a famous polynomial
interpolation problem due to Carl Runge. Let
f(x) =
1
1 + 25×2
,
and let Pn(x) denote the polynomial of degree n−1 that interpolates f(x) at
n equally spaced points on the interval −1 ≤ x ≤ 1. Runge asked whether,
as n increases, Pn(x) converges to f(x). The answer is yes for some x, but
no for others.
(a) For what x does Pn(x) → f(x) as n →∞?
(b) Change the distribution of the interpolation points so that they are not
equally spaced. How does this affect convergence? Can you find a distribution
so that Pn(x) → f(x) for all x in the interval?
3.10. We skipped from piecewise linear to piecewise cubic interpolation. How far
can you get with the development of piecewise quadratic interpolation?
3.11. Modify splinetx and pchiptx so that, if called with two output arguments,
they produce both the value of the interpolant and its first derivative. That
is,
[v,vprime] = pchiptx(x,y,u)
and
[v,vprime] = splinetx(x,y,u)
compute P (u) and P ′(u).
3.12. Modify splinetx and pchiptx so that, if called with only two input arguments,
they produce PP, the piecewise polynomial structure produced by the stan-
dard Matlab functions spline and pchip and used by ppval.
3.13. (a) Create two functions perpchip and perspline by modifying pchiptx
and splinetx to replace the one-sided and not-a-knot end conditions with
periodic boundary conditions. This requires that the given data have
yn = y1
and that the resulting interpolant be periodic. In other words, for all x,
P (x + ∆) = P (x),
where
∆ = xn − x1.
The algorithms for both pchip and spline involve calculations with yk, hk,
and δk to produce slopes dk. With the periodicity assumption, all of these
quantities become periodic functions, with period n − 1, of the subscript k.
In other words, for all k,
24 Chapter 3. Interpolation
yk = yk+n−1,
hk = hk+n−1,
δk = δk+n−1,
dk = dk+n−1.
This makes it possible to use the same calculations at the endpoints that
are used at the interior points in the nonperiodic situation. The special case
code for the end conditions can be eliminated and the resulting M-files are
actually much simpler.
For example, the slopes dk for pchip with equally spaced points are given by
dk = 0 if sign(δk−1) 6= sign(δk),
1
dk
=
1
2
(
1
δk−1
+
1
δk
)
if sign(δk−1) = sign(δk).
With periodicity, these formulas can also be used at the endpoints where
k = 1 and k = n because
δ0 = δn−1 and δn = δ1.
For spline, the slopes satisfy a system of simultaneous linear equations for
k = 2, . . . , n− 1:
hkdk−1 + 2(hk−1 + hk)dk + hk−1dk+1 = 3(hkδk−1 + hk−1δk).
With periodicity, this becomes
h1dn−1 + 2(hn−1 + h1)d1 + hn−1d2 = 3(h1δn−1 + hn−1δ1)
at k = 1 and
hndn−1 + 2(hn−1 + h1)dn + hn−1d2 = 3(h1δn−1 + hn−1δ1)
at k = n. The resulting matrix has two nonzero elements outside the tridi-
agonal structure. The additional nonzero elements in the first and last rows
are A1,n−1 = h1 and An,2 = hn−1.
(b) Demonstrate that your new functions work correctly on
x = 0:pi/4:2*pi;
y = cos(x);
u = 0:pi/50:2*pi;
v = your_function(x,y,u);
plot(x,y,’o’,u,v,’-’)
(c) Once you have perchip and perspline, you can use the NCM M-file
interp2dgui to investigate closed-curve interpolation in two dimensions.
You should find that the periodic boundary conditions do a better job of
reproducing symmetries of closed curves in the plane.
Exercises 25
3.14. (a) Modify splinetx so that it forms the full tridiagonal matrix
A = diag(a,-1) + diag(b,0) + diag(c,1)
and uses backslash to compute the slopes.
(b) Monitor condest(A) as the spline knots are varied with interpgui. What
happens if two of the knots approach each other? Find a data set that makes
condest(A) large.
3.15. Modify pchiptx so that it uses a weighted average of the slopes instead of
the weighted harmonic mean.
3.16. (a) Consider
x = -1:1/3:1
interpgui(1-x.^2)
Which, if any, of the four interpolants linear, spline, pchip, and polynomial
are the same? Why?
(b) Same questions for
interpgui(1-x.^4)
3.17. Why does interpgui(4) show only three graphs, not four, no matter where
you move the points?
3.18. (a) If you want to interpolate census data on the interval 1900 ≤ t ≤ 2000
with a polynomial,
P (t) = c1t
10 + c2t
9 + · · ·+ c10t + c11,
you might be tempted to use the Vandermonde matrix generated by
t = 1900:10:2000
V = vander(t)
Why is this a really bad idea?
(b) Investigate centering and scaling the independent variable. Plot some
data, pull down the Tools menu on the figure window, select Basic Fitting,
and find the check box about centering and scaling. What does this check
box do?
(c) Replace the variable t with
s =
t− µ
σ
.
This leads to a modified polynomial P̃ (s). How are its coefficients related to
those of P (t)? What happens to the Vandermonde matrix? What values of
µ and σ lead to a reasonably well conditioned Vandermonde matrix? One
possibility is
mu = mean(t)
sigma = std(t)
but are there better values?
26 Chapter 3. Interpolation
Bibliography
[1] C. de Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978.
[2] F. N. Fritsch and R. E. Carlson, Monotone Piecewise Cubic Interpolation,
SIAM Journal on Numerical Analysis, 17 (1980), pp. 238-246.
[3] D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software,
Prentice–Hall, Englewood Cliffs, NJ, 1989.
27
Chapter 4
Zeros and Roots
This chapter describes several basic methods for computing zeros of functions and
then combines three of the basic methods into a fast, reliable algorithm known as
‘zeroin’.
4.1 Bisection
Let’s compute
√
2. We will use interval bisection, which is a kind of systematic trial
and error. We know that
√
2 is between 1 and 2. Try x = 1 1
2
. Because x2 is greater
than 2, this x is too big. Try x = 1 1
4
. Because x2 is less than 2, this x is too small.
Continuing in this way, our approximations to
√
2 are
1
1
2
, 1
1
4
, 1
3
8
, 1
5
16
, 1
13
32
, 1
27
64
, . . . .
Here is a Matlab program, including a step counter.
M = 2
a = 1
b = 2
k = 0;
while b-a > eps
x = (a + b)/2;
if x^2 > M
b = x
else
a = x
end
k = k + 1;
end
February 15, 2008
1
2 Chapter 4. Zeros and Roots
We are sure that
√
2 is in the initial interval [a,b]. This interval is repeatedly cut
in half and always brackets the answer. The entire process requires 52 steps. Here
are the first few and the last few values.
b = 1.50000000000000
a = 1.25000000000000
a = 1.37500000000000
b = 1.43750000000000
a = 1.40625000000000
b = 1.42187500000000
a = 1.41406250000000
b = 1.41796875000000
b = 1.41601562500000
b = 1.41503906250000
b = 1.41455078125000
…..
b = 1.41421356237311
a = 1.41421356237299
a = 1.41421356237305
a = 1.41421356237308
a = 1.41421356237309
b = 1.41421356237310
b = 1.41421356237310
Using format hex, here are the final values of a and b.
a = 3ff6a09e667f3bcc
b = 3ff6a09e667f3bcd
They agree up to the last bit. We haven’t actually computed
√
2, which is irrational
and cannot be represented in floating point. But we have found two successive
floating-point numbers, one on either side of the theoretical result. We’ve come as
close as we can using floating-point arithmetic. The process takes 52 steps because
there are 52 bits in the fraction of an IEEE double-precision number. Each step
decreases the interval length by about one bit.
Interval bisection is a slow but sure algorithm for finding a zero of f(x), a
real-valued function of a real variable. All we assume about the function f(x) is
that we can write a Matlab program that evaluates it for any x. We also assume
that we know an interval [a, b] on which f(x) changes sign. If f(x) is actually a
continuous mathematical function, then there must be a point x∗ somewhere in the
interval where f(x∗) = 0. But the notion of continuity does not strictly apply to
floating-point computation. We might not be able to actually find a point where
f(x) is exactly zero. Our goal is as follows:
Find a very small interval, perhaps two successive floating-point num-
bers, on which the function changes sign.
The Matlab code for bisection is
4.2. Newton’s Method 3
k = 0;
while abs(b-a) > eps*abs(b)
x = (a + b)/2;
if sign(f(x)) == sign(f(b))
b = x;
else
a = x;
end
k = k + 1;
end
Bisection is slow. With the termination condition in the above code, it always
takes 52 steps for any function. But it is completely reliable. If we can find a
starting interval with a change of sign, then bisection cannot fail to reduce that
interval to two successive floating-point numbers that bracket the desired result.
4.2 Newton’s Method
Newton’s method for solving f(x) = 0 draws the tangent to the graph of f(x) at any
point and determines where the tangent intersects the x-axis. The method requires
one starting value, x0. The iteration is
xn+1 = xn −
f(xn)
f ′(xn)
.
The Matlab code is
k = 0;
while abs(x – xprev) > eps*abs(x)
xprev = x;
x = x – f(x)/fprime(x)
k = k + 1;
end
As a method for computing square roots, Newton’s method is particularly
elegant and effective. To compute
√
M , find a zero of
f(x) = x2 −M.
In this case, f ′(x) = 2x and
xn+1 = xn −
x2n −M
2xn
=
1
2
(
xn +
M
xn
)
.
The algorithm repeatedly averages x and M/x. The Matlab code is
4 Chapter 4. Zeros and Roots
while abs(x – xprev) > eps*abs(x)
xprev = x;
x = 0.5*(x + M/x);
end
Here are the results for
√
2, starting at x = 1.
1.50000000000000
1.41666666666667
1.41421568627451
1.41421356237469
1.41421356237309
1.41421356237309
Newton’s method takes only six iterations. In fact, it was done in five, but the sixth
iteration was needed to meet the termination condition.
When Newton’s method works as it does for square roots, it is very effective.
It is the basis for many powerful numerical methods. But, as a general-purpose
algorithm for finding zeros of functions, it has three serious drawbacks.
• The function f(x) must be smooth.
• It might not be convenient to compute the derivative f ′(x).
• The starting guess must be close to the final result.
In principle, the computation of the derivative f ′(x) can be done using a tech-
nique known as automatic differentiation. A Matlab function f(x) or a suitable
code in any other programming language, defines a mathematical function of its
arguments. By combining modern computer science parsing techniques with the
rules of calculus, especially the chain rule, it is theoretically possible to generate
the code for another function fprime(x), that computes f ′(x). However, the actual
implementation of such techniques is quite complicated and has not yet been fully
realized.
The local convergence properties of Newton’s method are very attractive. Let
x∗ be a zero of f(x) and let en = xn − x∗ be the error in the nth iterate. Assume
• f ′(x) and f ′′(x) exist and are continuous,
• x0 is close to x∗.
Then it is possible to prove [2] that
en+1 =
1
2
f ′′(ξ)
f ′(xn)
e2n,
where ξ is some point between xn and x∗. In other words,
en+1 = O(e
2
n).
4.3. A Perverse Example 5
0 0.5 1 1.5 2 2.5 3 3.5 4
−1.5
−1
−0.5
0
0.5
1
1.5
Figure 4.1. Newton’s method in an infinite loop.
This is called quadratic convergence. For nice, smooth functions, once you are close
enough to the zero, the error is roughly squared with each iteration. The number
of correct digits approximately doubles with each iteration. The results we saw for√
2 are typical.
When the assumptions underlying the local convergence theory are not sat-
isfied, Newton’s method might be unreliable. If f(x) does not have continuous,
bounded first and second derivatives, or if the starting point is not close enough to
the zero, then the local theory does not apply and we might get slow convergence,
or even no convergence at all. The next section provides one example of what might
happen.
4.3 A Perverse Example
Let’s see if we can get Newton’s method to iterate forever. The iteration
xn+1 = xn −
f(xn)
f ′(xn)
cycles back and forth around a point a if
xn+1 − a = −(xn − a).
This happens if f(x) satisfies
x− a− f(x)
f ′(x)
= −(x− a).
This is a separable ordinary differential equation:
f ′(x)
f(x)
=
1
2(x− a) .
6 Chapter 4. Zeros and Roots
The solution is
f(x) = sign(x− a)
√
|x− a|.
The zero of f(x) is, of course, at x∗ = a. A plot of f(x), Figure 4.1, with
a = 2, is obtained with
ezplot(’sign(x-2)*sqrt(abs(x-2))’,0,4)
If we draw the tangent to the graph at any point, it intersects the x-axis on the
opposite side of x = a. Newton’s method cycles forever, neither converging nor
diverging.
The convergence theory for Newton’s method fails in this case because f ′(x)
is unbounded as x → a. It is also interesting to apply the algorithms discussed in
the next sections to this function.
4.4 Secant Method
The secant method replaces the derivative evaluation in Newton’s method with a
finite difference approximation based on the two most recent iterates. Instead of
drawing a tangent to the graph of f(x) at one point, you draw a secant through
two points. The next iterate is the intersection of this secant with the x-axis.
The iteration requires two starting values, x0 and x1. The subsequent iterates
are given by
sn =
f(xn)− f(xn−1)
xn − xn−1
,
xn+1 = xn −
f(xn)
sn
.
This formulation makes it clear how Newton’s f ′(xn) is being replaced with the
slope of the secant, sn. The formulation in the following Matlab code is a little
more compact:
while abs(b-a) > eps*abs(b)
c = a;
a = b;
b = b + (b – c)/(f(c)/f(b)-1);
k = k + 1;
end
For
√
2, starting with a = 1 and b = 2, the secant method requires seven iterations,
compared with Newton’s six.
1.33333333333333
1.40000000000000
1.41463414634146
1.41421143847487
1.41421356205732
1.41421356237310
1.41421356237310
4.5. Inverse Quadratic Interpolation 7
The secant method’s primary advantage over Newton’s method is that it does
not require code to compute f ′(x). Its convergence properties are similar. Again,
assuming f ′(x) and f ′′(x) are continuous, it is possible to prove [2] that
en+1 =
1
2
f ′′(ξ)f ′(ξn)f ′(ξn−1)
f ′(ξ)3
enen−1,
where ξ is some point between xn and x∗. In other words,
en+1 = O(enen−1).
This is not quadratic convergence, but it is superlinear convergence. It turns out
that
en+1 = O(e
φ
n),
where φ is the golden ratio, (1 +
√
5)/2. Once you get close, the number of correct
digits is roughly multiplied by 1.6 with each iteration. That’s almost as fast as
Newton’s method and a whole lot faster than the one bit per step produced by
bisection.
We leave it to exercise 4.8 to investigate the behavior of the secant method
on the perverse function from the previous section:
f(x) = sign(x− a)
√
|x− a|.
4.5 Inverse Quadratic Interpolation
The secant method uses two previous points to get the next one, so why not use
three?
Suppose we have three values, a, b, and c, and corresponding function values,
f(a), f(b), and f(c). We could interpolate these values by a parabola, a quadratic
function of x, and take the next iterate to be the point where the parabola intersects
the x-axis. The difficulty is that the parabola might not intersect the x-axis; a
quadratic function does not necessarily have real roots. This could be regarded as
an advantage. An algorithm known as Muller’s method uses the complex roots of
the quadratic to produce approximations to complex zeros of f(x). But, for now,
we want to avoid complex arithmetic.
Instead of a quadratic in x, we can interpolate the three points with a quadratic
function in y. That’s a “sideways” parabola, P (y), determined by the interpolation
conditions
a = P (f(a)), b = P (f(b)), c = P (f(c)).
This parabola always intersects the x-axis, which is y = 0. So x = P (0) is the next
iterate.
This method is known as inverse quadratic interpolation. We will abbreviate
it with IQI. Here is Matlab code that illustrates the idea.
k = 0;
while abs(c-b) > eps*abs(c)
8 Chapter 4. Zeros and Roots
x = polyinterp([f(a),f(b),f(c)],[a,b,c],0)
a = b;
b = c;
c = x;
k = k + 1;
end
The trouble with this “pure” IQI algorithm is that polynomial interpolation
requires the abscissae, which in this case are f(a), f(b), and f(c), to be distinct.
We have no guarantee that they are. For example, if we try to compute
√
2 using
f(x) = x2 − 2 and start with a = −2, b = 0, c = 2, we are starting with f(a) = f(c)
and the first step is undefined. If we start near this singular situation, say with
a = −2.001, b = 0, c = 1.999, the next iterate is near x = 500.
So IQI is like an immature racehorse. It moves very quickly when it is near
the finish line, but its global behavior can be erratic. It needs a good trainer to
keep it under control.
4.6 Zeroin
The idea behind the zeroin algorithm is to combine the reliability of bisection with
the convergence speed of secant and IQI. T. J. Dekker and colleagues at the Math-
ematical Center in Amsterdam developed the first version of the algorithm in the
1960s [3]. Our implementation is based on a version by Richard Brent [1]. Here is
the outline:
• Start with a and b so that f(a) and f(b) have opposite signs.
• Use a secant step to give c between a and b.
• Repeat the following steps until |b− a| < ²|b| or f(b) = 0.
• Arrange a, b, and c so that
– f(a) and f(b) have opposite signs,
– |f(b)| ≤ |f(a)|,
– c is the previous value of b.
• If c 6= a, consider an IQI step.
• If c = a, consider a secant step.
• If the IQI or secant step is in the interval [a, b], take it.
• If the step is not in the interval, use bisection.
This algorithm is foolproof. It never loses track of the zero trapped in a
shrinking interval. It uses rapidly convergent methods when they are reliable. It
uses a slow, but sure, method when it is necessary.
4.7. fzerotx 9
0 5 10 15 20 25 30
−0.5
0
0.5
1
x
J
0
(x)
Figure 4.2. Zeros of J0(x).
4.7 fzerotx
The Matlab implementation of the zeroin algorithm is called fzero. It has several
features beyond the basic algorithm. A preamble takes a single starting guess and
searches for an interval with a sign change. The values returned by the function
f(x) are checked for infinities, NaNs, and complex numbers. Default tolerances can
be changed. Additional output, including a count of function evaluations, can be
requested. Our textbook version of zeroin is fzerotx. We have simplified fzero
by removing most of its additional features while retaining the essential features of
zeroin.
We can illustrate the use of fzerotx with the zeroth-order Bessel function of
the first kind, J0(x). This function is available in Matlab as besselj(0,x). The
following code finds the first 10 zeros of J0(x) and produces Figure 4.2 (except for
the red ‘x’, which we will add later).
J0 = @(x) besselj(0,x);
for n = 1:10
z(n) = fzerotx(J0,[(n-1) n]*pi);
end
x = 0:pi/50:10*pi;
y = J0(x);
plot(z,zeros(1,10),’o’,x,y,’-’)
line([0 10*pi],[0 0],’color’,’black’)
axis([0 10*pi -0.5 1.0])
You can see from the figure that the graph of J0(x) is like an amplitude and fre-
quency modulated version of cos x. The distance between successive zeros is close
to π.
10 Chapter 4. Zeros and Roots
The function fzerotx takes two arguments. The first specifies the function
F (x) whose zero is being sought and the second specifies the interval [a, b] to search.
fzerotx is an example of a Matlab function function, which is a function that
takes another function as an argument. ezplot is another example. Other chapters
of this book—Chapter 6, Quadrature; Chapter 7, Ordinary Differential Equations;
and even Chapter 9, Random Numbers—also describe “tx” and “gui” M-files that
are function functions.
A function can be passed as an argument to another function in two different
ways:
• function handle,
• anonymous function.
A function handle uses the ’@’ character preceding the name of a built-in
function or a function defined in an M-file. Examples include
@cos
@humps
@bessj0
where bessj0.m is the two-line M-file
function y = bessj0(x)
y = besselj(0,x)
These handles can then be used as arguments to function functions.
z = fzerotx(@bessj0,[0,pi])
Note that @besselj is also a valid function handle, but for a function of two argu-
ments.
Anonymous functions were introduced in Matlab 7. Examples include
F = @(t) cos(pi*t)
G = @(z) z^3-2*z-5
J0 = @(x) besselj(0,x)
These objects are called anonymous functions because the construction
@(arguments) expression
defines a function, but does not give it a name.
M-files and anonymous functions can define functions of more than one ar-
gument. In this case, the values of the extra arguments can be passed through
fzerotx to the objective function. These values remain constant during the zero
finding iteration. This allows us to find where a particular function takes on a
specified value y, instead of just finding a zero. For example, consider the equation
J0(ξ) = 0.5.
Define an anonymous function with two or three arguments.
4.7. fzerotx 11
F = @(x,y) besselj(0,x)-y
or
B = @(x,n,y) besselj(n,x)-y
Then either
xi = fzerotx(F,[0,2],.5)
or
xi = fzerotx(B,[0,2],0,.5)
produces
xi =
1.5211
The point (ξ, J0(ξ)) is marked with an ’x’ in Figure 4.2.
The preamble for fzerotx is as follows.
function b = fzerotx(F,ab,varargin);
%FZEROTX Textbook version of FZERO.
% x = fzerotx(F,[a,b]) tries to find a zero of F(x) between
% a and b. F(a) and F(b) must have opposite signs.
% fzerotx returns one endpoint of a small subinterval of
% [a,b] where F changes sign.
% Additional arguments, fzerotx(F,[a,b],p1,p2,...),
% are passed on, F(x,p1,p2,...).
The first section of code initializes the variables a, b, and c that characterize
the search interval. The function F is evaluated at the endpoints of the initial
interval.
a = ab(1);
b = ab(2);
fa = F(a,varargin{:});
fb = F(b,varargin{:});
if sign(fa) == sign(fb)
error(’Function must change sign on the interval’)
end
c = a;
fc = fa;
d = b - c;
e = d;
Here is the beginning of the main loop. At the start of each pass through the loop,
a, b, and c are rearranged to satisfy the conditions of the zeroin algorithm.
12 Chapter 4. Zeros and Roots
while fb ~= 0
% The three current points, a, b, and c, satisfy:
% f(x) changes sign between a and b.
% abs(f(b)) <= abs(f(a)).
% c = previous b, so c might = a.
% The next point is chosen from
% Bisection point, (a+b)/2.
% Secant point determined by b and c.
% Inverse quadratic interpolation point determined
% by a, b, and c if they are distinct.
if sign(fa) == sign(fb)
a = c; fa = fc;
d = b - c; e = d;
end
if abs(fa) < abs(fb)
c = b; b = a; a = c;
fc = fb; fb = fa; fa = fc;
end
This section is the convergence test and possible exit from the loop.
m = 0.5*(a - b);
tol = 2.0*eps*max(abs(b),1.0);
if (abs(m) <= tol) | (fb == 0.0),
break
end
The next section of code makes the choice between bisection and the two flavors of
interpolation.
% Choose bisection or interpolation
if (abs(e) < tol) | (abs(fc) <= abs(fb))
% Bisection
d = m;
e = m;
else
% Interpolation
s = fb/fc;
if (a == c)
% Linear interpolation (secant)
p = 2.0*m*s;
q = 1.0 - s;
else
% Inverse quadratic interpolation
q = fc/fa;
r = fb/fa;
4.8. fzerogui 13
p = s*(2.0*m*q*(q - r) - (b - c)*(r - 1.0));
q = (q - 1.0)*(r - 1.0)*(s - 1.0);
end;
if p > 0, q = -q; else p = -p; end;
% Is interpolated point acceptable
if (2.0*p < 3.0*m*q - abs(tol*q)) & (p < abs(0.5*e*q))
e = d;
d = p/q;
else
d = m;
e = m;
end;
end
The final section evaluates F at the next iterate.
% Next point
c = b;
fc = fb;
if abs(d) > tol
b = b + d;
else
b = b – sign(b-a)*tol;
end
fb = F(b,varargin{:});
end
4.8 fzerogui
The M-file fzerogui demonstrates the behavior of zeroin and fzerotx. At each
step of the iteration, you are offered a chance to choose the next point. The choice
always includes the bisection point, shown in red on the computer screen. When
there are three distinct points active, a, b, and c, the IQI point is shown in blue.
When a = c, so there are only two distinct points, the secant point is shown in
green. A plot of f(x) is also provided as a dotted line, but the algorithm does not
“know” these other function values. You can choose any point you like as the next
iterate. You do not have to follow the zeroin algorithm and choose the bisection or
interpolant point. You can even cheat by trying to pick the point where the dotted
line crosses the axis.
We can demonstrate how fzerogui behaves by seeking the first zero of the
Bessel function. It turns out that the first local minimum of J0(x) is located near
x = 3.83. So here are the first few steps of
fzerogui(’besselj(0,x)’,[0 3.83])
Initially, c = b, so the two choices are the bisection point and the secant point
(Figure 4.3).
14 Chapter 4. Zeros and Roots
0 0.5 1 1.5 2 2.5 3 3.5 4
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
a b
c
Figure 4.3. Initially, choose secant or bisection.
If you choose the secant point, then b moves there and J0(x) is evaluated at
x = b. We then have three distinct points, so the choice is between the bisection
point and the IQI point (Figure 4.4).
0 0.5 1 1.5 2 2.5 3 3.5 4
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
a b c
Figure 4.4. Choose IQI or bisection.
If you choose the IQI point, the interval shrinks, the GUI zooms in on the
reduced interval, and the choice is again between the bisection and secant points,
which now happen to be close together (Figure 4.5).
You can choose either point, or any other point close to them. After two more
steps, the interval shrinks again and the situation shown in Figure 4.6 is reached.
This is the typical configuration as we approach convergence. The graph of the
function looks nearly like a straight line and the secant or IQI point is much closer
4.9. Value Finding and Reverse Interpolation 15
2.2 2.3 2.4 2.5 2.6 2.7
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
ab
c
Figure 4.5. Secant and bisection points nearly coincide.
to the desired zero than the bisection point. It now becomes clear that choosing
secant or IQI will lead to much faster convergence than bisection.
After several more steps, the length of the interval containing a change of sign
is reduced to a tiny fraction of the original length, and the algorithm terminates,
returning the final b as its result.
2.15 2.2 2.25 2.3 2.35 2.4
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
a b c
Figure 4.6. Nearing convergence.
4.9 Value Finding and Reverse Interpolation
These two problems look very similar.
• Given a function F (x) and a value η, find ξ so that F (ξ) = η.
16 Chapter 4. Zeros and Roots
• Given data (xk, yk) that sample an unknown function F (x), and a value η,
find ξ so that F (ξ) = η.
For the first problem, we are able to evaluate F (x) at any x, so we can use a
zero finder on the translated function f(x) = F (x)− η. This gives us the desired ξ
so that f(ξ) = 0, and hence F (ξ) = η.
For the second problem, we need to do some kind of interpolation. The most
obvious approach is to use a zero finder on f(x) = P (x) − η, where P (x) is some
interpolant, such as pchiptx(xk,yk,x) or splinetx(xk,yk,x). This often works
well, but it can be expensive. The zero finder calls for repeated evaluation of
the interpolant. With the implementations we have in this book, that involves
repeated calculation of the interpolant’s parameters and repeated determination of
the appropriate interval index.
A sometimes preferable alternative, known as reverse interpolation, uses pchip
or spline with the roles of xk and yk reversed. This requires monotonicity in the
yk, or at least in a subset of the yk around the target value η. A different piecewise
polynomial, say Q(y), is created with the property that Q(yk) = xk. Now it is not
necessary to use a zero finder. We simply evaluate ξ = Q(y) at y = η.
The choice between these two alternatives depends on how the data are best
approximated by piecewise polynomials. Is it better to use x or y as the independent
variable?
4.10 Optimization and fmintx
The task of finding maxima and minima of functions is closely related to zero
finding. In this section, we describe an algorithm similar to zeroin that finds a
local minimum of a function of one variable. The problem specification involves a
function f(x) and an interval [a, b]. The objective is to find a value of x that gives
a local minimum of f(x) on the given interval. If the function is unimodular, that
is, has only one local minimum on the interval, then that minimum will be found.
But if there is more than one local minimum, only one of them will be found, and
that one will not necessarily be minimum for the entire interval. It is also possible
that one of the endpoints is a minimizer.
Interval bisection cannot be used. Even if we know the values of f(a), f(b),
and f((a + b)/2), we cannot decide which half of the interval to discard and still
keep the minimum enclosed.
Interval trisection is feasible, but inefficient. Let h = (b− a)/3, so u = a + h
and v = b − h divide the interval into three equal parts. Assume we find that
f(u) < f(v). Then we could replace b with v, thereby cutting the interval length by
a factor of two-thirds, and still be sure that the minimum is in the reduced interval.
However, u would be in the center of the new interval and would not be useful in
the next step. We would need to evaluate the function twice each step.
The natural minimization analogue of bisection is golden section search. The
idea is illustrated for a = 0 and b = 1 in Figure 4.7. Let h = ρ(b − a), where ρ
is a quantity a little bit larger than 1/3 that we have yet to determine. Then the
points u = a+h and v = b−h divide the interval into three unequal parts. For the
4.10. Optimization and fmintx 17
ρ 1−ρ
u v0 1
uv’0 1−ρ
Figure 4.7. Golden section search.
first step, evaluate both f(u) and f(v). Assume we find that f(u) < f(v). Then
we know the minimum is between a and v. We can replace b with v and repeat the
process. If we choose the right value for ρ, the point u is in the proper position to
be used in the next step. After the first step, the function has to be evaluated only
once each step.
The defining equation for ρ is
ρ
1− ρ =
1− ρ
1
,
or
ρ2 − 3ρ + 1 = 0.
The solution is
ρ = 2− φ = (3−
√
5)/2 ≈ 0.382.
Here φ is the golden ratio that we used to introduce Matlab in the first chapter
of this book.
With golden section search, the length of the interval is reduced by a factor
of φ− 1 ≈ 0.618 each step. It would take
−52
log2(φ− 1)
≈ 75
steps to reduce the interval length to roughly eps, the size of IEEE double-precision
roundoff error, times its original value.
After the first few steps, there is often enough history to give three distinct
points and corresponding function values in the active interval. If the minimum
of the parabola interpolating these three points is in the interval, then it, rather
than the golden section point, is usually a better choice for the next point. This
combination of golden section search and parabolic interpolation provides a reliable
and efficient method for one-dimensional optimization.
The proper stopping criteria for optimization searches can be tricky. At a
minimum of f(x), the first derivative f ′(x) is zero. Consequently, near a minimum,
f(x) acts like a quadratic with no linear term:
f(x) ≈ a + b(x− c)2 + · · · .
18 Chapter 4. Zeros and Roots
The minimum occurs at x = c and has the value f(c) = a. If x is close to c, say
x ≈ c + δ for small δ, then
f(x) ≈ a + bδ2.
Small changes in x are squared when computing function values. If a and b are com-
parable in size, and nonzero, then the stopping criterion should involve sqrt(eps)
because any smaller changes in x will not affect f(x). But if a and b have different
orders of magnitude, or if either a or c is nearly zero, then interval lengths of size
eps, rather than sqrt(eps), are appropriate.
Matlab includes a function function fminbnd that uses golden section search
and parabolic interpolation to find a local minimum of a real-valued function of
a real variable. The function is based upon a Fortran subroutine by Richard
Brent [1]. Matlab also includes a function function, fminsearch, that uses a tech-
nique known as the Nelder–Meade simplex algorithm to search for a local minimum
of a real-valued function of several real variables. The Matlab Optimization Tool-
box is a collection of programs for other kinds of optimization problems, including
constrained optimization, linear programming, and large-scale, sparse optimization.
Our NCM collection includes a function fmintx that is a simplified textbook
version of fminbnd. One of the simplifications involves the stopping criterion. The
search is terminated when the length of the interval becomes less than a specified
parameter tol. The default value of tol is 10−6. More sophisticated stopping
criteria involving relative and absolute tolerances in both x and f(x) are used in
the full codes.
−1 −0.5 0 0.5 1 1.5 2
−100
−80
−60
−40
−20
0
x
−humps(x)
Figure 4.8. Finding the minimum of -humps(x).
The Matlab demos directory includes a function named humps that is in-
tended to illustrate the behavior of graphics, quadrature, and zero-finding routines.
The function is
h(x) =
1
(x− 0.3)2 + 0.01 +
1
(x− 0.9)2 + 0.04 .
Exercises 19
The statements
F = @(x) -humps(x);
fmintx(F,-1,2,1.e-4)
take the steps shown in Figure 4.8 and in the following output. We see that golden
section search is used for the second, third, and seventh steps, and that parabolic
interpolation is used exclusively once the search nears the minimizer.
step x f(x)
init: 0.1458980337 -25.2748253202
gold: 0.8541019662 -20.9035150009
gold: -0.2917960675 2.5391843579
para: 0.4492755129 -29.0885282699
para: 0.4333426114 -33.8762343193
para: 0.3033578448 -96.4127439649
gold: 0.2432135488 -71.7375588319
para: 0.3170404333 -93.8108500149
para: 0.2985083078 -96.4666018623
para: 0.3003583547 -96.5014055840
para: 0.3003763623 -96.5014085540
para: 0.3003756221 -96.5014085603
Exercises
4.1. Use fzerogui to try to find a zero of each of the following functions in the
given interval. Do you see any interesting or unusual behavior?
x3 − 2x− 5 [0, 3]
sin x [1, 4]
x3 − 0.001 [−1, 1]
log (x + 2/3) [0, 1]
sign(x− 2)
√
|x− 2| [1, 4]
atan(x)− π/3 [0, 5]
1/(x− π) [0, 5]
4.2. Here is a little footnote to the history of numerical methods. The polynomial
x3 − 2x− 5
was used by Wallis when he first presented Newton’s method to the French
Academy. It has one real root, between x = 2 and x = 3, and a pair of
complex conjugate roots.
(a) Use the Symbolic Toolbox to find symbolic expressions for the three roots.
Warning: The results are not pretty. Convert the expressions to numerical
values.
20 Chapter 4. Zeros and Roots
(b) Use the roots function in Matlab to find numerical values for all three
roots.
(c) Use fzerotx to find the real root.
(d) Use Newton’s method starting with a complex initial value to find a
complex root.
(e) Can bisection be used to find the complex root? Why or why not?
4.3. Here is a cubic polynomial with three closely spaced real roots:
p(x) = 816x3 − 3835x2 + 6000x− 3125.
(a) What are the exact roots of p?
(b) Plot p(x) for 1.43 ≤ x ≤ 1.71. Show the location of the three roots.
(c) Starting with x0 = 1.5, what does Newton’s method do?
(d) Starting with x0 = 1 and x1 = 2, what does the secant method do?
(e) Starting with the interval [1, 2], what does bisection do?
(f) What is fzerotx(p,[1,2])? Why?
4.4. What causes fzerotx to terminate?
4.5. (a) How does fzerotx choose between the bisection point and the interpolant
point for its next iterate?
(b) Why is the quantity tol involved in the choice?
4.6. Derive the formula that fzerotx uses for IQI.
4.7. Hoping to find the zero of J0(x) in the interval 0 ≤ x ≤ π, we might try the
statement
z = fzerotx(@besselj,[0 pi],0)
This is legal usage of a function handle, and of fzerotx, but it produces
z = 3.1416. Why?
4.8. Investigate the behavior of the secant method on the function
f(x) = sign(x− a)
√
|x− a|.
4.9. Find the first ten positive values of x for which x = tan x.
4.10. (a) Compute the first ten zeros of J0(x). You can use our graph of J0(x) to
estimate their location.
(b) Compute the first ten zeros of Y0(x), the zeroth-order Bessel function of
the second kind.
(c) Compute all the values of x between 0 and 10π for which J0(x) = Y0(x).
(d) Make a composite plot showing J0(x) and Y0(x) for 0 ≤ x ≤ 10π, the
first ten zeros of both functions, and the points of intersection.
4.11. The gamma function is defined by an integral:
Γ(x + 1) =
∫ ∞
0
txe−tdt.
Integration by parts shows that, when evaluated at the integers, Γ(x) inter-
polates the factorial function
Γ(n + 1) = n!.
Exercises 21
Γ(x) and n! grow so rapidly that they generate floating-point overflow for
relatively small values of x and n. It is often more convenient to work with
the logarithms of these functions.
The Matlab functions gamma and gammaln compute Γ(x) and log Γ(x), re-
spectively. The quantity n! is easily computed by the expression
prod(1:n)
but many people expect there to be a function named factorial, so Matlab
has such a function.
(a) What is the largest value of n for which Γ(n + 1) and n! can be exactly
represented by a double-precision floating-point number?
(b) What is the largest value of n for which Γ(n + 1) and n! can be approxi-
mately represented by a double-precision floating-point number that does not
overflow?
4.12. Stirling’s approximation is a classical estimate for log Γ(x + 1):
log Γ(x + 1) ∼ x log(x)− x + 1
2
log(2πx).
Bill Gosper [4] has noted that a better approximation is
log Γ(x + 1) ∼ x log(x)− x + 1
2
log(2πx + π/3).
The accuracy of both approximations improves as x increases.
(a) What is the relative error in Stirling’s approximation and in Gosper’s
approximation when x = 2?
(b) How large must x be for Stirling’s approximation and for Gosper’s ap-
proximation to have a relative error less than 10−6?
4.13. The statements
y = 2:.01:10;
x = gammaln(y);
plot(x,y)
produce a graph of the inverse of the log Γ function.
(a) Write a Matlab function gammalninv that evaluates this function for
any x. That is, given x,
y = gammalninv(x)
computes y so that gammaln(y) is equal to x.
(b) What are the appropriate ranges of x and y for this function?
4.14. Here is a table of the distance, d, that a hypothetical vehicle requires to stop
if the brakes are applied when it is traveling with velocity v.
22 Chapter 4. Zeros and Roots
v(m/s) d(m)
0 0
10 5
20 20
30 46
40 70
50 102
60 153
What is the speed limit for this vehicle if it must be able to stop in at most
60 m? Compute the speed three different ways.
(a) piecewise linear interpolation,
(b) piecewise cubic interpolation with pchiptx,
(c) reverse piecewise cubic interpolation with pchiptx.
Because these are well-behaved data, the three values are close to each other,
but not identical.
4.15. Kepler’s model of planetary orbits includes a quantity E, the eccentricity
anomaly, that satisfies the equation
M = E − e sin E,
where M is the mean anomaly and e is the eccentricity of the orbit. For this
exercise, take M = 24.851090 and e = 0.1.
(a) Use fzerotx to solve for E. You can assign the appropriate values to M
and e and then use them in the definition of a function of E.
M = 24.851090
e = 0.1
F = @(E) E - e*sin(E) - M
Use F for the first argument to fzerotx.
(b) An “exact” formula for E is known:
E = M + 2
∞∑
m=1
1
m
Jm(me) sin(mM),
where Jm(x) is the Bessel function of the first kind of order m. Use this
formula, and besselj(m,x) in Matlab , to compute E. How many terms
are needed? How does this value of E compare to the value obtained with
fzerotx?
4.16. Utilities must avoid freezing water mains. If we assume uniform soil condi-
tions, the temperature T (x, t) at a distance x below the surface and time t
after the beginning of a cold snap is given approximately by
T (x, t)− Ts
Ti − Ts
= erf
(
x
2
√
αt
)
.
Here Ts is the constant surface temperature during the cold period, Ti is the
initial soil temperature before the cold snap, and α is the thermal conductivity
Exercises 23
of the soil. If x is measured in meters and t in seconds, then α = 0.138 ·
10−6 m2/s. Let Ti = 20◦C, and Ts = −15◦ C, and recall that water freezes at
0◦ C. Use fzerotx to determine how deep a water main should be buried so
that it will not freeze until at least 60 days’ exposure under these conditions.
4.17. Modify fmintx to provide printed and graphical output similar to that at the
end of section ??. Reproduce the results shown in Figure 4.8 for -humps(x).
4.18. Let f(x) = 9x2 − 6x + 2. What is the actual minimizer of f(x)? How close
to the actual minimizer can you get with fmintx? Why?
4.19. Theoretically, fmintx(@cos,2,4,eps) should return pi. How close does it
get? Why? On the other hand, fmintx(@cos,0,2*pi) does return pi. Why?
4.20. If you use tol = 0 with fmintx(@F,a,b,tol), does the iteration run forever?
Why or why not?
4.21. Derive the formulas for minimization by parabolic interpolation used in the
following portion of fmintx:
r = (x - w)*(fx - fv);
q = (x - v)*(fx - fw);
p = (x - v)*q - (x - w)*r;
s = 2.0*(q - r);
if s > 0.0, p = -p; end
s = abs(s);
% Is the parabola acceptable?
para = ( (abs(p)
Xβ ≈ y.
But this system is overdetermined—there are more equations than unknowns. So
we cannot expect to solve the system exactly. Instead, we solve it in the least
squares sense:
minβ‖Xβ − y‖.
A theoretical approach to solving the overdetermined system begins by mul-
tiplying both sides by XT . This reduces the system to a square, n-by-n system
known as the normal equations:
XT Xβ = XT y.
If there are thousands of observations and only a few parameters, the design matrix
X is quite large, but the matrix XT X is small. We have projected y onto the space
spanned by the columns of X. Continuing with this theoretical approach, if the
basis functions are independent, then XT X is nonsingular and
β = (XT X )−1XT y.
This formula for solving linear least squares problems appears in most text-
books on statistics and numerical methods. However, there are several undesirable
aspects to this theoretical approach. We have already seen that using a matrix
inverse to solve a system of equations is more work and less accurate than solving
the system by Gaussian elimination. But, more importantly, the normal equations
are always more badly conditioned than the original overdetermined system. In
fact, the condition number is squared:
κ(XT X ) = κ(X )2.
With finite-precision computation, the normal equations can actually become sin-
gular, and (XT X )−1 nonexistent, even though the columns of X are independent.
As an extreme example, consider the design matrix
X =
1 1
δ 0
0 δ
.
If δ is small, but nonzero, the two columns of X are nearly parallel but are still
linearly independent. The normal equations make the situation worse:
XT X =
(
1 + δ2 1
1 1 + δ2
)
.
5.5. The QR Factorization 9
If |δ| < 10−8, the matrix XT X computed with double-precision floating-point arith-
metic is exactly singular and the inverse required in the classic textbook formula
does not exist.
Matlab avoids the normal equations. The backslash operator not only solves
square, nonsingular systems, but it also computes the least squares solution to
rectangular, overdetermined systems:
β = X y.
Most of the computation is done by an orthogonalization algorithm known as
the QR factorization. The factorization is computed by the built-in function qr.
The NCM function qrsteps demonstrates the individual steps.
The two versions of the QR factorization are illustrated in Figure 5.3. Both
versions have
X = QR.
In the full version, R is the same size as X and Q is a square matrix with as many
rows as X. In the economy-sized version, Q is the same size as X and R is a square
matrix with as many columns as X. The letter “Q” is a substitute for the letter
“O” in “orthogonal” and the letter “R” is for “right” triangular matrix. The Gram–
Schmidt process described in many linear algebra texts is a related, but numerically
less satisfactory, algorithm that generates the same factorization.
X = Q R
X = Q R
Figure 5.3. Full and economy QR factorizations.
A sequence of Householder reflections is applied to the columns of X to pro-
duce the matrix R:
Hn · · ·H2H1X = R.
10 Chapter 5. Least Squares
The jth column of R is a linear combination of the first j columns of X. Conse-
quently, the elements of R below the main diagonal are zero.
If the same sequence of reflections is applied to the right-hand side, the equa-
tions
Xβ ≈ y
become
Rβ ≈ z,
where
Hn · · ·H2H1y = z.
The first n of these equations is a small, square, triangular system that can be
solved for β by back substitution with the subfunction backsubs in bslashtx. The
coefficients in the remaining m − n equations are all zero, so these equations are
independent of β and the corresponding components of z constitute the transformed
residual. This approach is preferable to the normal equations because Householder
reflections have impeccable numerical credentials and because the resulting trian-
gular system is ready for back substitution.
The matrix Q in the QR factorization is
Q = (Hn · · ·H2H1)T .
To solve least squares problems, we do not have to actually compute Q. In other
uses of the factorization, it may be convenient to have Q explicitly. If we compute
just the first n columns, we have the economy-sized factorization. If we compute
all m columns, we have the full factorization. In either case,
QT Q = I,
so Q has columns that are perpendicular to each other and have unit length. Such
a matrix is said to have orthonormal columns. For the full Q, it is also true that
QQT = I,
so the full Q is an orthogonal matrix.
Let’s illustrate this with a small version of the census example. We will fit the
last six observations with a quadratic:
y(s) ≈ β1s2 + β2s + β3.
The scaled time s = ((1950:10:2000)’ - 1950)/50 and the observations y are
s y
0.0000 150.6970
0.2000 179.3230
0.4000 203.2120
0.6000 226.5050
0.8000 249.6330
1.0000 281.4220
5.5. The QR Factorization 11
The design matrix is X = [s.*s s ones(size(s))].
0 0 1.0000
0.0400 0.2000 1.0000
0.1600 0.4000 1.0000
0.3600 0.6000 1.0000
0.6400 0.8000 1.0000
1.0000 1.0000 1.0000
The M-file qrsteps shows the steps in the QR factorization.
qrsteps(X,y)
The first step introduces zeros below the diagonal in the first column of X.
-1.2516 -1.4382 -1.7578
0 0.1540 0.9119
0 0.2161 0.6474
0 0.1863 0.2067
0 0.0646 -0.4102
0 -0.1491 -1.2035
The same Householder reflection is applied to y.
-449.3721
160.1447
126.4988
53.9004
-57.2197
-198.0353
Zeros are introduced in the second column.
-1.2516 -1.4382 -1.7578
0 -0.3627 -1.3010
0 0 -0.2781
0 0 -0.5911
0 0 -0.6867
0 0 -0.5649
The second Householder reflection is also applied to y.
-449.3721
-242.3136
-41.8356
-91.2045
-107.4973
-81.8878
Finally, zeros are introduced in the third column and the reflection applied to y.
This produces the triangular matrix R and a modified right-hand side z.
12 Chapter 5. Least Squares
R =
-1.2516 -1.4382 -1.7578
0 -0.3627 -1.3010
0 0 1.1034
0 0 0
0 0 0
0 0 0
z =
-449.3721
-242.3136
168.2334
-1.3202
-3.0801
4.0048
The system of equations Rβ = z is the same size as the original, 6 by 3. We
can solve the first three equations exactly (because R(1 : 3, 1 : 3) is nonsingular).
beta = R(1:3,1:3)z(1:3)
beta =
5.7013
121.1341
152.4745
This is the same solution beta that the backslash operator computes with
beta = Rz
or
beta = Xy
The last three equations in Rβ = z cannot be satisfied by any choice of β, so the
last three components of z represent the residual. In fact, the two quantities
norm(z(4:6))
norm(X*beta - y)
are both equal to 5.2219. Notice that even though we used the QR factorization,
we never actually computed Q.
The population in the year 2010 can be predicted by evaluating
β1s
2 + β2s + β3
at s = (2010− 1950)/50 = 1.2. This can be done with polyval.
p2010 = polyval(beta,1.2)
p2010 =
306.0453
Censusgui itself, fitting a quadratic to more data, predicts 311.5880. Which do
you think is going to be closer to the actual result of the 2010 census?
5.6. Pseudoinverse 13
5.6 Pseudoinverse
The definition of the pseudoinverse makes use of the Frobenius norm of a matrix:
‖A‖F =
( ∑
i
∑
j
a2i,j
)1/2
.
The Matlab expression norm(X,’fro’) computes the Frobenius norm. ‖A‖F is
the same as the 2-norm of the long vector formed from all the elements of A.
norm(A,’fro’) = norm(A(:))
The Moore–Penrose pseudoinverse generalizes and extends the usual matrix
inverse. The pseudoinverse is denoted by a dagger superscript,
Z = X†,
and computed by the Matlab pinv.
Z = pinv(X)
If X is square and nonsingular, then the pseudoinverse and the inverse are the
same:
X† = X−1.
If X is m by n with m > n and X has full rank, then its pseudoinverse is the matrix
involved in the normal equations:
X† = (XT X )−1XT .
The pseudoinverse has some, but not all, of the properties of the ordinary
inverse. X† is a left inverse because
X†X = (XT X )−1XT X = I
is the n-by-n identity. But X† is not a right inverse because
XX† = X(XT X )−1XT
only has rank n and so cannot be the m-by-m identity.
The pseudoinverse does get as close to a right inverse as possible in the sense
that, out of all the matrices Z that minimize
‖XZ − I‖F ,
Z = X† also minimizes
‖Z‖F .
It turns out that these minimization properties also define a unique pseudoinverse
even if X is rank deficient.
14 Chapter 5. Least Squares
Consider the 1-by-1 case. What is the inverse of a real (or complex) number
x? If x is not zero, then clearly x−1 = 1/x. But if x is zero, x−1 does not exist.
The pseudoinverse takes care of that because, in the scalar case, the unique number
that minimizes both
|xz − 1| and |z|
is
x† =
{
1/x : x 6= 0,
0 : x = 0.
The actual computation of the pseudoinverse involves the singular value de-
composition, which is described in a later chapter. You can edit pinv or type pinv
to see the code.
5.7 Rank Deficiency
If X is rank deficient, or has more columns than rows, the square matrix XT X is
singular and (XT X )−1 does not exist. The formula
β = (XT X )−1XT y
obtained from the normal equations breaks down completely.
In these degenerate situations, the least squares solution to the linear system
Xβ ≈ y
is not unique. A null vector of X is a nonzero solution to
Xη = 0.
Any multiple of any null vector can be added to β without changing how well Xβ
approximates y.
In Matlab, the solution to
Xβ ≈ y
can be computed with either backslash or the pseudoinverse, that is,
beta = Xy
or
beta = pinv(X)*y
In the full rank case, these two solutions are the same, although pinv does con-
siderably more computation to obtain it. But in degenerate situations these two
solutions are not the same.
The solution computed by backslash is called a basic solution. If r is the rank
of X, then at most r of the components of
beta = Xy
5.7. Rank Deficiency 15
are nonzero. Even the requirement of a basic solution does not guarantee unique-
ness. The particular basic computation obtained with backslash is determined by
details of the QR factorization.
The solution computed by pinv is the minimum norm solution. Out of all the
vectors β that minimize ‖Xβ − y‖, the vector
beta = pinv(X)*y
also minimizes ‖β‖. This minimum norm solution is unique.
For example, let
X =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
and
y =
16
17
18
19
20
.
The matrix X is rank deficient. The middle column is the average of the first and
last columns. The vector
η =
1
−2
1
is a null vector.
The statement
beta = Xy
produces a warning,
Warning: Rank deficient, rank = 2 tol = 2.4701e-014.
and the solution
beta =
-7.5000
0
7.8333
As promised, this solution is basic; it has only two nonzero components. However,
the vectors
beta =
0
-15.0000
15.3333
16 Chapter 5. Least Squares
and
beta =
-15.3333
15.6667
0
are also basic solutions.
The statement
beta = pinv(X)*y
produces the solution
beta =
-7.5556
0.1111
7.7778
without giving a warning about rank deficiency. The norm of the pseudoinverse
solution
norm(pinv(X)*y) = 10.8440
is slightly less than the norm of the backslash solution
norm(Xy) = 10.8449
Out of all the vectors β that minimize ‖Xβ − y‖, the pseudoinverse has found the
shortest. Notice that the difference between the two solutions,
Xy – pinv(X)*y =
0.0556
-0.1111
0.0556
is a multiple of the null vector η.
If handled with care, rank deficient least squares problems can be solved in
a satisfactory manner. Problems that are nearly, but not exactly, rank deficient
are more difficult to handle. The situation is similar to square linear systems that
are badly conditioned, but not exactly singular. Such problems are not well posed
numerically. Small changes in the data can lead to large changes in the computed
solution. The algorithms used by both backslash and pseudoinverse involve deci-
sions about linear independence and rank. These decisions use somewhat arbitrary
tolerances and are particularly susceptible to both errors in the data and roundoff
errors in the computation.
Which is “better,” backslash or pseudoinverse? In some situations, the under-
lying criteria of basic solution or minimum norm solution may be relevant. But most
problem formulations, particularly those involving curve fitting, do not include such
subtle distinctions. The important fact to remember is that the computed solutions
are not unique and are not well determined by the data.
5.8. Separable Least Squares 17
5.8 Separable Least Squares
Matlab provides several functions for solving nonlinear least squares problems.
Older versions of Matlab have one general-purpose, multidimensional, nonlinear
minimizer, fmins. In more recent versions of Matlab, fmins has been updated
and renamed fminsearch. The Optimization Toolbox provides additional capabil-
ities, including a minimizer for problems with constraints, fmincon; a minimizer
for unconstrained problems, fminunc; and two functions intended specifically for
nonlinear least squares, lsqnonlin and lsqcurvefit. The Curve Fitting Toolbox
provides a GUI to facilitate the solution of many different linear and nonlinear
fitting problems.
In this introduction, we focus on the use of fminsearch. This function uses a
direct search technique known as the Nelder–Meade algorithm. It does not attempt
to approximate any gradients or other partial derivatives. It is quite effective on
small problems involving only a few variables. Larger problems with more vari-
ables are better handled by the functions in the Optimization and Curve Fitting
Toolboxes.
Separable least squares curve-fitting problems involve both linear and nonlin-
ear parameters. We could ignore the linear portion and use fminsearch to search
for all the parameters. But if we take advantage of the separable structure, we ob-
tain a more efficient, robust technique. With this approach, fminsearch is used to
search for values of the nonlinear parameters that minimize the norm of the resid-
ual. At each step of the search process, the backslash operator is used to compute
values of the linear parameters.
Two blocks of Matlab code are required. One block can be a function,
a script, or a few lines typed directly in the Command Window. It sets up the
problem, establishes starting values for the nonlinear parameters, calls fminsearch,
processes the results, and usually produces a plot. The second block is the objective
function that is called by fminsearch. This function is given a vector of values of
the nonlinear parameters, alpha. It should compute the design matrix X for these
parameters, use backslash with X and the observations to compute values of the
linear parameters beta, and return the resulting residual norm.
Let’s illustrate all this with expfitdemo, which involves observations of ra-
dioactive decay. The task is to model the decay by a sum of two exponential terms
with unknown rates λj :
y ≈ β1e−λ1t + β2e−λ2t.
Consequently, in this example, there are two linear parameters and two nonlinear
parameters. The demo plots the various fits that are generated during the nonlinear
minimization process. Figure 5.4 shows plots of both the data and the final fit.
The main function begins by specifying 21 observations, t and y.
function expfitdemo
t = (0:.1:2)’;
y = [5.8955 3.5639 2.5173 1.9790 1.8990 1.3938 1.1359 …
1.0096 1.0343 0.8435 0.6856 0.6100 0.5392 0.3946 …
0.3903 0.5474 0.3459 0.1370 0.2211 0.1704 0.2636]’;
18 Chapter 5. Least Squares
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
1
2
3
4
5
6
1.4003 10.5865
Figure 5.4. expfitdemo.
The initial plot uses o’s for the observations, creates an all-zero placeholder
for what is going to become the evolving fit, and creates a title that will show the
values of lambda. The variable h holds the handles for these three graphics objects.
clf
shg
set(gcf,’doublebuffer’,’on’)
h = plot(t,y,’o’,t,0*t,’-’);
h(3) = title(’’);
axis([0 2 0 6.5])
The vector lambda0 specifies initial values for the nonlinear parameters. In this
example, almost any choice of initial values leads to convergence, but in other
situations, particularly with more nonlinear parameters, the choice of initial values
can be much more important. The call to fminsearch does most of the work.
The observations t and y, as well as the graphics handle h, are passed as extra
parameters.
lambda0 = [3 6]’;
lambda = fminsearch(@expfitfun,lambda0,[],t,y,h)
set(h(2),’color’,’black’)
The objective function is named expfitfun. It can handle n exponential
basis functions; we will be using n = 2. The first input parameter is a vector
5.9. Further Reading 19
provided by fminsearch that contains values of the n decay rates, λj . The other
parameters are vectors containing the independent and dependent variables, t and y,
and the graphics handle. The function computes the design matrix, uses backslash
to compute β, evaluates the resulting model, and returns the norm of the residual.
function res = expfitfun(lambda,t,y,h)
m = length(t);
n = length(lambda);
X = zeros(m,n);
for j = 1:n
X(:,j) = exp(-lambda(j)*t);
end
beta = Xy;
z = X*beta;
res = norm(z-y);
The objective function also updates the plot of the fit and the title and pauses long
enough for us to see the progress of the computation.
set(h(2),’ydata’,z);
set(h(3),’string’,sprintf(’%8.4f %8.4f’,lambda))
pause(.1)
5.9 Further Reading
The reference books on matrix computation [4, 5, 6, 7, 8, 9] discuss least squares.
An additional reference is Björck [1].
Exercises
5.1. Let X be the n-by-n matrix generated by
[I,J] = ndgrid(1:n);
X = min(I,J) + 2*eye(n,n) – 2;
(a) How does the condition number of X grow with n?
(b) Which, if any, of the triangular factorizations chol(X), lu(X), and qr(X)
reveal the poor conditioning?
5.2. In censusgui, change the 1950 population from 150.697 million to 50.697
million. This produces an extreme outlier in the data. Which models are the
most affected by this outlier? Which models are the least affected?
5.3. If censusgui is used to fit the U.S. Census data with a polynomial of degree
eight and the fit is extrapolated beyond the year 2000, the predicted popula-
tion actually becomes zero before the year 2020. On what year, month, and
day does that fateful event occur?
20 Chapter 5. Least Squares
5.4. Here are some details that we skipped over in our discussion of Householder
reflections. At the same time, we extend the description to include complex
matrices. The notation uT for transpose is replaced with the Matlab nota-
tion u′ for complex conjugate transpose. Let x be any nonzero m-by-1 vector
and let ek denote the kth unit vector, that is, the kth column of the m-by-m
identity matrix. The sign of a complex number z = reiθ is
sign(z) = z/|z| = eiθ.
Define σ by
σ = sign(xk)‖x‖.
Let
u = x + σek.
In other words, u is obtained from x by adding σ to its kth component.
(a) The definition of ρ uses σ̄, the complex conjugate of σ:
ρ = 1/(σ̄uk).
Show that
ρ = 2/‖u‖2.
(b) The Householder reflection generated by the vector x is
H = I − ρuu′.
Show that
H ′ = H
and that
H ′H = I.
(c) Show that all the components of Hx are zero, except for the kth. In other
words, show that
Hx = −σek.
(d) For any vector y, let
τ = ρu′y.
Show that
Hy = y − τu.
5.5. Let
x =
9
2
6
.
(a) Find the Householder reflection H that transforms x into
Hx =
−11
0
0
.
Exercises 21
(b) Find nonzero vectors u and v that satisfy
Hu = −u,
Hv = v.
5.6. Let
X =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
.
(a) Verify that X is rank deficient.
Consider three choices for the pseudoinverse of X.
Z = pinv(X) % The actual pseudoinverse
B = Xeye(5,5) % Backslash
S = eye(3,3)/X % Slash
(b) Compare the values of
‖Z‖F , ‖B‖F , and ‖S‖F ;
‖XZ − I‖F , ‖XB − I‖F , and ‖XS − I‖F ;
‖ZX − I‖F , ‖BX − I‖F , and ‖SX − I‖F .
Verify that the values obtained with Z are less than or equal to the values
obtained with the other two choices. Actually minimizing these quantities is
one way of characterizing the pseudoinverse.
(c) Verify that Z satisfies all four of the following conditions, and that B and
S fail to satisfy at least one of the conditions. These conditions are known as
the Moore–Penrose equations and are another way to characterize a unique
pseudoinverse.
XZ is symmetric.
ZX is symmetric.
XZX = X.
ZXZ = Z.
5.7. Generate 11 data points, tk = (k − 1)/10, yk = erf(tk), k = 1, . . . , 11.
(a) Fit the data in a least squares sense with polynomials of degrees 1 through
10. Compare the fitted polynomial with erf(t) for values of t between the data
points. How does the maximum error depend on the polynomial degree?
(b) Because erf(t) is an odd function of t, that is, erf(x) = −erf(−x), it is
reasonable to fit the data by a linear combination of odd powers of t:
erf(t) ≈ c1t + c2t3 + · · ·+ cnt2n−1.
Again, see how the error between data points depends on n.
22 Chapter 5. Least Squares
(c) Polynomials are not particularly good approximants for erf(t) because
they are unbounded for large t, whereas erf(t) approaches 1 for large t. So,
using the same data points, fit a model of the form
erf(t) ≈ c1 + e−t
2
(c2 + c3z + c4z
2 + c5z
3),
where z = 1/(1 + t). How does the error between the data points compare
with the polynomial models?
5.8. Here are 25 observations, yk, taken at equally spaced values of t.
t = 1:25
y = [ 5.0291 6.5099 5.3666 4.1272 4.2948
6.1261 12.5140 10.0502 9.1614 7.5677
7.2920 10.0357 11.0708 13.4045 12.8415
11.9666 11.0765 11.7774 14.5701 17.0440
17.0398 15.9069 15.4850 15.5112 17.6572]
y = y’;
y = y(:);
(a) Fit the data with a straight line, y(t) = β1 + β2t, and plot the residuals,
y(tk)− yk. You should observe that one of the data points has a much larger
residual than the others. This is probably an outlier.
(b) Discard the outlier, and fit the data again by a straight line. Plot the
residuals again. Do you see any pattern in the residuals?
(c) Fit the data, with the outlier excluded, by a model of the form
y(t) = β1 + β2t + β3 sin t.
(d) Evaluate the third fit on a finer grid over the interval [0, 26]. Plot the
fitted curve, using line style ’-’, together with the data, using line style ’o’.
Include the outlier, using a different marker, ’*’.
5.9. Statistical Reference Datasets. NIST, the National Institute of Standards and
Technology, is the branch of the U.S. Department of Commerce responsible
for setting national and international standards. NIST maintains Statistical
Reference Datasets, StRD, for use in testing and certifying statistical soft-
ware. The home page on the Web is [3]. Data sets for linear least squares are
under “Linear Regression.” This exercise involves two of the NIST reference
data sets:
• Norris: linear polynomial for calibration of ozone monitors;
• Pontius: quadratic polynomial for calibration of load cells.
For each of these data sets, follow the Web links labeled
• Data File (ASCII Format),
• Certified Values, and
• Graphics.
Exercises 23
Download each ASCII file. Extract the observations. Compute the polyno-
mial coefficients. Compare the coefficients with the certified values. Make
plots similar to the NIST plots of both the fit and the residuals.
5.10. Filip data set. One of the Statistical Reference Datasets from the NIST is the
“Filip” data set. The data consist of several dozen observations of a variable
y at different values of x. The task is to model y by a polynomial of degree
10 in x.
This data set is controversial. A search of the Web for “filip strd” will find
several dozen postings, including the original page at NIST [3]. Some math-
ematical and statistical packages are able to reproduce the polynomial coef-
ficients that NIST has decreed to be the “certified values.” Other packages
give warning or error messages that the problem is too badly conditioned to
solve. A few packages give different coefficients without warning. The Web
offers several opinions about whether or not this is a reasonable problem.
Let’s see what Matlab does with it.
The data set is available from the NIST Web site. There is one line for each
data point. The data are given with the first number on the line a value
of y, and the second number the corresponding x. The x-values are not
monotonically ordered, but it is not necessary to sort them. Let n be the
number of data points and p = 11 the number of polynomial coefficients.
(a) As your first experiment, load the data into Matlab, plot it with ‘.’
as the line type, and then invoke the Basic Fitting tool available under the
Tools menu on the figure window. Select the 10th-degree polynomial fit.
You will be warned that the polynomial is badly conditioned, but ignore that
for now. How do the computed coefficients compare with the certified values
on the NIST Web page? How does the plotted fit compare with the graphic
on the NIST Web page? The basic fitting tool also displays the norm of the
residuals, ‖r‖. Compare this with the NIST quantity “Residual Standard
Deviation,” which is
‖r‖√
n− p .
(b) Examine this data set more carefully by using six different methods to
compute the polynomial fit. Explain all the warning messages you receive
during these computations.
• Polyfit: Use polyfit(x,y,10).
• Backslash: Use Xy, where X is the n-by-p truncated Vandermonde
matrix with elements
Xi,j = x
p−j
i , i = 1, . . . , n, j = 1, . . . , p.
• Pseudoinverse: Use pinv(X)*y.
• Normal equations: Use inv(X’*X)*X’*y.
• Centering: Let µ = mean(x), σ = std(x), t = (x− µ)/σ.
Use polyfit(t,y,10).
24 Chapter 5. Least Squares
• Certified coefficients: Obtain the coefficients from the NIST Web page.
(c) What are the norms of the residuals for the fits computed by the six
different methods?
(d) Which one of the six methods gives a very poor fit? (Perhaps the packages
that are criticized on the Web for reporting bad results are using this method.)
(e) Plot the five good fits. Use dots, ’.’, at the data values and curves
obtained by evaluating the polynomials at a few hundred points over the
range of the x’s. The plot should look like Figure 5.5. There are five different
plots, but only two visually distinct ones. Which methods produce which
plots?
(f) Why do polyfit and backslash give different results?
−9 −8 −7 −6 −5 −4 −3
0.76
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
NIST Filip data set
Data
Rank 11
Rank 10
Figure 5.5. NIST Filip standard reference data set.
5.11. Longley data set. The Longley data set of labor statistics was one of the first
used to test the accuracy of least squares computations. You don’t need to
go to the NIST Web site to do this problem, but if you are interested in the
background, you should see the Longley page at [3]. The data set is available
in NCM in the file longley.dat. You can bring the data into Matlab with
load longley.dat
y = longley(:,1);
X = longley(:,2:7);
There are 16 observations of 7 variables, gathered over the years 1947 to 1962.
The variable y and the 6 variables making up the columns of the data matrix
Exercises 25
X are
y = Total Derived Employment,
x1 = GNP Implicit Price Deflater,
x2 = Gross National Product,
x3 = Unemployment,
x4 = Size of Armed Forces,
x5 = Noninstitutional Population Age 14 and Over,
x6 = Year.
The objective is to predict y by a linear combination of a constant and the
six x’s:
y ≈ β0 +
6∑
1
βkxk.
(a) Use the Matlab backslash operator to compute β0, β1, . . . , β6. This
involves augmenting X with a column of all 1’s, corresponding to the constant
term.
(b) Compare your β’s with the certified values [3].
(c) Use errorbar to plot y with error bars whose magnitude is the difference
between y and the least squares fit.
(d) Use corrcoef to compute the correlation coefficients for X without the
column of 1’s. Which variables are highly correlated?
(e) Normalize the vector y so that its mean is zero and its standard deviation
is one. You can do this with
y = y – mean(y);
y = y/std(y)
Do the same thing to the columns of X. Now plot all seven normalized vari-
ables on the same axis. Include a legend.
5.12. Planetary orbit [2]. The expression z = ax2+bxy+cy2+dx+ey+f is known
as a quadratic form. The set of points (x, y), where z = 0, is a conic section.
It can be an ellipse, a parabola, or a hyperbola, depending on the sign of
the discriminant b2 − 4ac. Circles and lines are special cases. The equation
z = 0 can be normalized by dividing the quadratic form by any nonzero
coefficient. For example, if f 6= 0, we can divide all the other coefficients by
f and obtain a quadratic form with the constant term equal to one. You can
use the Matlab meshgrid and contour functions to plot conic sections. Use
meshgrid to create arrays X and Y. Evaluate the quadratic form to produce
Z. Then use contour to plot the set of points where Z is zero.
[X,Y] = meshgrid(xmin:deltax:xmax,ymin:deltay:ymax);
Z = a*X.^2 + b*X.*Y + c*Y.^2 + d*X + e*Y + f;
contour(X,Y,Z,[0 0])
A planet follows an elliptical orbit. Here are ten observations of its position
in the (x, y) plane:
26 Chapter 5. Least Squares
x = [1.02 .95 .87 .77 .67 .56 .44 .30 .16 .01]’;
y = [0.39 .32 .27 .22 .18 .15 .13 .12 .13 .15]’;
(a) Determine the coefficients in the quadratic form that fits these data in
the least squares sense by setting one of the coefficients equal to one and
solving a 10-by-5 overdetermined system of linear equations for the other
five coefficients. Plot the orbit with x on the x-axis and y on the y-axis.
Superimpose the ten data points on the plot.
(b) This least squares problem is nearly rank deficient. To see what effect this
has on the solution, perturb the data slightly by adding to each coordinate
of each data point a random number uniformly distributed in the interval
[−.0005, .0005]. Compute the new coefficients resulting from the perturbed
data. Plot the new orbit on the same plot with the old orbit. Comment on
your comparison of the sets of coefficients and the orbits.
Bibliography
[1] A. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadel-
phia, 1996.
[2] M. T. Heath, Scientific Computing: An Introductory Survey, McGraw–Hill,
New York, 1997.
[3] National Institute of Standards and Technology, Statistical Reference
Datasets.
http://www.itl.nist.gov/div898/strd
http://www.itl.nist.gov/div898/strd/lls/lls.shtml
http://www.itl.nist.gov/div898/strd/lls/data/Longley.shtml
[4] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J.
Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney,
and D. Sorensen, LAPACK Users’ Guide, Third Edition, SIAM, Philadelphia,
1999.
http://www.netlib.org/lapack
[5] J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
[6] G. H. Golub and C. F. Van Loan, Matrix Computations, Third Edition,
The Johns Hopkins University Press, Baltimore, 1996.
[7] G. W. Stewart, Introduction to Matrix Computations, Academic Press, New
York, 1973.
[8] G. W. Stewart, Matrix Algorithms: Basic Decompositions, SIAM, Philadel-
phia, 1998.
[9] L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM,
Philadelphia, 1997.
27
Chapter 6
Quadrature
The term numerical integration covers several different tasks, including numerical
evaluation of integrals and numerical solution of ordinary differential equations. So
we use the somewhat old-fashioned term quadrature for the simplest of these, the
numerical evaluation of a definite integral. Modern quadrature algorithms auto-
matically vary an adaptive step size.
6.1 Adaptive Quadrature
Let f(x) be a real-valued function of a real variable, defined on a finite interval
a ≤ x ≤ b. We seek to compute the value of the integral,
∫ b
a
f(x)dx.
The word “quadrature” reminds us of an elementary technique for finding this
area—plot the function on graph paper and count the number of little squares that
lie underneath the curve.
In Figure 6.1, there are 148 little squares underneath the curve. If the area
of one little square is 3/512, then a rough estimate of the integral is 148× 3/512 =
0.8672.
Adaptive quadrature involves careful selection of the points where f(x) is sam-
pled. We want to evaluate the function at as few points as possible while approx-
imating the integral to within some specified accuracy. A fundamental additive
property of a definite integral is the basis for adaptive quadrature. If c is any point
between a and b, then
∫ b
a
f(x)dx =
∫ c
a
f(x)dx +
∫ b
c
f(x)dx.
The idea is that if we can approximate each of the two integrals on the right to
within a specified tolerance, then the sum gives us the desired result. If not, we
February 15, 2008
1
2 Chapter 6. Quadrature
Figure 6.1. Quadrature.
can recursively apply the additive property to each of the intervals [a, c] and [c, b].
The resulting algorithm will adapt to the integrand automatically, partitioning the
interval into subintervals with fine spacing where the integrand is varying rapidly
and coarse spacing where the integrand is varying slowly.
6.2 Basic Quadrature Rules
The derivation of the quadrature rule used by our Matlab function begins with
two of the basic quadrature rules shown in Figure 6.2: the midpoint rule and the
trapezoid rule. Let h = b− a be the length of the interval. The midpoint rule, M ,
approximates the integral by the area of a rectangle whose base has length h and
whose height is the value of the integrand at the midpoint:
M = hf
(
a + b
2
)
.
The trapezoid rule, T , approximates the integral by the area of a trapezoid with
base h and sides equal to the values of the integrand at the two endpoints:
T = h
f(a) + f(b)
2
.
The accuracy of a quadrature rule can be predicted in part by examining its
behavior on polynomials. The order of a quadrature rule is the degree of the lowest
degree polynomial that the rule does not integrate exactly. If a quadrature rule of
order p is used to integrate a smooth function over a small interval of length h, then
a Taylor series analysis shows that the error is proportional to hp. The midpoint
6.2. Basic Quadrature Rules 3
Midpoint rule Trapezoid rule
Simpson’s rule Composite Simpson’s rule
Figure 6.2. Four quadrature rules.
rule and the trapezoid rule are both exact for constant and linear functions of x,
but neither of them is exact for a quadratic in x, so they both have order two. (The
order of a rectangle rule with height f(a) or f(b) instead of the midpoint is only
one.)
The accuracy of the two rules can be compared by examining their behavior
on the simple integral ∫ 1
0
x2dx =
1
3
.
The midpoint rule gives
M = 1
(
1
2
)2
=
1
4
.
The trapezoid rule gives
T = 1
(
0 + 12
2
)
=
1
2
.
So the error in M is 1/12, while the error in T is −1/6. The errors have oppo-
site signs and, perhaps surprisingly, the midpoint rule is twice as accurate as the
trapezoid rule.
4 Chapter 6. Quadrature
This turns out to be true more generally. For integrating smooth functions
over short intervals, M is roughly twice as accurate as T and the errors have opposite
signs. Knowing these error estimates allows us to combine the two and get a rule
that is usually more accurate than either one separately. If the error in T were
exactly −2 times the error in M , then solving
S − T = −2(S −M)
for S would give us the exact value of the integral. In any case, the solution
S =
2
3
M +
1
3
T
is usually a more accurate approximation than either M or T alone. This rule
is known as Simpson’s rule. It can also be derived by integrating the quadratic
function that interpolates the integrand at the two endpoints, a and b, and the
midpoint, c = (a + b)/2:
S =
h
6
(f(a) + 4f(c) + f(b)).
It turns out that S also integrates cubics exactly, but not quartics, so its order
is four.
We can carry this process one step further using the two halves of the interval,
[a, c] and [c, b]. Let d and e be the midpoints of these two subintervals: d = (a+c)/2
and e = (c+ b)/2. Apply Simpson’s rule to each subinterval to obtain a quadrature
rule over [a, b]:
S2 =
h
12
(f(a) + 4f(d) + 2f(c) + 4f(e) + f(b)).
This is an example of a composite quadrature rule. See Figure 6.2.
S and S2 approximate the same integral, so their difference can be used as an
estimate of the error:
E = (S2 − S ).
Moreover, the two can be combined to get an even more accurate approximation,
Q. Both rules are of order four, but the S2 step size is half the S step size, so S2 is
roughly 24 times as accurate. Thus, Q is obtained by solving
Q− S = 16(Q− S2).
The result is
Q = S2 + (S2 − S )/15.
Exercise 6.2 asks you to express Q as a weighted combination of the five function
values f(a) through f(e) and to establish that its order is six. The rule is known
as Weddle’s rule, the sixth-order Newton–Cotes rule, and also as the first step of
Romberg integration. We will simply call it the extrapolated Simpson’s rule because
it uses Simpson’s rule for two different values of h and then extrapolates toward
h = 0.
6.3. quadtx, quadgui 5
6.3 quadtx, quadgui
The Matlab function quad uses the extrapolated Simpson’s rule in an adaptive
recursive algorithm. Our textbook function quadtx is a simplified version of quad.
The function quadgui provides a graphical demonstration of the behavior of
quad and quadtx. It produces a dynamic plot of the function values selected by the
adaptive algorithm. The count of function evaluations is shown in the title position
on the plot.
The initial portion of quadtx evaluates the integrand f(x) three times to
give the first, unextrapolated, Simpson’s rule estimate. A recursive subfunction,
quadtxstep, is then called to complete the computation.
function [Q,fcount] = quadtx(F,a,b,tol,varargin)
%QUADTX Evaluate definite integral numerically.
% Q = QUADTX(F,A,B) approximates the integral of F(x)
% from A to B to within a tolerance of 1.e-6.
%
% Q = QUADTX(F,A,B,tol) uses tol instead of 1.e-6.
%
% The first argument, F, is a function handle or
% an anonymous function that defines F(x).
%
% Arguments beyond the first four,
% Q = QUADTX(F,a,b,tol,p1,p2,…), are passed on to the
% integrand, F(x,p1,p2,..).
%
% [Q,fcount] = QUADTX(F,…) also counts the number of
% evaluations of F(x).
%
% See also QUAD, QUADL, DBLQUAD, QUADGUI.
% Default tolerance
if nargin < 4 | isempty(tol)
tol = 1.e-6;
end
% Initialization
c = (a + b)/2;
fa = F(a,varargin{:});
fc = F(c,varargin{:});
fb = F(b,varargin{:});
% Recursive call
[Q,k] = quadtxstep(F, a, b, tol, fa, fc, fb, varargin{:});
fcount = k + 3;
6 Chapter 6. Quadrature
Each recursive call of quadtxstep combines three previously computed func-
tion values with two more to obtain the two Simpson’s approximations for a par-
ticular interval. If their difference is small enough, they are combined to return the
extrapolated approximation for that interval. If their difference is larger than the
tolerance, the recursion proceeds on each of the two half intervals.
function [Q,fcount] = quadtxstep(F,a,b,tol,fa,fc,fb,varargin)
% Recursive subfunction used by quadtx.
h = b - a;
c = (a + b)/2;
fd = F((a+c)/2,varargin{:});
fe = F((c+b)/2,varargin{:});
Q1 = h/6 * (fa + 4*fc + fb);
Q2 = h/12 * (fa + 4*fd + 2*fc + 4*fe + fb);
if abs(Q2 - Q1) <= tol
Q = Q2 + (Q2 - Q1)/15;
fcount = 2;
else
[Qa,ka] = quadtxstep(F, a, c, tol, fa, fd, fc, varargin{:});
[Qb,kb] = quadtxstep(F, c, b, tol, fc, fe, fb, varargin{:});
Q = Qa + Qb;
fcount = ka + kb + 2;
end
The choice of tolerance for comparison with the error estimates is important,
but a little tricky. If a tolerance is not specified as the fourth argument to the
function, then 10−6 is used as the default.
The tricky part is how to specify the tolerance in the recursive calls. How
small does the tolerance in each recursive call have to be in order for the final result
to have the desired accuracy? One approach would cut the tolerance in half with
each level in the recursion. The idea is that if both Qa and Qb have errors less than
tol/2, then their sum certainly has an error less than tol. If we did this, the two
statements
[Qa,ka] = quadtxstep(F, a, c, tol, fa, fd, fc, varargin{:});
[Qb,kb] = quadtxstep(F, c, b, tol, fc, fe, fb, varargin{:});
would have tol/2 in place of tol.
However, this approach is too conservative. We are estimating the error in
the two separate Simpson’s rules, not their extrapolated combination. So the actual
error is almost always much less than the estimate. More importantly, the subinter-
vals where the actual error is close to the estimate are usually fairly rare. We can
allow one of the two recursive calls to have an error close to the tolerance because
the other subinterval will probably have a much smaller error. For these reasons,
the same value of tol is used in each recursive call.
6.4. Specifying Integrands 7
Our textbook function does have one serious defect: there is no provision for
failure. It is possible to try to evaluate integrals that do not exist. For example,
∫ 1
0
1
3x− 1dx
has a nonintegrable singularity. Attempting to evaluate this integral with quadtx
results in a computation that runs for a long time and eventually terminates with
an error message about the maximum recursion limit. It would be better to have
diagnostic information about the singularity.
6.4 Specifying Integrands
Matlab has several different ways of specifying the function to be integrated by
a quadrature routine. The anonymous function facility is convenient for a simple,
one-line formula. For example,
∫ 1
0
1√
1 + x4
dx
can be computed with the statements
f = @(x) 1./sqrt(1+x^4)
Q = quadtx(f,0,1)
If we want to compute ∫ π
0
sinx
x
dx,
we could try
f = @(x) sin(x)./x
Q = quadtx(f,0,pi)
Unfortunately, this results in a division by zero message when f(0) is evaluated
and, eventually, a recursion limit error. One remedy is to change the lower limit of
integration from 0 to the smallest positive floating-point number, realmin.
Q = quadtx(f,realmin,pi)
The error made by changing the lower limit is many orders of magnitude smaller
than roundoff error because the integrand is bounded by one and the length of the
omitted interval is less than 10−300.
Another remedy is to use an M-file instead of an anonymous function. Create
a file named sinc.m that contains the text
function f = sinc(x)
if x == 0
f = 1;
else
f = sin(x)/x;
end
8 Chapter 6. Quadrature
Then the statement
Q = quadtx(@sinc,0,pi)
uses a function handle and computes the integral with no difficulty.
Integrals that depend on parameters are encountered frequently. An example
is the beta function, defined by
β(z, w) =
∫ 1
0
tz−1(1− t)w−1dt.
Matlab already has a beta function, but we can use this example to illustrate how
to handle parameters. Create an anonymous function with three arguments.
F = @(t,z,w) t^(z-1)*(1-t)^(w-1)
Or use an M-file with a name like betaf.m.
function f = betaf(t,z,w)
f = t^(z-1)*(1-t)^(w-1)
As with all functions, the order of the arguments is important. The functions used
with quadrature routines must have the variable of integration as the first argument.
Values for the parameters are then given as extra arguments to quadtx. To compute
β(8/3, 10/3), you should set
z = 8/3;
w = 10/3;
tol = 1.e-6;
and then use
Q = quadtx(F,0,1,tol,z,w);
or
Q = quadtx(@betaf,0,1,tol,z,w);
The function functions in Matlab itself usually expect the first argument to
be in vectorized form. This means, for example, that the mathematical expression
sin x
1 + x2
should be specified with Matlab array notation.
sin(x)./(1 + x.^2)
Without the two dots,
sin(x)/(1 + x^2)
6.5. Performance 9
calls for linear algebraic vector operations that are not appropriate here. The Mat-
lab function vectorize transforms a scalar expression into something that can be
used as an argument to function functions.
Many of the function functions in Matlab require the specification of an
interval of the x-axis. Mathematically, we have two possible notations, a ≤ x ≤ b
or [a, b]. With Matlab, we also have two possibilities. The endpoints can be given
as two separate arguments, a and b, or can be combined into one vector argument,
[a,b]. The quadrature functions quad and quadl use two separate arguments. The
zero finder, fzero, uses a single argument because either a single starting point or
a two-element vector can specify the interval. The ordinary differential equation
solvers that we encounter in the next chapter also use a single argument because a
many-element vector can specify a set of points where the solution is to be evaluated.
The easy plotting function, ezplot, accepts either one or two arguments.
6.5 Performance
The Matlab demos directory includes a function named humps that is intended
to illustrate the behavior of graphics, quadrature, and zero-finding routines. The
function is
h(x) =
1
(x− 0.3)2 + 0.01 +
1
(x− 0.9)2 + 0.04 .
The statement
ezplot(@humps,0,1)
produces a graph of h(x) for 0 ≤ x ≤ 1. There is a fairly strong peak near x = 0.3
and a more modest peak near x = 0.9.
The default problem for quadgui is
quadgui(@humps,0,1,1.e-4)
You can see in Figure 6.3 that with this tolerance, the adaptive algorithm has
evaluated the integrand 93 times at points clustered near the two peaks.
With the Symbolic Toolbox, it is possible to analytically integrate h(x). The
statements
syms x
h = 1/((x-.3)^2+.01) + 1/((x-.9)^2+.04) - 6
I = int(h)
produce the indefinite integral
I = 10*atan(10*x-3)+5*atan(5*x-9/2)-6*x
The statements
D = simple(int(h,0,1))
Qexact = double(D)
produce a definite integral
10 Chapter 6. Quadrature
10
20
30
40
50
60
70
80
90
100
93
Figure 6.3. Adaptive quadrature.
D = 5*atan(16/13)+10*pi-6
and its floating-point numerical value
Qexact = 29.85832539549867
The effort required by a quadrature routine to approximate an integral within
a specified accuracy can be measured by counting the number of times the integrand
is evaluated. Here is one experiment involving humps and quadtx.
for k = 1:12
tol = 10^(-k);
[Q,fcount] = quadtx(@humps,0,1,tol);
err = Q - Qexact;
ratio = err/tol;
fprintf(’%8.0e %21.14f %7d %13.3e %9.3f
’, ...
tol,Q,fcount,err,ratio)
end
The results are
tol Q fcount err err/tol
1.e-01 29.83328444174863 25 -2.504e-02 -0.250
1.e-02 29.85791444629948 41 -4.109e-04 -0.041
6.6. Integrating Discrete Data 11
1.e-03 29.85834299237636 69 1.760e-05 0.018
1.e-04 29.85832444437543 93 -9.511e-07 -0.010
1.e-05 29.85832551548643 149 1.200e-07 0.012
1.e-06 29.85832540194041 265 6.442e-09 0.006
1.e-07 29.85832539499819 369 -5.005e-10 -0.005
1.e-08 29.85832539552631 605 2.763e-11 0.003
1.e-09 29.85832539549603 1061 -2.640e-12 -0.003
1.e-10 29.85832539549890 1469 2.274e-13 0.002
1.e-11 29.85832539549866 2429 -7.105e-15 -0.001
1.e-12 29.85832539549867 4245 0.000e+00 0.000
We see that as the tolerance is decreased, the number of function evaluations in-
creases and the error decreases. The error is always less than the tolerance, usually
by a considerable factor.
6.6 Integrating Discrete Data
So far, this chapter has been concerned with computing an approximation to the
definite integral of a specified function. We have assumed the existence of a Mat-
lab program that can evaluate the integrand at any point in a given interval.
However, in many situations, the function is only known at a finite set of points,
say (xk, yk), k = 1, . . . , n. Assume the x’s are sorted in increasing order, with
a = x1 < x2 < · · · < xn = b.
How can we approximate the integral
∫ b
a
f(x)dx?
Since it is not possible to evaluate y = f(x) at any other points, the adaptive
methods we have described are not applicable.
The most obvious approach is to integrate the piecewise linear function that
interpolates the data. This leads to the composite trapezoid rule
T =
n−1∑
k=1
hk
yk+1 + yk
2
,
where hk = xk+1 − xk. The trapezoid rule can be implemented by a one-liner.
T = sum(diff(x).*(y(1:end-1)+y(2:end))/2)
The Matlab function trapz also provides an implementation.
An example with equally spaced x’s is shown in Figure 6.4.
x = 1:6
y = [6 8 11 7 5 2]
For these data, the trapezoid rule gives
12 Chapter 6. Quadrature
trapezoid
area = 35.00
spline
area = 35.25
Figure 6.4. Integrating discrete data.
T = 35
The trapezoid rule is often satisfactory in practice, and more complicated
methods may not be necessary. Nevertheless, methods based on higher order inter-
polation can give other estimates of the integral. Whether or not they are “more
accurate” is impossible to decide without further assumptions about the origin of
the data.
Recall that both the spline and pchip interpolants are based on the Hermite
interpolation formula:
P (x) =
3hs2 − 2s3
h3
yk+1 +
h3 − 3hs2 + 2s3
h3
yk
+
s2(s− h)
h2
dk+1 +
s(s− h)2
h2
dk,
where xk ≤ x ≤ xk+1, s = x−xk, and h = hk. This is a cubic polynomial in s, and
hence in x, that satisfies four interpolation conditions, two on function values and
two on derivative values:
P (xk) = yk, P (xk+1) = yk+1,
P ′(xk) = dk, P
′(xk+1) = dk+1.
The slopes dk are computed in splinetx or pchiptx.
Exercise 6.20 asks you to show that
∫ xk+1
xk
P (x)dx = hk
yk+1 + yk
2
− h2k
dk+1 − dk
12
.
Consequently, ∫ b
a
P (x)dx = T −D,
where T is the trapezoid rule and
D =
n−1∑
k=1
h2k
dk+1 − dk
12
.
6.7. Further Reading 13
The quantity D is a higher order correction to the trapezoid rule that makes use of
the slopes computed by splinetx or pchiptx.
If the x’s are equally spaced, most of the terms in the sum cancel each other.
Then D becomes a simple end correction involving just the first and last slopes:
D = h2
dn − d1
12
.
For the sample data shown in Figure 6.4, the area obtained by linear interpola-
tion is 35.00 and by spline interpolation is 35.25. We haven’t shown shape-preserving
Hermite interpolation, but its area is 35.41667. The integration process averages
out the variation in the interpolants, so even though the three graphs might have
rather different shapes, the resulting approximations to the integral are often quite
close to each other.
6.7 Further Reading
For background on quad and quadl, see Gander and Gautschi [3].
Exercises
6.1. Use quadgui to try to find the integrals of each of the following functions
over the given interval and with the given tolerance. How many function
evaluations are required for each problem and where are the evaluation points
concentrated?
f(x) a b tol
humps(x) 0 1 10−4
humps(x) 0 1 10−6
humps(x) −1 2 10−4
sin x 0 π 10−8
cosx 0 (9/2)π 10−6√
x 0 1 10−8√
x log x eps 1 10−8
tan(sin x)− sin(tanx) 0 π 10−8
1/(3x− 1) 0 1 10−4
t8/3(1− t)10/3 0 1 10−8
t25(1− t)2 0 1 10−8
6.2. Express Q as a weighted combination of the five function values f(a) through
f(e) and establish that its order is six. (See section 6.2.)
6.3. The composite trapezoid rule with n equally spaced points is
Tn(f ) =
h
2
f(a) + h
n−2∑
k=1
f(a + kh) +
h
2
f(b),
14 Chapter 6. Quadrature
where
h =
b− a
n− 1 .
Use Tn(f ) with various values of n to compute π by approximating
π =
∫ 1
−1
2
1 + x2
dx.
How does the accuracy vary with n?
6.4. Use quadtx with various tolerances to compute π by approximating
π =
∫ 1
−1
2
1 + x2
dx.
How do the accuracy and the function evaluation count vary with tolerance?
6.5. Use the Symbolic Toolbox to find the exact value of
∫ 1
0
x4(1− x)4
1 + x2
dx.
(a) What famous approximation does this integral bring to mind?
(b) Does numerical evaluation of this integral present any difficulties?
6.6. The error function erf(x) is defined by an integral:
erf(x) =
2√
π
∫ x
0
e−x
2
dx.
Use quadtx to tabulate erf(x) for x = 0.1, 0.2, . . . , 1.0. Compare the results
with the built-in Matlab function erf(x).
6.7. The beta function, β(z, w), is defined by an integral:
β(z, w) =
∫ 1
0
tz−1(1− t)w−1dt.
Write an M-file mybeta that uses quadtx to compute β(z, w). Compare your
function with the built-in Matlab function beta(z,w).
6.8. The gamma function, Γ(x), is defined by an integral:
Γ(x) =
∫ ∞
0
tx−1e−tdt.
Trying to compute Γ(x) by evaluating this integral with numerical quadrature
can be both inefficient and unreliable. The difficulties are caused by the
infinite interval and the wide variation of values of the integrand.
Write an M-file mygamma that tries to use quadtx to compute Γ(x). Compare
your function with the built-in Matlab function gamma(x). For what x is
your function reasonably fast and accurate? For what x does your function
become slow or unreliable?
Exercises 15
6.9. (a) What is the exact value of
∫ 4π
0
cos2 x dx?
(b) What does quadtx compute for this integral? Why is it wrong?
(c) How does quad overcome the difficulty?
6.10. (a) Use ezplot to plot x sin 1
x
for 0 ≤ x ≤ 1.
(b) Use the Symbolic Toolbox to find the exact value of
∫ 1
0
x sin
1
x
dx.
(c) What happens if you try
quadtx(@(x) x*sin(1/x),0,1)
(d) How can you overcome this difficulty?
6.11. (a) Use ezplot to plot xx for 0 ≤ x ≤ 1.
(b) What happens if you try to use the Symbolic Toolbox to find an analytic
expression for ∫ 1
0
xxdx?
(c) Try to find the numerical value of this integral as accurately as you can.
(d) What do you think is the error in the answer you have given?
6.12. Let
f(x) = log(1 + x) log(1− x).
(a) Use ezplot to plot f(x) for −1 ≤ x ≤ 1.
(b) Use the Symbolic Toolbox to find an analytic expression for
∫ 1
−1
f(x)dx.
(c) Find the numerical value of the analytic expression from (b).
(d) What happens if you try to find the integral numerically with
quadtx(@(x)log(1+x)*log(1-x),-1,1)
(e) How do you work around this difficulty? Justify your solution.
(f ) Use quadtx and your workaround with various tolerances. Plot error
versus tolerance. Plot function evaluation count versus tolerance.
6.13. Let
f(x) = x10 − 10x8 + 33x6 − 40x4 + 16x2.
(a) Use ezplot to plot f(x) for −2 ≤ x ≤ 2.
(b) Use the Symbolic Toolbox to find an analytic expression for
∫ 2
−2
f(x)dx.
16 Chapter 6. Quadrature
(c) Find the numerical value of the analytic expression.
(d) What happens if you try to find the integral numerically with
F = @(x) x^10-10*x^8+33*x^6-40*x^4+16*x^2
quadtx(F,-2,2)
Why?
(e) How do you work around this difficulty?
6.14. (a) Use quadtx to evaluate
∫ 2
−1
1
sin(
√
|t|)
dt.
(b) Why don’t you encounter division-by-zero difficulties at t = 0?
6.15. Definite integrals sometimes have the property that the integrand becomes
infinite at one or both of the endpoints, but the integral itself is finite. In
other words, limx→a |f(x)| = ∞ or limx→b |f(x)| = ∞, but
∫ b
a
f(x) dx
exists and is finite.
(a) Modify quadtx so that, if an infinite value of f(a) or f(b) is detected,
an appropriate warning message is displayed and f(x) is reevaluated at a
point very near to a or b. This allows the adaptive algorithm to proceed and
possibly converge. (You might want to see how quad does this.)
(b) Find an example that triggers the warning, but has a finite integral.
6.16. (a) Modify quadtx so that the recursion is terminated and an appropriate
warning message is displayed whenever the function evaluation count exceeds
10,000. Make sure that the warning message is only displayed once.
(b) Find an example that triggers the warning.
6.17. The Matlab function quadl uses adaptive quadrature based on methods
that have higher order than Simpson’s method. As a result, for integrating
smooth functions, quadl requires fewer function evaluations to obtain a spec-
ified accuracy. The “l” in the function name comes from Lobatto quadrature,
which uses unequal spacing to obtain higher order. The Lobatto rule used in
quadl is of the form
∫ 1
−1
f(x) dx = w1f(−1) + w2f(−x1) + w2f(x1) + w1f(1).
The symmetry in this formula makes it exact for monic polynomials of odd
degree f(x) = xp, p = 1, 3, 5, . . . . Requiring the formula to be exact for
even degrees x0, x2, and x4 leads to three nonlinear equations in the three
parameters w1, w2, and x1. In addition to this basic Lobatto rule, quadl
employs even higher order Kronrod rules, involving other abscissae, xk, and
weights, wk.
Exercises 17
(a) Derive and solve the equations for the Lobatto parameters w1, w2, and
x1.
(b) Find where these values occur in quadl.m.
6.18. Let
Ek =
∫ 1
0
xkex−1 dx.
(a) Show that
E0 = 1− 1/e
and that
Ek = 1− kEk−1.
(b) Suppose we want to compute E1, . . . , En for n = 20. Which of the
following approaches is the fastest and most accurate?
• For each k, use quadtx to evaluate Ek numerically.
• Use forward recursion:
E0 = 1− 1/e;
for k = 2, . . . , n, Ek = 1− kEk−1.
• Use backward recursion, starting at N = 32 with a completely inaccu-
rate value for EN :
EN = 0;
for k = N, . . . , 2, Ek−1 = (1− Ek)/k;
ignore En+1, . . . , EN .
6.19. An article by Prof. Nick Trefethen of Oxford University in the January/February
2002 issue of SIAM News is titled “A Hundred-dollar, Hundred-digit Chal-
lenge” [2]. Trefethen’s challenge consists of ten computational problems, each
of whose answers is a single real number. He asked for each answer to be com-
puted to ten significant digits and offered a $100 prize to the person or group
who managed to calculate the greatest number of correct digits. Ninety-four
teams from 25 countries entered the computation. Much to Trefethen’s sur-
prise, 20 teams scored a perfect 100 points and five more teams scored 99
points. A follow-up book has recently been published [1].
Trefethen’s first problem is to find the value of
T = lim
²→0
∫ 1
²
x−1 cos (x−1 log x) dx.
(a) Why can’t we simply use one of the Matlab numerical quadrature rou-
tines to compute this integral with just a few lines of code?
Here is one way to compute T to several significant digits. Express the
integral as an infinite sum of integrals over intervals where the integrand
does not change sign:
T =
∞∑
k=1
Tk,
18 Chapter 6. Quadrature
where
Tk =
∫ xk−1
xk
x−1 cos (x−1 log x)dx.
Here x0 = 1, and, for k > 0, the xk’s are the successive zeros of cos (x−1 log x),
ordered in decreasing order, x1 > x2 > · · · . In other words, for k > 0, xk
solves the equation
log xk
xk
= −
(
k − 1
2
)
π.
You can use a zero finder such as fzerotx or fzero to compute the xk’s.
If you have access to the Symbolic Toolbox, you can also use lambertw to
compute the xk’s. For each xk, Tk can be computed by numerical quadrature
with quadtx, quad, or quadl. The Tk’s are alternately positive and negative,
and hence the partial sums of the series are alternately greater than and less
than the infinite sum. Moreover, the average of two successive partial sums
is a more accurate approximation to the final result than either sum by itself.
(b) Use this approach to compute T as accurately as you can with a reasonable
amount of computer time. Try to get at least four or five digits. You may be
able to get more. In any case, indicate how accurate you think your result is.
(c) Investigate the use of Aitken’s δ2 acceleration
T̃k = Tk −
(Tk+1 − Tk)2
Tk+1 − 2Tk + Tk−1
.
6.20. Show that the integral of the Hermite interpolating polynomial
P (s) =
3hs2 − 2s3
h3
yk+1 +
h3 − 3hs2 + 2s3
h3
yk
+
s2(s− h)
h2
dk+1 +
s(s− h)2
h2
dk
over one subinterval is
∫ h
0
P (s)ds = h
yk+1 + yk
2
− h2 dk+1 − dk
12
.
6.21. (a) Modify splinetx and pchiptx to create splinequad and pchipquad that
integrate discrete data using spline and pchip interpolation.
(b) Use your programs, as well as trapz, to integrate the discrete data set
x = 1:6
y = [6 8 11 7 5 2]
(c) Use your programs, as well as trapz, to approximate the integral
∫ 1
0
4
1 + x2
dx.
Generate random discrete data sets using the statements
Exercises 19
x = round(100*[0 sort(rand(1,6)) 1])/100
y = round(400./(1+x.^2))/100
With infinitely many infinitely accurate points, the integrals would all equal
π. But these data sets have only eight points, rounded to only two decimal
digits of accuracy.
6.22. This program uses functions in the Spline Toolbox. What does it do?
x = 1:6
y = [6 8 11 7 5 2]
for e = [’c’,’n’,’p’,’s’,’v’]
disp(e)
ppval(fnint(csape(x,y,e)),x(end))
end
6.23. How large is your hand? Figure 6.5 shows three different approaches to
computing the area enclosed by the data that you obtained for exercise 3.3.
Q = 0.3991 Q = 0.4075 Q = 0.4141
Figure 6.5. The area of a hand.
(a) Area of a polygon. Connect successive data points with straight lines and
connect the last data point to the first. If none of these lines intersect, the
result is a polygon with n vertices, (xi, yi). A classic, but little known, fact
is that the area of this polygon is
(x1y2 − x2y1 + x2y3 − x3y2 + · · ·+ xny1 − x1yn)/2.
If x and y are column vectors, this can be computed with the Matlab one-
liner
(x’*y([2:n 1]) – x([2:n 1])’*y)/2
(b) Simple quadrature. The Matlab function inpolygon determines which
of a set of points is contained in a given polygonal region in the plane. The
polygon is specified by the two arrays x and y containing the coordinates of
the vertices. The set of points can be a two-dimensional square grid with
spacing h.
[u,v] = meshgrid(xmin:h:xmax,ymin:h:ymax)
20 Chapter 6. Quadrature
The statement
k = inpolygon(u,v,x,y)
returns an array the same size as u and v whose elements are one for the
points in the polygon and zero for the points outside. The total number of
points in the region is the number of nonzeros in k, that is, nnz(k), so the
area of the corresponding portion of the grid is
h^2*nnz(k)
(c) Two-dimensional adaptive quadrature. The characteristic function of the
region χ(u, v) is equal to one for points (u, v) in the region and zero for points
outside. The area of the region is
∫ ∫
χ(u, v)dudv.
The Matlab function inpolygon(u,v,x,y) computes the characteristic func-
tion if u and v are scalars, or arrays of the same size. But the quadrature
functions have one of them a scalar and the other an array. So we need an
M-file, chi.m, containing
function k = chi(u,v,x,y)
if all(size(u) == 1), u = u(ones(size(v))); end
if all(size(v) == 1), v = v(ones(size(u))); end
k = inpolygon(u,v,x,y);
Two-dimensional adaptive numerical quadrature is obtained with
dblquad(@chi,xmin,xmax,ymin,ymax,tol,[],x,y)
This is the least efficient of the three methods. Adaptive quadrature expects
the integrand to be reasonably smooth, but χ(u, v) is certainly not smooth.
Consequently, values of tol smaller than 10−4 or 10−5 require a lot of com-
puter time.
Figure 6.5 shows that the estimates of the area obtained by these three meth-
ods agree to about two digits, even with fairly large grid sizes and tolerances.
Experiment with your own data, use a moderate amount of computer time,
and see how close the three estimates can be to each other.
Bibliography
[1] F. Bornemann, D. Laurie, S. Wagon, and J. Waldvogel, The SIAM
100-Digit Challenge: A Study in High-Accuracy Numerical Computing, SIAM,
Philadelphia, 2004.
[2] L. N. Trefethen, A hundred-dollar, hundred-digit challenge, SIAM News,
35(1)(2002). Society of Industrial and Applied Mathematics.
http://www.siam.org/books/100digitchallenge
[3] W. Gander and W. Gautschi, Adaptive Quadrature—Revisited, BIT Nu-
merical Mathematics, 40 (2000), pp. 84–101.
http://www.inf.ethz.ch/personal/gander
21
Chapter 7
Ordinary Differential
Equations
Matlab has several different functions for the numerical solution of ordinary dif-
ferential equations. This chapter describes the simplest of these functions and then
compares all of the functions for efficiency, accuracy, and special features. Stiffness
is a subtle concept that plays an important role in these comparisons.
7.1 Integrating Differential Equations
The initial value problem for an ordinary differential equation involves finding a
function y(t) that satisfies
dy(t)
dt
= f(t, y(t))
together with the initial condition
y(t0) = y0.
A numerical solution to this problem generates a sequence of values for the indepen-
dent variable, t0, t1, . . . , and a corresponding sequence of values for the dependent
variable, y0, y1, . . . , so that each yn approximates the solution at tn:
yn ≈ y(tn), n = 0, 1, . . . .
Modern numerical methods automatically determine the step sizes
hn = tn+1 − tn
so that the estimated error in the numerical solution is controlled by a specified
tolerance.
The fundamental theorem of calculus gives us an important connection be-
tween differential equations and integrals:
y(t + h) = y(t) +
∫ t+h
t
f(s, y(s))ds.
February 16, 2008
1
2 Chapter 7. Ordinary Differential Equations
We cannot use numerical quadrature directly to approximate the integral because we
do not know the function y(s) and so cannot evaluate the integrand. Nevertheless,
the basic idea is to choose a sequence of values of h so that this formula allows us
to generate our numerical solution.
One special case to keep in mind is the situation where f(t, y) is a function of
t alone. The numerical solution of such simple differential equations is then just a
sequence of quadratures:
yn+1 = yn +
∫ tn+1
tn
f(s)ds.
Throughout this chapter, we frequently use “dot” notation for derivatives:
ẏ =
dy(t)
dt
and ÿ =
d2y(t)
dt2
.
7.2 Systems of Equations
Many mathematical models involve more than one unknown function, and second-
and higher order derivatives. These models can be handled by making y(t) a vector-
valued function of t. Each component is either one of the unknown functions or one
of its derivatives. The Matlab vector notation is particularly convenient here.
For example, the second-order differential equation describing a simple har-
monic oscillator
ẍ(t) = −x(t)
becomes two first-order equations. The vector y(t) has two components, x(t) and
its first derivative ẋ(t):
y(t) =
[
x(t)
ẋ(t)
]
.
Using this vector, the differential equation is
ẏ(t) =
[
ẋ(t)
−x(t)
]
=
[
y2(t)
−y1(t)
]
.
The Matlab function defining the differential equation has t and y as input
arguments and should return f(t, y) as a column vector. For the harmonic oscillator,
the function could be an M-file containing
function ydot = harmonic(t,y)
ydot = [y(2); -y(1)]
A more compact version uses matrix multiplication in an anonymous function,
f = @(t,y) [0 1; -1 0]*y
7.3. Linearized Differential Equations 3
In both cases, the variable t has to be included as the first argument, even though
it is not explicitly involved in the differential equation.
A slightly more complicated example, the two-body problem, describes the
orbit of one body under the gravitational attraction of a much heavier body. Using
Cartesian coordinates, u(t) and v(t), centered in the heavy body, the equations are
ü(t) = −u(t)/r(t)3,
v̈(t) = −v(t)/r(t)3,
where
r(t) =
√
u(t)2 + v(t)2.
The vector y(t) has four components:
y(t) =
u(t)
v(t)
u̇(t)
v̇(t)
.
The differential equation is
ẏ(t) =
u̇(t)
v̇(t)
−u(t)/r(t)3
−v(t)/r(t)3
.
The Matlab function could be
function ydot = twobody(t,y)
r = sqrt(y(1)^2 + y(2)^2);
ydot = [y(3); y(4); -y(1)/r^3; -y(2)/r^3];
A more compact Matlab function is
ydot = @(t,y) [y(3:4); -y(1:2)/norm(y(1:2))^3]
Despite the use of vector operations, the second M-file is not significantly more
efficient than the first.
7.3 Linearized Differential Equations
The local behavior of the solution to a differential equation near any point (tc, yc)
can be analyzed by expanding f(t, y) in a two-dimensional Taylor series:
f(t, y) = f(tc, yc) + α(t− tc) + J(y − yc) + · · · ,
where
α =
∂f
∂t
(tc, yc), J =
∂f
∂y
(tc, yc).
4 Chapter 7. Ordinary Differential Equations
The most important term in this series is usually the one involving J , the Jacobian.
For a system of differential equations with n components,
d
dt
y1(t)
y2(t)
…
yn(t)
=
f1(t, y1, . . . , yn)
f2(t, y1, . . . , yn)
…
fn(t, y1, . . . , yn)
,
the Jacobian is an n-by-n matrix of partial derivatives:
J =
∂f1
∂y1
∂f1
∂y2
. . . ∂f1
∂yn
∂f2
∂y1
∂f2
∂y2
. . . ∂f2
∂yn
…
…
…
∂fn
∂y1
∂fn
∂y2
. . . ∂fn
∂yn
.
The influence of the Jacobian on the local behavior is determined by the
solution to the linear system of ordinary differential equations
ẏ = Jy.
Let λk = µk + iνk be the eigenvalues of J and Λ = diag(λk) the diagonal eigenvalue
matrix. If there is a linearly independent set of corresponding eigenvectors V , then
J = V ΛV −1.
The linear transformation
V x = y
transforms the local system of equations into a set of decoupled equations for the
individual components of x:
ẋk = λkxk.
The solutions are
xk(t) = e
λk(t−tc)x(tc).
A single component xk(t) grows with t if µk is positive, decays if µk is negative,
and oscillates if νk is nonzero. The components of the local solution y(t) are linear
combinations of these behaviors.
For example, the harmonic oscillator
ẏ =
[
0 1
−1 0
]
y
is a linear system. The Jacobian is simply the matrix
J =
[
0 1
−1 0
]
.
The eigenvalues of J are ±i and the solutions are purely oscillatory linear combi-
nations of eit and e−it.
7.4. Single-Step Methods 5
A nonlinear example is the two-body problem
ẏ(t) =
y3(t)
y4(t)
−y1(t)/r(t)3
−y2(t)/r(t)3
,
where
r(t) =
√
y1(t)2 + y2(t)2.
In exercise 7.6, we ask you to show that the Jacobian for this system is
J =
1
r5
0 0 r5 0
0 0 0 r5
2y21 − y22 3y1y2 0 0
3y1y2 2y22 − y21 0 0
.
It turns out that the eigenvalues of J just depend on the radius r(t):
λ =
1
r3/2
√
2
i
−
√
2
−i
.
We see that one eigenvalue is real and positive, so the corresponding component
of the solution is growing. One eigenvalue is real and negative, corresponding to a
decaying component. Two eigenvalues are purely imaginary, corresponding to os-
cillatory components. However, the overall global behavior of this nonlinear system
is quite complicated and is not described by this local linearized analysis.
7.4 Single-Step Methods
The simplest numerical method for the solution of initial value problems is Euler’s
method. It uses a fixed step size h and generates the approximate solution by
yn+1 = yn + hf(tn, yn),
tn+1 = tn + h.
The Matlab code would use an initial point t0, a final point tfinal, an initial
value y0, a step size h, and a function f. The primary loop would simply be
t = t0;
y = y0;
while t <= tfinal
y = y + h*f(t,y)
t = t + h
end
6 Chapter 7. Ordinary Differential Equations
Note that this works perfectly well if y0 is a vector and f returns a vector.
As a quadrature rule for integrating f(t), Euler’s method corresponds to a
rectangle rule where the integrand is evaluated only once, at the left-hand endpoint
of the interval. It is exact if f(t) is constant, but not if f(t) is linear. So the error
is proportional to h. Tiny steps are needed to get even a few digits of accuracy.
But, from our point of view, the biggest defect of Euler’s method is that it does not
provide an error estimate. There is no automatic way to determine what step size
is needed to achieve a specified accuracy.
If Euler’s method is followed by a second function evaluation, we begin to
get a viable algorithm. There are two natural possibilities, corresponding to the
midpoint rule and the trapezoid rule for quadrature. The midpoint analogue uses
Euler to step halfway across the interval, evaluates the function at this intermediate
point, then uses that slope to take the actual step:
s1 = f(tn, yn),
s2 = f
(
tn +
h
2
, yn +
h
2
s1
)
,
yn+1 = yn + hs2,
tn+1 = tn + h.
The trapezoid analogue uses Euler to take a tentative step across the interval,
evaluates the function at this exploratory point, then averages the two slopes to
take the actual step:
s1 = f(tn, yn),
s2 = f(tn + h, yn + hs1),
yn+1 = yn + h
s1 + s2
2
,
tn+1 = tn + h.
If we were to use both of these methods simultaneously, they would produce
two different values for yn+1. The difference between the two values would provide
an error estimate and a basis for picking the step size. Furthermore, an extrapolated
combination of the two values would be more accurate than either one individually.
Continuing with this approach is the idea behind single-step methods for in-
tegrating ordinary differential equations. The function f(t, y) is evaluated several
times for values of t between tn and tn+1 and values of y obtained by adding linear
combinations of the values of f to yn. The actual step is taken using another linear
combination of the function values. Modern versions of single-step methods use yet
another linear combination of function values to estimate error and determine step
size.
Single-step methods are often called Runge–Kutta methods, after the two Ger-
man applied mathematicians who first wrote about them around 1905. The classical
Runge–Kutta method was widely used for hand computation before the invention
of digital computers and is still popular today. It uses four function evaluations per
7.4. Single-Step Methods 7
step:
s1 = f(tn, yn),
s2 = f
(
tn +
h
2
, yn +
h
2
s1
)
,
s3 = f
(
tn +
h
2
, yn +
h
2
s2
)
,
s4 = f(tn + h, yn + hs3),
yn+1 = yn +
h
6
(s1 + 2s2 + 2s3 + s4),
tn+1 = tn + h.
If f(t, y) does not depend on y, then classical Runge–Kutta has s2 = s3 and the
method reduces to Simpson’s quadrature rule.
Classical Runge–Kutta does not provide an error estimate. The method is
sometimes used with a step size h and again with step size h/2 to obtain an error
estimate, but we now know more efficient methods.
Several of the ordinary differential equation solvers in Matlab, including the
textbook solver we describe later in this chapter, are single-step or Runge–Kutta
solvers. A general single-step method is characterized by a number of parameters,
αi, βi,j , γi, and δi. There are k stages. Each stage computes a slope, si, by
evaluating f(t, y) for a particular value of t and a value of y obtained by taking
linear combinations of the previous slopes:
si = f
tn + αih, yn + h
i−1∑
j=1
βi,jsj
, i = 1, . . . , k.
The proposed step is also a linear combination of the slopes:
yn+1 = yn + h
k∑
i=1
γisi.
An estimate of the error that would occur with this step is provided by yet another
linear combination of the slopes:
en+1 = h
k∑
i=1
δisi.
If this error is less than the specified tolerance, then the step is successful and yn+1
is accepted. If not, the step is a failure and yn+1 is rejected. In either case, the
error estimate is used to compute the step size h for the next step.
The parameters in these methods are determined by matching terms in Taylor
series expansions of the slopes. These series involve powers of h and products of
various partial derivatives of f(t, y). The order of a method is the exponent of the
smallest power of h that cannot be matched. It turns out that one, two, three, and
8 Chapter 7. Ordinary Differential Equations
four stages yield methods of order one, two, three, and four, respectively. But it
takes six stages to obtain a fifth-order method. The classical Runge–Kutta method
has four stages and is fourth order.
The names of the Matlab ordinary differential equation solvers are all of the
form odennxx with digits nn indicating the order of the underlying method and
a possibly empty xx indicating some special characteristic of the method. If the
error estimate is obtained by comparing formulas with different orders, the digits nn
indicate these orders. For example, ode45 obtains its error estimate by comparing
a fourth-order and a fifth-order formula.
7.5 The BS23 Algorithm
Our textbook function ode23tx is a simplified version of the function ode23 that is
included with Matlab. The algorithm is due to Bogacki and Shampine [3, 6]. The
“23” in the function names indicates that two simultaneous single-step formulas,
one of second order and one of third order, are involved.
The method has three stages, but there are four slopes si because, after the
first step, the s1 for one step is the s4 from the previous step. The essentials are
s1 = f(tn, yn),
s2 = f
(
tn +
h
2
, yn +
h
2
s1
)
,
s3 = f
(
tn +
3
4
h, yn +
3
4
hs2
)
,
tn+1 = tn + h,
yn+1 = yn +
h
9
(2s1 + 3s2 + 4s3),
s4 = f(tn+1, yn+1),
en+1 =
h
72
(−5s1 + 6s2 + 8s3 − 9s4).
The simplified pictures in Figure 7.1 show the starting situation and the three
stages. We start at a point (tn, yn) with an initial slope s1 = f(tn, yn) and an
estimate of a good step size, h. Our goal is to compute an approximate solution
yn+1 at tn+1 = tn + h that agrees with the true solution y(tn+1) to within the
specified tolerances.
The first stage uses the initial slope s1 to take an Euler step halfway across
the interval. The function is evaluated there to get the second slope, s2. This slope
is used to take an Euler step three-quarters of the way across the interval. The
function is evaluated again to get the third slope, s3. A weighted average of the
three slopes,
s =
1
9
(2s1 + 3s2 + 4s3),
is used for the final step all the way across the interval to get a tentative value for
yn+1. The function is evaluated once more to get s4. The error estimate then uses
7.5. The BS23 Algorithm 9
tn tn+h
yn
s1
tn tn+h/2
yn
s1
s2
tn tn+3*h/4
yn
s2
s3
tn tn+h
yn
ynp1
s
s4
Figure 7.1. BS23 algorithm.
all four slopes:
en+1 =
h
72
(−5s1 + 6s2 + 8s3 − 9s4).
If the error is within the specified tolerance, then the step is successful, the tentative
value of yn+1 is accepted, and s4 becomes the s1 of the next step. If the error is
too large, then the tentative yn+1 is rejected and the step must be redone. In either
case, the error estimate en+1 provides the basis for determining the step size h for
the next step.
The first input argument of ode23tx specifies the function f(t, y). This argu-
ment can be either
• a function handle, or
• an anonymous function.
The function should accept two arguments—usually, but not necessarily, t and y.
The result of evaluating the character string or the function should be a column
vector containing the values of the derivatives, dy/dt.
The second input argument of ode23tx is a vector, tspan, with two compo-
nents, t0 and tfinal. The integration is carried out over the interval
t0 ≤ t ≤ tfinal.
One of the simplifications in our textbook code is this form of tspan. Other Mat-
lab ordinary differential equation solvers allow more flexible specifications of the
integration interval.
10 Chapter 7. Ordinary Differential Equations
The third input argument is a column vector, y0, providing the initial value
of y0 = y(t0). The length of y0 tells ode23tx the number of differential equations
in the system.
A fourth input argument is optional and can take two different forms. The
simplest, and most common, form is a scalar numerical value, rtol, to be used
as the relative error tolerance. The default value for rtol is 10−3, but you can
provide a different value if you want more or less accuracy. The more complicated
possibility for this optional argument is the structure generated by the Matlab
function odeset. This function takes pairs of arguments that specify many different
options for the Matlab ordinary differential equation solvers. For ode23tx, you
can change the default values of three quantities: the relative error tolerance, the
absolute error tolerance, and the M-file that is called after each successful step. The
statement
opts = odeset(’reltol’,1.e-5, ’abstol’,1.e-8, ...
’outputfcn’,@myodeplot)
creates a structure that specifies the relative error tolerance to be 10−5, the absolute
error tolerance to be 10−8, and the output function to be myodeplot.
The output produced by ode23tx can be either graphic or numeric. With no
output arguments, the statement
ode23tx(F,tspan,y0);
produces a dynamic plot of all the components of the solution. With two output
arguments, the statement
[tout,yout] = ode23tx(F,tspan,y0);
generates a table of values of the solution.
7.6 ode23tx
Let’s examine the code for ode23tx. Here is the preamble.
function [tout,yout] = ode23tx(F,tspan,y0,arg4,varargin)
%ODE23TX Solve non-stiff differential equations.
% Textbook version of ODE23.
%
% ODE23TX(F,TSPAN,Y0) with TSPAN = [T0 TFINAL]
% integrates the system of differential equations
% dy/dt = f(t,y) from t = T0 to t = TFINAL.
% The initial condition is y(T0) = Y0.
%
% The first argument, F, is a function handle or an
% anonymous function that defines f(t,y). This function
% must have two input arguments, t and y, and must
% return a column vector of the derivatives, dy/dt.
7.6. ode23tx 11
%
% With two output arguments, [T,Y] = ODE23TX(...)
% returns a column vector T and an array Y where Y(:,k)
% is the solution at T(k).
%
% With no output arguments, ODE23TX plots the solution.
%
% ODE23TX(F,TSPAN,Y0,RTOL) uses the relative error
% tolerance RTOL instead of the default 1.e-3.
%
% ODE23TX(F,TSPAN,Y0,OPTS) where OPTS = ...
% ODESET(’reltol’,RTOL,’abstol’,ATOL,’outputfcn’,@PLTFN)
% uses relative error RTOL instead of 1.e-3,
% absolute error ATOL instead of 1.e-6, and calls PLTFN
% instead of ODEPLOT after each step.
%
% More than four input arguments, ODE23TX(F,TSPAN,Y0,
% RTOL,P1,P2,..), are passed on to F, F(T,Y,P1,P2,..).
%
% ODE23TX uses the Runge-Kutta (2,3) method of
% Bogacki and Shampine.
%
% Example
% tspan = [0 2*pi];
% y0 = [1 0]’;
% F = ’[0 1; -1 0]*y’;
% ode23tx(F,tspan,y0);
%
% See also ODE23.
Here is the code that parses the arguments and initializes the internal variables.
rtol = 1.e-3;
atol = 1.e-6;
plotfun = @odeplot;
if nargin >= 4 & isnumeric(arg4)
rtol = arg4;
elseif nargin >= 4 & isstruct(arg4)
if ~isempty(arg4.RelTol), rtol = arg4.RelTol; end
if ~isempty(arg4.AbsTol), atol = arg4.AbsTol; end
if ~isempty(arg4.OutputFcn),
plotfun = arg4.OutputFcn; end
end
t0 = tspan(1);
tfinal = tspan(2);
tdir = sign(tfinal – t0);
plotit = (nargout == 0);
12 Chapter 7. Ordinary Differential Equations
threshold = atol / rtol;
hmax = abs(0.1*(tfinal-t0));
t = t0;
y = y0(:);
% Initialize output.
if plotit
plotfun(tspan,y,’init’);
else
tout = t;
yout = y.’;
end
The computation of the initial step size is a delicate matter because it requires some
knowledge of the overall scale of the problem.
s1 = F(t, y, varargin{:});
r = norm(s1./max(abs(y),threshold),inf) + realmin;
h = tdir*0.8*rtol^(1/3)/r;
Here is the beginning of the main loop. The integration starts at t = t0 and
increments t until it reaches tfinal. It is possible to go “backward,” that is, have
tfinal < t0.
while t ~= tfinal
hmin = 16*eps*abs(t);
if abs(h) > hmax, h = tdir*hmax; end
if abs(h) < hmin, h = tdir*hmin; end
% Stretch the step if t is close to tfinal.
if 1.1*abs(h) >= abs(tfinal – t)
h = tfinal – t;
end
Here is the actual computation. The first slope s1 has already been computed. The
function defining the differential equation is evaluated three more times to obtain
three more slopes.
s2 = F(t+h/2, y+h/2*s1, varargin{:});
s3 = F(t+3*h/4, y+3*h/4*s2, varargin{:});
tnew = t + h;
ynew = y + h*(2*s1 + 3*s2 + 4*s3)/9;
s4 = F(tnew, ynew, varargin{:});
Here is the error estimate. The norm of the error vector is scaled by the ratio of the
absolute tolerance to the relative tolerance. The use of the smallest floating-point
number, realmin, prevents err from being exactly zero.
7.7. Examples 13
e = h*(-5*s1 + 6*s2 + 8*s3 – 9*s4)/72;
err = norm(e./max(max(abs(y),abs(ynew)),threshold),
… inf) + realmin;
Here is the test to see if the step is successful. If it is, the result is plotted or
appended to the output vector. If it is not, the result is simply forgotten.
if err <= rtol
t = tnew;
y = ynew;
if plotit
if plotfun(t,y,’’);
break
end
else
tout(end+1,1) = t;
yout(end+1,:) = y.’;
end
s1 = s4; % Reuse final function value to start new step.
end
The error estimate is used to compute a new step size. The ratio rtol/err is
greater than one if the current step is successful, or less than one if the current step
fails. A cube root is involved because the BS23 is a third-order method. This means
that changing tolerances by a factor of eight will change the typical step size, and
hence the total number of steps, by a factor of two. The factors 0.8 and 5 prevent
excessive changes in step size.
% Compute a new step size.
h = h*min(5,0.8*(rtol/err)^(1/3));
Here is the only place where a singularity would be detected.
if abs(h) <= hmin
warning(sprintf( ...
’Step size %e too small at t = %e.
’,h,t));
t = tfinal;
end
end
That ends the main loop. The plot function might need to finish its work.
if plotit
plotfun([],[],’done’);
end
7.7 Examples
Please sit down in front of a computer running Matlab. Make sure ode23tx is in
your current directory or on your Matlab path. Start your session by entering
14 Chapter 7. Ordinary Differential Equations
F = @(t,x) 0 ; ode23tx(F,[0 10],1)
This should produce a plot of the solution of the initial value problem
dy
dt
= 0,
y(0) = 1,
0 ≤ t ≤ 10.
The solution, of course, is a constant function, y(t) = 1.
Now you can press the up arrow key, use the left arrow key to space over to
the 0, and change it to something more interesting. Here are some examples. At
first, we’ll change just the 0 and leave the [0 10] and 1 alone.
F Exact solution
0 1
t 1+t^2/2
y exp(t)
-y exp(-t)
1/(1-3*t) 1-log(1-3*t)/3 (Singular)
2*y-y^2 2/(1+exp(-2*t))
Make up some of your own examples. Change the initial condition. Change the
accuracy by including 1.e-6 as the fourth argument.
Now let’s try the harmonic oscillator, a second-order differential equation writ-
ten as a pair of two first-order equations. First, create a function to specify the
equations. Use either
F = @(t,y) [y(2); -y(1)];
or
F = @(t,y) [0 1; -1 0]*y;
Then the statement
ode23tx(F,[0 2*pi],[1; 0])
plots two functions of t that you should recognize. If you want to produce a phase
plane plot, you have two choices. One possibility is to capture the output and plot
it after the computation is complete.
[t,y] = ode23tx(F,[0 2*pi],[1; 0])
plot(y(:,1),y(:,2),’-o’)
axis([-1.2 1.2 -1.2 1.2])
axis square
The more interesting possibility is to use a function that plots the solution
while it is being computed. Matlab provides such a function in odephas2.m. It is
accessed by using odeset to create an options structure.
7.7. Examples 15
opts = odeset(’reltol’,1.e-4,’abstol’,1.e-6, ...
’outputfcn’,@odephas2);
If you want to provide your own plotting function, it should be something like
function flag = phaseplot(t,y,job)
persistent p
if isequal(job,’init’)
p = plot(y(1),y(2),’o’,’erasemode’,’none’);
axis([-1.2 1.2 -1.2 1.2])
axis square
flag = 0;
elseif isequal(job,’’)
set(p,’xdata’,y(1),’ydata’,y(2))
drawnow
flag = 0;
end
This is with
opts = odeset(’reltol’,1.e-4,’abstol’,1.e-6, ...
’outputfcn’,@phaseplot);
Once you have decided on a plotting function and created an options structure, you
can compute and simultaneously plot the solution with
ode23tx(F,[0 2*pi],[1; 0],opts)
Try this with other values of the tolerances.
Issue the command type twobody to see if there is an M-file twobody.m on
your path. If not, find the two or three lines of code earlier in this chapter and
create your own M-file. Then try
ode23tx(@twobody,[0 2*pi],[1; 0; 0; 1]);
The code, and the length of the initial condition, indicate that the solution has four
components. But the plot shows only three. Why? Hint: Find the zoom button on
the figure window toolbar and zoom in on the blue curve.
You can vary the initial condition of the two-body problem by changing the
fourth component.
y0 = [1; 0; 0; change_this];
ode23tx(@twobody,[0 2*pi],y0);
Graph the orbit, and the heavy body at the origin, with
y0 = [1; 0; 0; change_this];
[t,y] = ode23tx(@twobody,[0 2*pi],y0);
plot(y(:,1),y(:,2),’-’,0,0,’ro’)
axis equal
You might also want to use something other than 2π for tfinal.
16 Chapter 7. Ordinary Differential Equations
7.8 Lorenz Attractor
One of the world’s most extensively studied ordinary differential equations is the
Lorenz chaotic attractor. It was first described in 1963 by Edward Lorenz, an
M.I.T. mathematician and meteorologist who was interested in fluid flow models of
the earth’s atmosphere. An excellent reference is a book by Colin Sparrow [8].
We have chosen to express the Lorenz equations in a somewhat unusual way
involving a matrix-vector product:
ẏ = Ay.
The vector y has three components that are functions of t:
y(t) =
y1(t)
y2(t)
y3(t)
.
Despite the way we have written it, this is not a linear system of differential equa-
tions. Seven of the nine elements in the 3-by-3 matrix A are constant, but the other
two depend on y2(t):
A =
−β 0 y2
0 −σ σ
−y2 ρ −1
.
The first component of the solution, y1(t), is related to the convection in the atmo-
spheric flow, while the other two components are related to horizontal and vertical
temperature variation. The parameter σ is the Prandtl number, ρ is the normal-
ized Rayleigh number, and β depends on the geometry of the domain. The most
popular values of the parameters, σ = 10, ρ = 28, and β = 8/3, are outside the
ranges associated with the earth’s atmosphere.
The deceptively simple nonlinearity introduced by the presence of y2 in the
system matrix A changes everything. There are no random aspects to these equa-
tions, so the solutions y(t) are completely determined by the parameters and the
initial conditions, but their behavior is very difficult to predict. For some values of
the parameters, the orbit of y(t) in three-dimensional space is known as a strange
attractor. It is bounded, but not periodic and not convergent. It never intersects
itself. It ranges chaotically back and forth around two different points, or attractors.
For other values of the parameters, the solution might converge to a fixed point,
diverge to infinity, or oscillate periodically. See Figures 7.2 and 7.3.
Let’s think of η = y2 as a free parameter, restrict ρ to be greater than one,
and study the matrix
A =
−β 0 η
0 −σ σ
−η ρ −1
.
It turns out that A is singular if and only if
η = ±
√
β(ρ− 1).
7.8. Lorenz Attractor 17
0 5 10 15 20 25 30
y3
y2
y1
t
Figure 7.2. Three components of Lorenz attractor.
−25 −20 −15 −10 −5 0 5 10 15 20 25
−25
−20
−15
−10
−5
0
5
10
15
20
25
y2
y
3
Figure 7.3. Phase plane plot of Lorenz attractor.
The corresponding null vector, normalized so that its second component is equal to
η, is
ρ− 1
η
η
.
With two different signs for η, this defines two points in three-dimensional space.
18 Chapter 7. Ordinary Differential Equations
These points are fixed points for the differential equation. If
y(t0) =
ρ− 1
η
η
,
then, for all t,
ẏ(t) =
0
0
0
,
and so y(t) never changes. However, these points are unstable fixed points. If y(t)
does not start at one of these points, it will never reach either of them; if it tries to
approach either point, it will be repulsed.
We have provided an M-file, lorenzgui.m, that facilitates experiments with
the Lorenz equations. Two of the parameters, β = 8/3 and σ = 10, are fixed. A
uicontrol offers a choice among several different values of the third parameter, ρ.
A simplified version of the program for ρ = 28 would begin with
rho = 28;
sigma = 10;
beta = 8/3;
eta = sqrt(beta*(rho-1));
A = [ -beta 0 eta
0 -sigma sigma
-eta rho -1 ];
The initial condition is taken to be near one of the attractors.
yc = [rho-1; eta; eta];
y0 = yc + [0; 0; 3];
The time span is infinite, so the integration will have to be stopped by another
uicontrol.
tspan = [0 Inf];
opts = odeset(’reltol’,1.e-6,’outputfcn’,@lorenzplot);
ode45(@lorenzeqn, tspan, y0, opts, A);
The matrix A is passed as an extra parameter to the integrator, which sends it on to
lorenzeqn, the subfunction defining the differential equation. The extra parameter
machinery included in the function functions allows lorenzeqn to be written in a
particularly compact manner.
function ydot = lorenzeqn(t,y,A)
A(1,3) = y(2);
A(3,1) = -y(2);
ydot = A*y;
Most of the complexity of lorenzgui is contained in the plotting subfunction,
lorenzplot. It not only manages the user interface controls, it must also anticipate
the possible range of the solution in order to provide appropriate axis scaling.
7.9. Stiffness 19
7.9 Stiffness
Stiffness is a subtle, difficult, and important concept in the numerical solution of
ordinary differential equations. It depends on the differential equation, the initial
conditions, and the numerical method. Dictionary definitions of the word “stiff”
involve terms like “not easily bent,” “rigid,” and “stubborn.” We are concerned
with a computational version of these properties.
A problem is stiff if the solution being sought varies slowly, but there are
nearby solutions that vary rapidly, so the numerical method must take
small steps to obtain satisfactory results.
Stiffness is an efficiency issue. If we weren’t concerned with how much time a
computation takes, we wouldn’t be concerned about stiffness. Nonstiff methods
can solve stiff problems; they just take a long time to do it.
A model of flame propagation provides an example. We learned about this
example from Larry Shampine, one of the authors of the Matlab ordinary differ-
ential equation suite. If you light a match, the ball of flame grows rapidly until it
reaches a critical size. Then it remains at that size because the amount of oxygen
being consumed by the combustion in the interior of the ball balances the amount
available through the surface. The simple model is
ẏ = y2 − y3,
y(0) = δ,
0 ≤ t ≤ 2/δ.
The scalar variable y(t) represents the radius of the ball. The y2 and y3 terms come
from the surface area and the volume. The critical parameter is the initial radius,
δ, which is “small.” We seek the solution over a length of time that is inversely
proportional to δ.
At this point, we suggest that you start up Matlab and actually run our
examples. It is worthwhile to see them in action. We will start with ode45, the
workhorse of the Matlab ordinary differential equation suite. If δ is not very small,
the problem is not very stiff. Try δ = 0.01 and request a relative error of 10−4.
delta = 0.01;
F = @(t,y) y^2 - y^3’;
opts = odeset(’RelTol’,1.e-4);
ode45(F,[0 2/delta],delta,opts);
With no output arguments, ode45 automatically plots the solution as it is computed.
You should get a plot of a solution that starts at y = 0.01, grows at a modestly
increasing rate until t approaches 100, which is 1/δ, then grows rapidly until it
reaches a value close to 1, where it remains.
Now let’s see stiffness in action. Decrease δ by three orders of magnitude. (If
you run only one example, run this one.)
delta = 0.00001;
ode45(F,[0 2/delta],delta,opts);
20 Chapter 7. Ordinary Differential Equations
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10
4
0
0.2
0.4
0.6
0.8
1
ode45
0.98 1 1.02 1.04 1.06 1.08 1.1 1.12
x 10
4
0.9999
1
1.0001
Figure 7.4. Stiff behavior of ode45.
You should see something like Figure 7.4, although it will take a long time
to complete the plot. If you get tired of watching the agonizing progress, click
the stop button in the lower left corner of the window. Turn on zoom, and use
the mouse to explore the solution near where it first approaches steady state. You
should see something like the detail in Figure 7.4. Notice that ode45 is doing its
job. It’s keeping the solution within 10−4 of its nearly constant steady state value.
But it certainly has to work hard to do it. If you want an even more dramatic
demonstration of stiffness, decrease the tolerance to 10−5 or 10−6.
This problem is not stiff initially. It only becomes stiff as the solution ap-
proaches steady state. This is because the steady state solution is so “rigid.” Any
solution near y(t) = 1 increases or decreases rapidly toward that solution. (We
should point out that “rapidly” here is with respect to an unusually long time
scale.)
What can be done about stiff problems? You don’t want to change the dif-
ferential equation or the initial conditions, so you have to change the numerical
method. Methods intended to solve stiff problems efficiently do more work per
step, but can take much bigger steps. Stiff methods are implicit. At each step they
use Matlab matrix operations to solve a system of simultaneous linear equations
that helps predict the evolution of the solution. For our flame example, the matrix
is only 1 by 1, but even here, stiff methods do more work per step than nonstiff
methods.
7.9. Stiffness 21
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10
4
0
0.2
0.4
0.6
0.8
1
ode23s
0.98 1 1.02 1.04 1.06 1.08 1.1 1.12
x 10
4
0.9999
1
1.0001
Figure 7.5. Stiff behavior of ode23s.
Let’s compute the solution to our flame example again, this time with one of
the ordinary differential equation solvers in Matlab whose name ends in “s” for
“stiff.”
delta = 0.00001;
ode23s(F,[0 2/delta],delta,opts);
Figure 7.5 shows the computed solution and the zoom detail. You can see that
ode23s takes many fewer steps than ode45. This is actually an easy problem for a
stiff solver. In fact, ode23s takes only 99 steps and uses just 412 function evalua-
tions, while ode45 takes 3040 steps and uses 20179 function evaluations. Stiffness
even affects graphical output. The print files for the ode45 figures are much larger
than those for the ode23s figures.
Imagine you are returning from a hike in the mountains. You are in a narrow
canyon with steep slopes on either side. An explicit algorithm would sample the
local gradient to find the descent direction. But following the gradient on either side
of the trail will send you bouncing back and forth across the canyon, as with ode45.
You will eventually get home, but it will be long after dark before you arrive. An
implicit algorithm would have you keep your eyes on the trail and anticipate where
each step is taking you. It is well worth the extra concentration.
This flame problem is also interesting because it involves the Lambert W
function, W (z). The differential equation is separable. Integrating once gives an
22 Chapter 7. Ordinary Differential Equations
0 20 40 60 80 100 120 140 160 180 200
0
0.2
0.4
0.6
0.8
1
t
1/(lambertw(99 exp(99−t))+1)
Figure 7.6. Exact solution for the flame example.
implicit equation for y as a function of t:
1
y
+ log
(
1
y
− 1
)
=
1
δ
+ log
(
1
δ
− 1
)
− t.
This equation can be solved for y. The exact analytical solution to the flame model
turns out to be
y(t) =
1
W (aea−t) + 1
,
where a = 1/δ− 1. The function W (z), the Lambert W function, is the solution to
W (z)eW (z) = z.
With Matlab and the Symbolic Math Toolbox, the statements
y = dsolve(’Dy = y^2 - y^3’,’y(0) = 1/100’);
y = simplify(y);
pretty(y)
ezplot(y,0,200)
produce
1
----------------------------
lambertw(99 exp(99 - t)) + 1
and the plot of the exact solution shown in Figure 7.6. If the initial value 1/100 is
decreased and the time span 0 ≤ t ≤ 200 increased, the transition region becomes
narrower.
The Lambert W function is named after J. H. Lambert (1728–1777). Lambert
was a colleague of Euler and Lagrange’s at the Berlin Academy of Sciences and
is best known for his laws of illumination and his proof that π is irrational. The
function was “rediscovered” a few years ago by Corless, Gonnet, Hare, and Jeffrey,
working on Maple, and by Don Knuth [4].
7.10. Events 23
7.10 Events
So far, we have been assuming that the tspan interval, t0 ≤ t ≤ tfinal, is a given
part of the problem specification, or we have used an infinite interval and a GUI
button to terminate the computation. In many situations, the determination of
tfinal is an important aspect of the problem.
One example is a body falling under the force of gravity and encountering
air resistance. When does it hit the ground? Another example is the two-body
problem, the orbit of one body under the gravitational attraction of a much heavier
body. What is the period of the orbit? The events feature of the Matlab ordinary
differential equation solvers provides answers to such questions.
Events detection in ordinary differential equations involves two functions,
f(t, y) and g(t, y), and an initial condition, (t0, y0). The problem is to find a function
y(t) and a final value t∗ so that
ẏ = f(t, y),
y(t0) = y0,
and
g(t∗, y(t∗)) = 0.
A simple model for the falling body is
ÿ = −1 + ẏ2,
with initial conditions y(0) = 1, ẏ(0) = 0. The question is, for what t does y(t) = 0?
The code for the function f(t, y) is
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.2
0.4
0.6
0.8
1
t
y
Falling body
tfinal = 1.6585
Figure 7.7. Event handling for falling object.
24 Chapter 7. Ordinary Differential Equations
function ydot = f(t,y)
ydot = [y(2); -1+y(2)^2];
With the differential equation written as a first-order system, y becomes a vector
with two components and so g(t, y) = y1. The code for g(t, y) is
function [gstop,isterminal,direction] = g(t,y)
gstop = y(1);
isterminal = 1;
direction = [];
The first output, gstop, is the value that we want to make zero. Setting the second
output, isterminal, to one indicates that the ordinary differential equation solver
should terminate when gstop is zero. Setting the third output, direction, to
the empty matrix indicates that the zero can be approached from either direction.
With these two functions available, the following statements compute and plot the
trajectory shown in Figure 7.7.
opts = odeset(’events’,@g);
y0 = [1; 0];
[t,y,tfinal] = ode45(@f,[0 Inf],y0,opts);
tfinal
plot(t,y(:,1),’-’,[0 tfinal],[1 0],’o’)
axis([-.1 tfinal+.1 -.1 1.1])
xlabel(’t’)
ylabel(’y’)
title(’Falling body’)
text(1.2, 0, [’tfinal = ’ num2str(tfinal)])
The terminating value of t is found to be tfinal = 1.6585.
The three sections of code for this example can be saved in three separate
M-files, with two functions and one script, or they can all be saved in one function
M-file. In the latter case, f and g become subfunctions and have to appear after
the main body of code.
Events detection is particularly useful in problems involving periodic phenom-
ena. The two-body problem provides a good example. Here is the first portion of a
function M-file, orbit.m. The input parameter is reltol, the desired local relative
tolerance.
function orbit(reltol)
y0 = [1; 0; 0; 0.3];
opts = odeset(’events’,@gstop,’reltol’,reltol);
[t,y,te,ye] = ode45(@twobody,[0 2*pi],y0,opts,y0);
tfinal = te(end)
yfinal = ye(end,1:2)
plot(y(:,1),y(:,2),’-’,0,0,’ro’)
axis([-.1 1.05 -.35 .35])
The function ode45 is used to compute the orbit. The first input argument is
a function handle, @twobody, that references the function defining the differential
7.10. Events 25
equations. The second argument to ode45 is any overestimate of the time interval
required to complete one period. The third input argument is y0, a 4-vector that
provides the initial position and velocity. The light body starts at (1, 0), which
is a point with a distance 1 from the heavy body, and has initial velocity (0, 0.3),
which is perpendicular to the initial position vector. The fourth input argument is
an options structure created by odeset that overrides the default value for reltol
and that specifies a function gstop that defines the events we want to locate. The
last argument is y0, an “extra” argument that ode45 passes on to both twobody
and gstop.
The code for twobody has to be modified to accept a third argument, even
though it is not used.
function ydot = twobody(t,y,y0)
r = sqrt(y(1)^2 + y(2)^2);
ydot = [y(3); y(4); -y(1)/r^3; -y(2)/r^3];
The ordinary differential equation solver calls the gstop function at every step
during the integration. This function tells the solver whether or not it is time to
stop.
function [val,isterm,dir] = gstop(t,y,y0)
d = y(1:2)-y0(1:2);
v = y(3:4);
val = d’*v;
isterm = 1;
dir = 1;
The 2-vector d is the difference between the current position and the starting point.
The 2-vector v is the velocity at the current position. The quantity val is the inner
product between these two vectors. Mathematically, the stopping function is
g(t, y) = ḋ(t)T d(t),
where
d = (y1(t)− y1(0), y2(t)− y2(0))T .
Points where g(t, y(t)) = 0 are the local minimum or maximum of d(t). By setting
dir = 1, we indicate that the zeros of g(t, y) must be approached from above, so
they correspond to minima. By setting isterm = 1, we indicate that computation
of the solution should be terminated at the first minimum. If the orbit is truly
periodic, then any minima of d occur when the body returns to its starting point.
Calling orbit with a very loose tolerance
orbit(2.0e-3)
produces
tfinal =
2.35087197761898
26 Chapter 7. Ordinary Differential Equations
yfinal =
0.98107659901079 -0.00012519138559
and plots Figure 7.8.
0 0.2 0.4 0.6 0.8 1
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Figure 7.8. Periodic orbit computed with loose tolerance.
You can see from both the value of yfinal and the graph that the orbit does
not quite return to the starting point. We need to request more accuracy.
orbit(1.0e-6)
produces
tfinal =
2.38025846171805
yfinal =
0.99998593905521 0.00000000032240
Now the value of yfinal is close enough to y0 that the graph of the orbit is effec-
tively closed.
7.11 Multistep Methods
A single-step numerical method has a short memory. The only information passed
from one step to the next is an estimate of the proper step size and, perhaps, the
value of f(tn, yn) at the point the two steps have in common.
As the name implies, a multistep method has a longer memory. After an initial
start-up phase, a pth-order multistep method saves up to perhaps a dozen values of
the solution, yn−p+1, yn−p+2, . . . , yn−1, yn, and uses them all to compute yn+1. In
fact, these methods can vary both the order, p, and the step size, h.
7.12. The Matlab ODE Solvers 27
Multistep methods tend to be more efficient than single-step methods for
problems with smooth solutions and high accuracy requirements. For example, the
orbits of planets and deep space probes are computed with multistep methods.
7.12 The Matlab ODE Solvers
This section is derived from the Algorithms portion of the Matlab Reference Man-
ual page for the ordinary differential equation solvers.
ode45 is based on an explicit Runge–Kutta (4, 5) formula, the Dormand–
Prince pair. It is a one-step solver. In computing y(tn+1), it needs only the solution
at the immediately preceding time point, y(tn). In general, ode45 is the first func-
tion to try for most problems.
ode23 is an implementation of an explicit Runge–Kutta (2, 3) pair of Bogacki
and Shampine’s. It is often more efficient than ode45 at crude tolerances and in
the presence of moderate stiffness. Like ode45, ode23 is a one-step solver.
ode113 uses a variable-order Adams–Bashforth–Moulton predictor-corrector
algorithm. It is often more efficient than ode45 at stringent tolerances and if the
ordinary differential equation file function is particularly expensive to evaluate.
ode113 is a multistep solver—it normally needs the solutions at several preceding
time points to compute the current solution.
The above algorithms are intended to solve nonstiff systems. If they appear
to be unduly slow, try using one of the stiff solvers below.
ode15s is a variable-order solver based on the numerical differentiation formu-
las (NDFs). Optionally, it uses the backward differentiation formulas (BDFs, also
known as Gear’s method), which are usually less efficient. Like ode113, ode15s is
a multistep solver. Try ode15s if ode45 fails or is very inefficient and you suspect
that the problem is stiff, or if you are solving a differential-algebraic problem.
ode23s is based on a modified Rosenbrock formula of order two. Because it is
a one-step solver, it is often more efficient than ode15s at crude tolerances. It can
solve some kinds of stiff problems for which ode15s is not effective.
ode23t is an implementation of the trapezoidal rule using a “free” interpolant.
Use this solver if the problem is only moderately stiff and you need a solution
without numerical damping. ode23t can solve differential-algebraic equations.
ode23tb is an implementation of TR-BDF2, an implicit Runge–Kutta formula
with a first stage that is a trapezoidal rule step and a second stage that is a BDF
of order two. By construction, the same iteration matrix is used in evaluating
both stages. Like ode23s, this solver is often more efficient than ode15s at crude
tolerances.
Here is a summary table from the Matlab Reference Manual. For each
function, it lists the appropriate problem type, the typical accuracy of the method,
and the recommended area of usage.
• ode45. Nonstiff problems, medium accuracy. Use most of the time. This
should be the first solver you try.
• ode23. Nonstiff problems, low accuracy. Use for large error tolerances or
moderately stiff problems.
28 Chapter 7. Ordinary Differential Equations
• ode113. Nonstiff problems, low to high accuracy. Use for stringent error tol-
erances or computationally intensive ordinary differential equation functions.
• ode15s. Stiff problems, low to medium accuracy. Use if ode45 is slow (stiff
systems) or there is a mass matrix.
• ode23s. Stiff problems, low accuracy. Use for large error tolerances with stiff
systems or with a constant mass matrix.
• ode23t. Moderately stiff problems, low accuracy. Use for moderately stiff
problems where you need a solution without numerical damping.
• ode23tb. Stiff problems, low accuracy. Use for large error tolerances with
stiff systems or if there is a mass matrix.
7.13 Errors
Errors enter the numerical solution of the initial value problem from two sources:
• discretization error,
• roundoff error.
Discretization error is a property of the differential equation and the numerical
method. If all the arithmetic could be performed with infinite precision, discretiza-
tion error would be the only error present. Roundoff error is a property of the
computer hardware and the program. It is usually far less important than the
discretization error, except when we try to achieve very high accuracy.
Discretization error can be assessed from two points of view, local and global.
Local discretization error is the error that would be made in one step if the previous
values were exact and if there were no roundoff error. Let un(t) be the solution of
the differential equation determined not by the original initial condition at t0 but
by the value of the computed solution at tn. That is, un(t) is the function of t
defined by
u̇n = f(t, un),
un(tn) = yn.
The local discretization error dn is the difference between this theoretical solution
and the computed solution (ignoring roundoff) determined by the same data at tn:
dn = yn+1 − un(tn+1).
Global discretization error is the difference between the computed solution,
still ignoring roundoff, and the true solution determined by the original initial con-
dition at t0, that is,
en = yn − y(tn).
The distinction between local and global discretization error can be easily seen
in the special case where f(t, y) does not depend on y. In this case, the solution
7.13. Errors 29
is simply an integral, y(t) =
∫ t
t0
f(τ)dτ . Euler’s method becomes a scheme for
numerical quadrature that might be called the “composite lazy man’s rectangle
rule.” It uses function values at the left-hand ends of the subintervals rather than
at the midpoints: ∫ tN
t0
f(τ)dτ ≈
N−1∑
0
hnf(tn).
The local discretization error is the error in one subinterval:
dn = hnf(tn)−
∫ tn+1
tn
f(τ)dτ,
and the global discretization error is the total error:
eN =
N−1∑
n=0
hnf(tn)−
∫ tN
t0
f(τ)dτ.
In this special case, each of the subintegrals is independent of the others (the sum
could be evaluated in any order), so the global error is the sum of the local errors:
eN =
N−1∑
n=0
dn.
In the case of a genuine differential equation where f(t, y) depends on y, the
error in any one interval depends on the solutions computed for earlier intervals.
Consequently, the relationship between the global error and the local errors is related
to the stability of the differential equation. For a single scalar equation, if the partial
derivative ∂f/∂y is positive, then the solution y(t) grows as t increases and the
global error will be greater than the sum of the local errors. If ∂f/∂y is negative,
then the global error will be less than the sum of the local errors. If ∂f/∂y changes
sign, or if we have a nonlinear system of equations where ∂f/∂y is a varying matrix,
the relationship between eN and the sum of the dn can be quite complicated and
unpredictable.
Think of the local discretization error as the deposits made to a bank account
and the global error as the overall balance in the account. The partial derivative
∂f/∂y acts like an interest rate. If it is positive, the overall balance is greater than
the sum of the deposits. If it is negative, the final error balance might well be less
than the sum of the errors deposited at each step.
Our code ode23tx, like all the production codes in Matlab, only attempts
to control the local discretization error. Solvers that try to control estimates of the
global discretization error are much more complicated, are expensive to run, and
are not very successful.
A fundamental concept in assessing the accuracy of a numerical method is its
order. The order is defined in terms of the local discretization error obtained if the
method is applied to problems with smooth solutions. A method is said to be of
order p if there is a number C so that
|dn| ≤ Chp+1n .
30 Chapter 7. Ordinary Differential Equations
The number C might depend on the partial derivatives of the function defining the
differential equation and on the length of the interval over which the solution is
sought, but it should be independent of the step number n and the step size hn.
The above inequality can be abbreviated using “big-oh notation”:
dn = O(h
p+1
n ).
For example, consider Euler’s method:
yn+1 = yn + hnf(tn, yn).
Assume the local solution un(t) has a continuous second derivative. Then, using
Taylor series near the point tn,
un(t) = un(tn) + (t− tn)u′n(tn) + O((t− tn)2).
Using the differential equation and the initial condition defining un(t),
un(tn+1) = yn + hnf(tn, yn) + O(h
2
n).
Consequently,
dn = yn+1 − un(tn+1) = O(h2n).
We conclude that p = 1, so Euler’s method is first order. The Matlab naming
conventions for ordinary differential equation solvers would imply that a function
using Euler’s method by itself, with fixed step size and no error estimate, should be
called ode1.
Now consider the global discretization error at a fixed point t = tf . As ac-
curacy requirements are increased, the step sizes hn will decrease, and the total
number of steps N required to reach tf will increase. Roughly, we shall have
N =
tf − t0
h
,
where h is the average step size. Moreover, the global error eN can be expressed as
a sum of N local errors coupled by factors describing the stability of the equations.
These factors do not depend in a strong way on the step sizes, and so we can say
roughly that if the local error is O(hp+1), then the global error will be N ·O(hp+1) =
O(hp). This is why p + 1 was used instead of p as the exponent in the definition of
order.
For Euler’s method, p = 1, so decreasing the average step size by a factor of
2 decreases the average local error by a factor of roughly 2p+1 =4, but about twice
as many steps are required to reach tf , so the global error is decreased by a factor
of only 2p =2. With higher order methods, the global error for smooth solutions is
reduced by a much larger factor.
It should be pointed out that in discussing numerical methods for ordinary
differential equations, the word “order” can have any of several different meanings.
The order of a differential equation is the index of the highest derivative appearing.
For example, d2y/dt2 = −y is a second-order differential equation. The order of
7.13. Errors 31
a system of equations sometimes refers to the number of equations in the system.
For example, ẏ = 2y − yz, ż = −z + yz is a second-order system. The order of a
numerical method is what we have been discussing here. It is the power of the step
size that appears in the expression for the global error.
One way of checking the order of a numerical method is to examine its behavior
if f(t, y) is a polynomial in t and does not depend on y. If the method is exact for
tp−1, but not for tp, then its order is not more than p. (The order could be less
than p if the method’s behavior for general functions does not match its behavior
for polynomials.) Euler’s method is exact if f(t, y) is constant, but not if f(t, y) = t,
so its order is not greater than one.
With modern computers, using IEEE floating-point double-precision arith-
metic, the roundoff error in the computed solution only begins to become important
if very high accuracies are requested or the integration is carried out over a long
interval. Suppose we integrate over an interval of length L = tf − t0. If the roundoff
error in one step is of size ², then the worst the roundoff error can be after N steps
of size h = L
N
is something like
N² =
L²
h
.
For a method with global discretization error of size Chp, the total error is something
like
Chp +
L²
h
.
For the roundoff error to be comparable with the discretization error, we need
h ≈
(
L²
C
) 1
p+1
.
The number of steps taken with this step size is roughly
N ≈ L
(
C
L²
) 1
p+1
.
Here are the numbers of steps for various orders p if L = 20: C = 100, and
² = 2−52:
p N
1 4.5 · 1017
3 5,647,721
5 37,285
10 864
These values of p are the orders for Euler’s method and for the Matlab
functions ode23 and ode45, and a typical choice for the order in the variable-
order method used by ode113. We see that the low-order methods have to take an
impractically large number of steps before this worst-case roundoff error estimate
becomes significant. Even more steps are required if we assume the roundoff error at
each step varies randomly. The variable-order multistep function ode113 is capable
of achieving such high accuracy that roundoff error can be a bit more significant
with it.
32 Chapter 7. Ordinary Differential Equations
7.14 Performance
We have carried out an experiment to see how all this applies in practice. The
differential equation is the harmonic oscillator
ẍ(t) = −x(t)
with initial conditions x(0) = 1, ẋ(0) = 0, over the interval 0 ≤ t ≤ 10π. The
interval is five periods of the periodic solution, so the global error can be computed
simply as the difference between the initial and final values of the solution. Since
the solution neither grows nor decays with t, the global error should be roughly
proportional to the local error.
The following Matlab script uses odeset to change both the relative and the
absolute tolerances. The refinement level is set so that one step of the algorithm
generates one row of output.
y0 = [1 0];
for k = 1:13
tol = 10^(-k);
opts = odeset(’reltol’,tol,’abstol’,tol,’refine’,1);
tic
[t,y] = ode23(@harmonic,[0 10*pi],y0’,opts);
time = toc;
steps = length(t)-1;
err = max(abs(y(end,:)-y0));
end
The differential equation is defined in harmonic.m.
function ydot = harmonic(t,y)
ydot = [y(2); -y(1)];
The script was run three times, with ode23, ode45, and ode113. The first
plot in Figure 7.9 shows how the global error varies with the requested tolerance
for the three routines. We see that the actual error tracks the requested tolerance
quite well. For ode23, the global error is about 36 times the tolerance; for ode45,
it is about 4 times the tolerance; and for ode113, it varies between 1 and 45 times
the tolerance.
The second plot in Figure 7.9 shows the numbers of steps required. The results
also fit our model quite well. Let τ denote the tolerance 10−k. For ode23, the
number of steps is about 10τ−1/3, which is the expected behavior for a third-order
method. For ode45, the number of steps is about 9τ−1/5, which is the expected
behavior for a fifth-order method. For ode113, the number of steps reflects the fact
that the solution is very smooth, so the method was often able to use its maximum
order, 13.
The third plot in Figure 7.9 shows the execution times, in seconds, on an
800 MHz Pentium III laptop. For this problem, ode45 is the fastest method for
tolerances of roughly 10−6 or larger, while ode113 is the fastest method for more
7.14. Performance 33
10
−12
10
−8
10
−4
10
0
e
rr
o
r
10
2
10
3
10
4
10
5
st
e
p
s
ode23
ode45
ode113
10
−13
10
−10
10
−7
10
−4
10
−1
10
−1
10
0
10
1
10
2
tim
e
tol
Figure 7.9. Performance of ordinary differential equation solvers.
stringent tolerances. The low-order method, ode23, takes a very long time to obtain
high accuracy.
This is just one experiment, on a problem with a very smooth and stable
solution.
34 Chapter 7. Ordinary Differential Equations
7.15 Further Reading
The Matlab ordinary differential equation suite is described in [7]. Additional
material on the numerical solution of ordinary differential equations, and especially
stiffness, is available in Ascher and Petzold [1], Brennan, Campbell, and Petzold [2],
and Shampine [6].
Exercises
7.1. The standard form of an ODE initial value problem is:
ẏ = f(t, y), y(t0) = y0.
Express this ODE problem in the standard form.
ü =
v
1 + t2
− sin r,
v̈ =
−u
1 + t2
+ cos r,
where r =
√
u̇2 + v̇2. The initial conditions are
u(0) = 1, v(0) = u̇(0) = v̇(0) = 0.
7.2. You invest $100 in a savings account paying 6% interest per year. Let y(t)
be the amount in your account after t years. If the interest is compounded
continuously, then y(t) solves the ODE initial value problem
Exercises 35
ẏ = ry, r = .06
y(0) = 100.
Compounding interest at a discrete time interval, h, corresponds to using a
finite difference method to approximate the solution to the differential equa-
tion. The time interval h is expressed as a fraction of a year. For example,
compounding monthly has h = 1/12. The quantity yn, the balance after n
time intervals, approximates the continuously compounded balance y(nh).
The banking industry effectively uses Euler’s method to compute compound
interest.
y0 = y(0),
yn+1 = yn + hryn.
This exercise asks you to investigate the use of higher order difference methods
to compute compound interest. What is the balance in your account after 10
years with each of the following methods of compounding interest?
Euler’s method, yearly.
Euler’s method, monthly.
Midpoint rule, monthly.
Trapezoid rule, monthly.
BS23 algorithm, monthly.
Continuous compounding.
7.3. (a) Show experimentally or algebraically that the BS23 algorithm is exact for
f(t, y) = 1, f(t, y) = t, and f(t, y) = t2, but not for f(t, y) = t3.
(b) When is the ode23 error estimator exact?
7.4. The error function erf(x) is usually defined by an integral,
erf(x) =
2√
π
∫ x
0
e−x
2
dx,
but it can also be defined as the solution to the differential equation
y′(x) =
2√
π
e−x
2
,
y(0) = 0.
Use ode23tx to solve this differential equation on the interval 0 ≤ x ≤ 2.
Compare the results with the built-in Matlab function erf(x) at the points
chosen by ode23tx.
7.5. (a) Write an M-file named myrk4.m, in the style of ode23tx.m, that imple-
ments the classical Runge–Kutta fixed step size algorithm. Instead of an
optional fourth argument rtol or opts, the required fourth argument should
be the step size h. Here is the proposed preamble.
36 Chapter 7. Ordinary Differential Equations
% function [tout,yout] = myrk4(F,tspan,y0,h,varargin)
% MYRK4 Classical fourth-order Runge-Kutta.
% Usage is the same as ODE23TX except the fourth
% argument is a fixed step size h.
% MYRK4(F,TSPAN,Y0,H) with TSPAN = [T0 TF] integrates
% the system of differential equations y’ = f(t,y)
% from t = T0 to t = TF. The initial condition
% is y(T0) = Y0.
% With no output arguments, MYRK4 plots the solution.
% With two output arguments, [T,Y] = MYRK4(..) returns
% T and Y so that Y(:,k) is the approximate solution at
% T(k). More than four input arguments,
% MYRK4(..,P1,P2,..), are passed on to F,
% F(T,Y,P1,P2,...).
(b) Roughly, how should the error behave if the step size h for classical Runge–
Kutta is cut in half? (Hint: Why is there a “4” in the name of myrk4?) Run
an experiment to illustrate this behavior.
(c) If you integrate the simple harmonic oscillator ÿ = −y over one full period,
0 ≤ t ≤ 2π, you can compare the initial and final values of y to get a measure
of the global accuracy. If you use your myrk4 with a step size h = π/50,
you should find that it takes 100 steps and computes a result with an error
of about 10−6. Compare this with the number of steps required by ode23,
ode45, and ode113 if the relative tolerance is set to 10−6 and the refinement
level is set to one. This is a problem with a very smooth solution, so you
should find that ode23 requires more steps, while ode45 and ode113 require
fewer.
7.6. The ordinary differential equation problem
ẏ = −1000(y − sin t) + cos t, y(0) = 1,
on the interval 0 ≤ t ≤ 1 is mildly stiff.
(a) Find the exact solution, either by hand or using dsolve from the Symbolic
Toolbox.
(b) Compute the solution with ode23tx. How many steps are required?
(c) Compute the solution with the stiff solver ode23s. How many steps are
required?
(d) Plot the two computed solutions on the same graph, with line style ’.’
for the ode23tx solution and ’o’ for the ode23s solution.
(e) Zoom in, or change the axis settings, to show a portion of the graph where
the solution is varying rapidly. You should see that both solvers are taking
small steps.
(f) Show a portion of the graph where the solution is varying slowly. You
should see that ode23tx is taking much smaller steps than ode23s.
Exercises 37
7.7. The following problems all have the same solution on 0 ≤ t ≤ π/2:
ẏ = cos t, y(0) = 0,
ẏ =
√
1− y2, y(0) = 0,
ÿ = −y, y(0) = 0, ẏ(0) = 1,
ÿ = − sin t, y(0) = 0, ẏ(0) = 1.
(a) What is the common solution y(t)?
(b) Two of the problems involve second derivatives, ÿ. Rewrite these problems
as first-order systems, ẏ = f(t, y), involving vectors y and f .
(c) What is the Jacobian, J = ∂f
∂y
, for each problem? What happens to each
Jacobian as t approaches π/2?
(d) The work required by a Runge–Kutta method to solve an initial value
problem ẏ = f(t, y) depends on the function f(t, y), not just the solution,
y(t). Use odeset to set both reltol and abstol to 10−9. How much work
does ode45 require to solve each problem? Why are some problems more
work than others?
(e) What happens to the computed solutions if the interval is changed to
0 ≤ t ≤ π?
(f) What happens on 0 ≤ t ≤ π if the second problem is changed to
ẏ =
√
|1− y2|, y(0) = 0.
7.8. Use the jacobian and eig functions in the Symbolic Toolbox to verify that
the Jacobian for the two-body problem is
J =
1
r5
0 0 r5 0
0 0 0 r5
2y21 − y22 3y1y2 0 0
3y1y2 2y22 − y21 0 0
and that its eigenvalues are
λ =
1
r3/2
√
2
i
−
√
2
−i
.
7.9. Verify that the matrix in the Lorenz equations
A =
−β 0 η
0 −σ σ
−η ρ −1
is singular if and only if
η = ±
√
β(ρ− 1).
Verify that the corresponding null vector is
ρ− 1
η
η
.
38 Chapter 7. Ordinary Differential Equations
7.10. The Jacobian matrix J for the Lorenz equations is not A, but is closely related
to A. Find J , compute its eigenvalues at one of the fixed points, and verify
that the fixed point is unstable.
7.11. Find the largest value of ρ in the Lorenz equations for which the fixed point
is stable.
7.12. All the values of ρ available with lorenzgui except ρ = 28 give trajectories
that eventually settle down to stable periodic orbits. In his book on the
Lorenz equations, Sparrow classifies a periodic orbit by what we might call
its signature, a sequence of +’s and −’s specifying the order of the critical
points that the trajectory circles during one period. A single + or − would
be the signature of a trajectory that circles just one critical point, except that
no such orbits exist. The signature ‘+−’ indicates that the trajectory circles
each critical point once. The signature ‘+ + + − + − −−’ would indicate a
very fancy orbit that circles the critical points a total of eight times before
repeating itself.
What are the signatures of the four different periodic orbits generated by
lorenzgui? Be careful—each of the signatures is different, and ρ = 99.65 is
particularly delicate.
7.13. What are the periods of the periodic orbits generated for the different values
of ρ available with lorenzgui?
7.14. The Matlab demos directory contains an M-file, orbitode, that uses ode45
to solve an instance of the restricted three-body problem. This involves the
orbit of a light object around two heavier objects, such as an Apollo capsule
around the earth and the moon. Run the demo and then locate its source
code with the statements
orbitode
which orbitode
Make your own copy of orbitode.m. Find these two statements:
tspan = [0 7];
y0 = [1.2; 0; 0; -1.04935750983031990726];
These statements set the time interval for the integration and the initial
position and velocity of the light object. Our question is, Where do these
values come from? To answer this question, find the statement
[t,y,te,ye,ie] = ode45(@f,tspan,y0,options);
Remove the semicolon and insert three more statements after it:
te
ye
ie
Run the demo again. Explain how the values of te, ye, and ie are related
to tspan and y0.
Exercises 39
7.15. A classical model in mathematical ecology is the Lotka–Volterra predator-
prey model. Consider a simple ecosystem consisting of rabbits that have an
infinite supply of food and foxes that prey on the rabbits for their food. This
is modeled by a pair of nonlinear, first-order differential equations:
dr
dt
= 2r − αrf, r(0) = r0,
df
dt
= −f + αrf, f(0) = f0,
where t is time, r(t) is the number of rabbits, f(t) is the number of foxes,
and α is a positive constant. If α = 0, the two populations do not interact,
the rabbits do what rabbits do best, and the foxes die off from starvation. If
α > 0, the foxes encounter the rabbits with a probability that is proportional
to the product of their numbers. Such an encounter results in a decrease
in the number of rabbits and (for less obvious reasons) an increase in the
number of foxes.
The solutions to this nonlinear system cannot be expressed in terms of other
known functions; the equations must be solved numerically. It turns out that
the solutions are always periodic, with a period that depends on the initial
conditions. In other words, for any r(0) and f(0), there is a value t = tp when
both populations return to their original values. Consequently, for all t,
r(t + tp) = r(t), f(t + tp) = f(t).
(a) Compute the solution with r0 = 300, f0 = 150, and α = 0.01. You should
find that tp is close to 5. Make two plots, one of r and f as functions of t
and one a phase plane plot with r as one axis and f as the other.
(b) Compute and plot the solution with r0 = 15, f0 = 22, and α = 0.01. You
should find that tp is close to 6.62.
(c) Compute and plot the solution with r0 = 102, f0 = 198, and α = 0.01.
Determine the period tp either by trial and error or with an event handler.
(d) The point (r0, f0) = (1/α, 2/α) is a stable equilibrium point. If the popu-
lations have these initial values, they do not change. If the initial populations
are close to these values, they do not change very much. Let u(t) = r(t)−1/α
and v(t) = f(t)− 2/α. The functions u(t) and v(t) satisfy another nonlinear
system of differential equations, but if the uv terms are ignored, the system
becomes linear. What is this linear system? What is the period of its periodic
solutions?
7.16. Many modifications of the Lotka–Volterra predator-prey model (see previous
problem) have been proposed to more accurately reflect what happens in
nature. For example, the number of rabbits can be prevented from growing
indefinitely by changing the first equation as follows:
dr
dt
= 2
(
1− r
R
)
r − αrf, r(0) = r0,
df
dt
= −f + αrf, y(0) = y0,
40 Chapter 7. Ordinary Differential Equations
where t is time, r(t) is the number of rabbits, f(t) is the number of foxes, α
is a positive constant, and R is a positive constant. Because α is positive, dr
dt
is negative whenever r ≥ R. Consequently, the number of rabbits can never
exceed R.
For α = 0.01, compare the behavior of the original model with the behavior
of this modified model with R = 400. In making this comparison, solve the
equations with r0 = 300 and f0 = 150 over 50 units of time. Make four
different plots:
• number of foxes and number of rabbits versus time for the original model,
• number of foxes and number of rabbits versus time for the modified
model,
• number of foxes versus number of rabbits for the original model,
• number of foxes versus number of rabbits for the modified model.
For all plots, label all curves and all axes and put a title on the plot. For
the last two plots, set the aspect ratio so that equal increments on the x- and
y-axes are equal in size.
7.17. An 80-kg paratrooper is dropped from an airplane at a height of 600m. After
5 s the chute opens. The paratrooper’s height as a function of time, y(t), is
given by
ÿ = −g + α(t)/m,
y(0) = 600 m,
ẏ(0) = 0 m/s,
where g = 9.81 m/s2 is the acceleration due to gravity and m = 80 kg is
the paratrooper’s mass. The air resistance α(t) is proportional to the square
of the velocity, with different proportionality constants before and after the
chute opens.
α(t) =
{
K1ẏ(t)2, t < 5 s, K2ẏ(t)2, t ≥ 5 s. (a) Find the analytical solution for free-fall, K1 = 0, K2 = 0. At what height does the chute open? How long does it take to reach the ground? What is the impact velocity? Plot the height versus time and label the plot appropriately. (b) Consider the case K1 = 1/15,K2 = 4/15. At what height does the chute open? How long does it take to reach the ground? What is the impact veloc- ity? Make a plot of the height versus time and label the plot appropriately. 7.18. Determine the trajectory of a spherical cannonball in a stationary Cartesian coordinate system that has a horizontal x-axis, a vertical y-axis, and an origin at the launch point. The initial velocity of the projectile in this coordinate system has magnitude v0 and makes an angle with respect to the x-axis of θ0 radians. The only forces acting on the projectile are gravity and the aerodynamic drag, D, which depends on the projectile’s speed relative to any wind that might be present. The equations describing the motion of the Exercises 41 projectile are ẋ = v cos θ, ẏ = v sin θ, θ̇ = −g v cos θ, v̇ = −D m − g sin θ. Constants for this problem are the acceleration of gravity, g = 9.81m/s2, the mass, m = 15 kg, and the initial speed, v0 = 50 m/s. The wind is assumed to be horizontal and its speed is a specified function of time, w(t). The aerodynamic drag is proportional to the square of the projectile’s velocity relative to the wind: D(t) = cρs 2 ( (ẋ− w(t))2 + ẏ2 ) , where c = 0.2 is the drag coefficient, ρ = 1.29 kg/m3 is the density of air, and s = 0.25 m2 is the projectile’s cross-sectional area. Consider four different wind conditions. • No wind. w(t) = 0 for all t. • Steady headwind. w(t) = −10 m/s for all t. • Intermittent tailwind. w(t) = 10 m/s if the integer part of t is even, and zero otherwise. • Gusty wind. w(t) is a Gaussian random variable with mean zero and standard deviation 10 m/s. The integer part of a real number t is denoted by btc and is computed in Mat- lab by floor(t). A Gaussian random variable with mean 0 and standard deviation σ is generated by sigma*randn (see Chapter 9, Random Numbers). For each of these four wind conditions, carry out the following computations. Find the 17 trajectories whose initial angles are multiples of 5 degrees, that is, θ0 = kπ/36 radians, k = 1, 2, . . . , 17. Plot all 17 trajectories on one figure. Determine which of these trajectories has the greatest downrange distance. For that trajectory, report the initial angle in degrees, the flight time, the downrange distance, the impact velocity, and the number of steps required by the ordinary differential equation solver. Which of the four wind conditions requires the most computation? Why? 7.19. In the 1968 Olympic games in Mexico City, Bob Beamon established a world record with a long jump of 8.90 m. This was 0.80 m longer than the previous world record. Since 1968, Beamon’s jump has been exceeded only once in competition, by Mike Powell’s jump of 8.95 m in Tokyo in 1991. After Bea- mon’s remarkable jump, some people suggested that the lower air resistance at Mexico City’s 2250m altitude was a contributing factor. This problem examines that possibility. The mathematical model is the same as the cannonball trajectory in the previous exercise. The fixed Cartesian coordinate system has a horizontal x-axis, a vertical y-axis, and an origin at the takeoff board. The jumper’s initial velocity has magnitude v0 and makes an angle with respect to the x-axis of θ0 radians. The only forces acting after takeoff are gravity and the 42 Chapter 7. Ordinary Differential Equations aerodynamic drag, D, which is proportional to the square of the magnitude of the velocity. There is no wind. The equations describing the jumper’s motion are ẋ = v cos θ, ẏ = v sin θ, θ̇ = −g v cos θ, v̇ = −D m − g sin θ. The drag is D = cρs 2 ( ẋ2 + ẏ2 ) . Constants for this exercise are the acceleration of gravity, g = 9.81 m/s2, the mass, m = 80 kg, the drag coefficient, c = 0.72, the jumper’s cross-sectional area, s = 0.50 m2, and the takeoff angle, θ0 = 22.5◦ = π/8 radians. Compute four different jumps, with different values for initial velocity, v0, and air density, ρ. The length of each jump is x(tf ), where the air time, tf , is determined by the condition y(tf ) = 0. (a) “Nominal” jump at high altitude. v0 = 10 m/s and ρ = 0.94 kg/m3. (b) “Nominal” jump at sea level. v0 = 10 m/s and ρ = 1.29 kg/m3. (c) Sprinter’s approach at high altitude. ρ = 0.94 kg/m3. Determine v0 so that the length of the jump is Beamon’s record, 8.90 m. (d) Sprinter’s approach at sea level. ρ = 1.29 kg/m3 and v0 is the value determined in (c). Present your results by completing the following table. v0 theta0 rho distance 10.0000 22.5000 0.9400 ??? 10.0000 22.5000 1.2900 ??? ??? 22.5000 0.9400 8.9000 ??? 22.5000 1.2900 ??? Which is more important, the air density or the jumper’s initial velocity? 7.20. A pendulum is a point mass at the end of a weightless rod of length L supported by a frictionless pin. If gravity is the only force acting on the pendulum, its oscillation is modeled by θ̈ = −(g/L) sin θ. Here θ is the angular position of the rod, with θ = 0 if the rod is hanging down from the pin and θ = π if the rod is precariously balanced above the pin. Take L = 30 cm and g = 981 cm/s2. The initial conditions are θ(0) = θ0, θ̇(0) = 0. If the initial angle θ0 is not too large, then the approximation sin θ ≈ θ Exercises 43 leads to a linearized equation θ̈ = −(g/L)θ that is easily solved. (a) What is the period of oscillation for the linearized equation? If we do not make the assumption that θ0 is small and do not replace sin θ by θ, then it turns out that the period T of the oscillatory motion is given by T (θ0) = 4(L/g) 1/2K(sin2 (θ0/2)), where K(s2) is the complete elliptic integral of the first kind, given by K(s2) = ∫ 1 0 dt√ 1− s2t2 √ 1− t2 . (b) Compute and plot T (θ0) for 0 ≤ θ0 ≤ 0.9999π two different ways. Use the Matlab function ellipke and also use numerical quadrature with quadtx. Verify that the two methods yield the same results, to within the quadrature tolerance. (c) Verify that for small θ0 the linear equation and the nonlinear equation have approximately the same period. (d) Compute the solutions to the nonlinear model over one period for several different values of θ0, including values near 0 and near π. Superimpose the phase plane plots of the solutions on one graph. 7.21. What effect does the burning of fossil fuels have on the carbon dioxide in the earth’s atmosphere? Even though today carbon dioxide accounts for only about 350 parts per million of the atmosphere, any increase has profound implications for our climate. An informative background article is available at a Web site maintained by the Lighthouse Foundation [5]. A model developed by J. C. G. Walker [9] was brought to our attention by Eric Roden. The model simulates the interaction of the various forms of carbon that are stored in three regimes: the atmosphere, the shallow ocean, and the deep ocean. The five principal variables in the model are all functions of time: p, partial pressure of carbon dioxide in the atmosphere; σs, total dissolved carbon concentration in the shallow ocean; σd, total dissolved carbon concentration in the deep ocean; αs, alkalinity in the shallow ocean; αd, alkalinity in the deep ocean. Three additional quantities are involved in equilibrium equations in the shal- low ocean: hs, hydrogen carbonate in the shallow ocean; cs, carbonate in the shallow ocean; 44 Chapter 7. Ordinary Differential Equations ps, partial pressure of gaseous carbon dioxide in the shallow ocean. The rate of change of the five principal variables is given by five ordinary differential equations. The exchange between the atmosphere and the shallow ocean involves a constant characteristic transfer time d and a source term f(t): dp dt = ps − p d + f(t) µ1 . The equations describing the exchange between the shallow and deep oceans involve vs and vd, the volumes of the two regimes: dσs dt = 1 vs ( (σd − σs)w − k1 − ps − p d µ2 ) , dσd dt = 1 vd (k1 − (σd − σs)w) , dαs dt = 1 vs ((αd − αs)w − k2) , dαd dt = 1 vd (k2 − (αd − αs)w) . The equilibrium between carbon dioxide and the carbonates dissolved in the shallow ocean is described by three nonlinear algebraic equations: hs = σs − ( σ2s − k3αs(2σs − αs) )1/2 k3 , cs = αs − hs 2 , ps = k4 h2s cs . The numerical values of the constants involved in the model are d = 8.64, µ1 = 4.95 · 102, µ2 = 4.95 · 10−2, vs = 0.12, vd = 1.23, w = 10−3, k1 = 2.19 · 10−4, k2 = 6.12 · 10−5, k3 = 0.997148, k4 = 6.79 · 10−2. Exercises 45 The source term f(t) describes the burning of fossil fuels in the modern industrial era. We will use a time interval that starts about a thousand years ago and extends a few thousand years into the future: 1000 ≤ t ≤ 5000. The initial values at t = 1000, p = 1.00, σs = 2.01, σd = 2.23, αs = 2.20, αd = 2.26, represent preindustrial equilibrium and remain nearly constant as long as the source term f(t) is zero. The following table describes one scenario for a source term f(t) that models the release of carbon dioxide from burning fossil fuels, especially gasoline. The amounts begin to be significant after 1850, peak near the end of this century, and then decrease until the supply is exhausted. year rate 1000 0.0 1850 0.0 1950 1.0 1980 4.0 2000 5.0 2050 8.0 2080 10.0 2100 10.5 2120 10.0 2150 8.0 2225 3.5 2300 2.0 2500 0.0 5000 0.0 Figure 7.10 shows this source term and its effect on the atmosphere and the ocean. The three graphs in the lower half of the figure show the atmospheric, shallow ocean, and deep ocean carbon. (The two alkalinity values are not plotted at all because they are almost constant throughout this entire simu- lation.) Initially, the carbon in the three regimes is nearly at equilibrium and so the amounts hardly change before 1850. Over the period 1850 ≤ t ≤ 2500, the upper half of Figure 7.10 shows the additional carbon produced by burning fossil fuels entering the system, and the lower half shows the system response. The atmosphere is the first to be affected, showing more than a fourfold increase in 500 years. Almost half of the carbon is then slowly transferred to the shallow ocean and eventually to the deep ocean. 46 Chapter 7. Ordinary Differential Equations (a) Reproduce Figure 7.10. Use pchiptx to interpolate the fuel table and ode23tx with the default tolerances to solve the differential equations. (b) How do the amounts of carbon in the three regimes at year 5000 compare with the amounts at year 1000? (c) When does the atmospheric carbon dioxide reach its maximum? (d) These equations are mildly stiff, because the various chemical reactions take place on very different time scales. If you zoom in on some portions of the graphs, you should see a characteristic sawtooth behavior caused by the small time steps required by ode23tx. Find such a region. (e) Experiment with other Matlab ordinary differential equation solvers, including ode23, ode45, ode113, ode23s, and ode15s. Try various tolerances and report computational costs by using something like odeset(’RelTol’,1.e-6,’AbsTol’,1.e-6,’stats’,’on’); Which method is preferable for this problem? 7.22. This problem makes use of quadrature, ordinary differential equations, and zero finding to study a nonlinear boundary value problem. The function y(x) is defined on the interval 0 ≤ x ≤ 1 by y′′ = y2 − 1, 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 1 2 3 4 time (yr) c a rb o n 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 5 10 15 fu e l Carbon in the atmosphere and ocean atmosphere shallow deep fossil fuel Figure 7.10. Carbon in the atmosphere and ocean. Exercises 47 y(0) = 0, y(1) = 1. This problem can be solved four different ways. Plot the four solutions ob- tained on a single figure, using subplot(2,2,1),..., subplot(2,2,4). (a) Shooting method. Suppose we know the value of η = y′(0). Then we could use an ordinary differential equation solver like ode23tx or ode45 to solve the initial value problem y′′ = y2 − 1, y(0) = 0, y′(0) = η. on the interval 0 ≤ x ≤ 1. Each value of η determines a different solution y(x; η) and corresponding value for y(1; η). The desired boundary condition y(1) = 1 leads to the definition of a function of η: f(η) = y(1; η)− 1. Write a Matlab function whose argument is η. This function should solve the ordinary differential equation initial problem and return f(η). Then use fzero or fzerotx to find a value η∗ so that f(η∗) = 0. Finally, use this η∗ in the initial value problem to get the desired y(x). Report the value of η∗ you obtain. (b) Quadrature. Observe that y′′ = y2 − 1 can be written d dx ( (y′)2 2 − y 3 3 + y ) = 0. This means that the expression κ = (y′)2 2 − y 3 3 + y is actually constant. Because y(0) = 0, we have y′(0) = √ 2κ. So, if we could find the constant κ, the boundary value problem would be converted into an initial value problem. Integrating the equation dx dy = 1√ 2(κ + y3/3− y) gives x = ∫ y 0 h(y, κ) dy, where h(y, κ) = 1√ 2(κ + y3/3− y) . 48 Chapter 7. Ordinary Differential Equations This, together with the boundary condition y(1) = 1, leads to the definition of a function g(κ): g(κ) = ∫ 1 0 h(y, κ) dy − 1. You need two Matlab functions, one that computes h(y, κ) and one that computes g(κ). They can be two separate M-files, but a better idea is to make h(y, κ) a function within g(κ). The function g(κ) should use quadtx to evaluate the integral of h(y, κ). The parameter κ is passed as an extra argument from g, through quadtx, to h. Then fzerotx can be used to find a value κ∗ so that g(κ∗) = 0. Finally, this κ∗ provides the second initial value necessary for an ordinary differential equation solver to compute y(x). Report the value of κ∗ you obtain. (c and d) Nonlinear finite differences. Partition the interval into n + 1 equal subintervals with spacing h = 1/(n + 1): xi = ih, i = 0, . . . , n + 1. Replace the differential equation with a nonlinear system of difference equa- tions involving n unknowns, y1, y2, . . . , yn: yi+1 − 2yi + yi−1 = h2(y2i − 1), i = 1, . . . , n. The boundary conditions are y0 = 0 and yn+1 = 1. A convenient way to compute the vector of second differences involves the n-by-n tridiagonal matrix A with −2’s on the diagonal, 1’s on the super- and subdiagonals, and 0’s elsewhere. You can generate a sparse form of this matrix with e = ones(n,1); A = spdiags([e -2*e e],[-1 0 1],n,n); The boundary conditions y0 = 0 and yn+1 = 1 can be represented by the n-vector b, with bi = 0, i = 1, . . . , n− 1, and bn = 1. The vector formulation of the nonlinear difference equation is Ay + b = h2(y2 − 1), where y2 is the vector containing the squares of the elements of y, that is, the Matlab element-by-element power y.^2. There are at least two ways to solve this system. (c) Linear iteration. This is based on writing the difference equation in the form Ay = h2(y2 − 1)− b. Start with an initial guess for the solution vector y. The iteration consists of plugging the current y into the right-hand side of this equation and then solving the resulting linear system for a new y. This makes repeated use of the sparse backslash operator with the iterated assignment statement Exercises 49 y = A(h^2*(y.^2 - 1) - b) It turns out that this iteration converges linearly and provides a robust method for solving the nonlinear difference equations. Report the value of n you use and the number of iterations required. (d) Newton’s method. This is based on writing the difference equation in the form F (y) = Ay + b− h2(y2 − 1) = 0. Newton’s method for solving F (y) = 0 requires a many-variable analogue of the derivative F ′(y). The analogue is the Jacobian, the matrix of partial derivatives J = ∂Fi ∂yj = A− h2diag(2y). In Matlab, one step of Newton’s method would be F = A*y + b - h^2*(y.^2 - 1); J = A - h^2*spdiags(2*y,0,n,n); y = y - JF; With a good starting guess, Newton’s method converges in a handful of iter- ations. Report the value of n you use and the number of iterations required. 7.23. The double pendulum is a classical physics model system that exhibits chaotic motion if the initial angles are large enough. The model, shown in Figure 7.11, involves two weights, or bobs, attached by weightless, rigid rods to each other and to a fixed pivot. There is no friction, so once initiated, the motion continues forever. The motion is fully described by the two angles θ1 and θ2 that the rods make with the negative y-axis. x y θ 1 θ 2 Figure 7.11. Double pendulum. Let m1 and m2 be the masses of the bobs and `1 and `2 be the lengths of the 50 Chapter 7. Ordinary Differential Equations rods. The positions of the bobs are x1 = `1 sin θ1, y1 = −`1 cos θ1, x2 = `1 sin θ1 + `2 sin θ2, y2 = −`1 cos θ1 − `2 cos θ2. The only external force is gravity, denoted by g. Analysis based on the Lagrangian formulation of classical mechanics leads to a pair of coupled, second-order, nonlinear ordinary differential equations for the two angles θ1(t) and θ2(t): (m1 + m2)`1θ̈1 + m2`2θ̈2 cos (θ1 − θ2) = −g(m1 + m2) sin θ1 −m2`2θ̇22 sin (θ1 − θ2), m2`1θ̈1 cos (θ1 − θ2) + m2`2θ̈2 = −gm2 sin θ2 + m2`1θ̇21 sin (θ1 − θ2). To rewrite these equations as a first-order system, introduce the 4-by-1 col- umn vector u(t): u = [θ1, θ2, θ̇1, θ̇2] T . With m1 = m2 = `1 = `2 = 1, c = cos (u1 − u2), and s = sin (u1 − u2), the equations become u̇1 = u3, u̇2 = u4, 2u̇3 + cu̇4 = −g sin u1 − su24, cu̇3 + u̇4 = −g sin u2 + su23. Let M = M(u) denote the 4-by-4 mass matrix M = 1 0 0 0 0 1 0 0 0 0 2 c 0 0 c 1 and let f = f(u) denote the 4-by-1 nonlinear force function f = u3 u4 −g sin u1 − su24 −g sin u2 + su23 . In matrix-vector notation, the equations are simply Mu̇ = f. This is an implicit system of differential equations involving a nonconstant, nonlinear mass matrix. The double pendulum problem is usually formulated without the mass matrix, but larger problems, with more degrees of freedom, Exercises 51 are frequently in implicit form. In some situations, the mass matrix is singular and it is not possible to write the equations in explicit form. The NCM M-file swinger provides an interactive graphical implementation of these equations. The initial position is determined by specifying the starting coordinates of the second bob, (x2, y2), either as arguments to swinger or by using the mouse. In most situations, this does not uniquely determine the starting position of the first bob, but there are only two possibilities and one of them is chosen arbitrarily. The initial velocities, θ̇1 and θ̇2, are zero. The numerical solution is carried out by ode23 because our textbook code, ode23tx, cannot handle implicit equations. The call to ode23 involves using odeset to specify the functions that generate the mass matrix and do the plotting opts = odeset(’mass’,@swingmass, ... ’outputfcn’,@swingplot); ode23(@swingrhs,tspan,u0,opts); The mass matrix function is function M = swingmass(t,u) c = cos(u(1)-u(2)); M = [1 0 0 0; 0 1 0 0; 0 0 2 c; 0 0 c 1]; The driving force function is function f = swingrhs(t,u) g = 1; s = sin(u(1)-u(2)); f = [u(3); u(4); -2*g*sin(u(1))-s*u(4)^2; -g*sin(u(2))+s*u(3)^2]; It would be possible to have just one ordinary differential equation function that returns Mf, but we want to emphasize the implicit facility. An internal function swinginit converts a specified starting point (x, y) to a pair of angles (θ1, θ2). If (x, y) is outside the circle √ x2 + y2 > `1 + `2,
then the pendulum cannot reach the specified point. In this case, we straighten
out the pendulum with θ1 = θ2 and point it in the given direction. If (x, y) is
inside the circle of radius two, we return one of the two possible configurations
that reach to that point.
Here are some questions to guide your investigation of swinger.
(a) When the initial point is outside the circle of radius two, the two rods
start out as one. If the initial angle is not too large, the double pendulum
continues to act pretty much like a single pendulum. But if the initial angles
are large enough, chaotic motion ensues. Roughly what initial angles lead to
chaotic motion?
(b) The default initial condition is
52 Chapter 7. Ordinary Differential Equations
swinger(0.862,-0.994)
Why is this orbit interesting? Can you find any similar orbits?
(c) Run swinger for a while, then click on its stop button. Go to the Matlab
command line and type get(gcf,’userdata’). What is returned?
(d) Modify swinginit so that, when the initial point is inside the circle of
radius two, the other possible initial configuration is chosen.
(e) Modify swinger so that masses other than m1 = m2 = 1 are possible.
(f) Modify swinger so that lengths other than `1 = `2 = 1 are possible. This
is trickier than changing the masses because the initial geometry is involved.
(g) What role does gravity play? How would the behavior of a double pendu-
lum change if you could take it to the moon? How does changing the value of
g in swingrhs affect the speed of the graphics display, the step sizes chosen
by the ordinary differential equation solver, and the computed values of t?
(h) Combine swingmass and swingrhs into one function, swingode. Elimi-
nate the mass option and use ode23tx instead of ode23.
(i) Are these equations stiff?
(j) This is a difficult question. The statement swinger(0,2) tries to deli-
cately balance the pendulum above its pivot point. The pendulum does stay
there for a while, but then loses its balance. Observe the value of t displayed
in the title for swinger(0,2). What force knocks the pendulum away from
the vertical position? At what value of t does this force become noticeable?
Bibliography
[1] U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Dif-
ferential Equations and Differential-Algebraic Equations, SIAM, Philadelphia,
1998.
[2] K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of
Initial Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia,
1996.
[3] P. Bogacki and L. F. Shampine, A 3(2) pair of Runge-Kutta formulas, Ap-
plied Mathematics Letters, 2 (1989), pp. 1–9.
[4] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E.
Knuth, On the Lambert W function, Advances in Computational Mathematics,
5 (1996), pp. 329–359.
http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW
[5] Lighthouse Foundation.
http://www.lighthouse-foundation.org
[6] L. F. Shampine, Numerical Solution of Ordinary Differential Equations, Chap-
man and Hall, New York, 1994.
[7] L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM
Journal on Scientific Computing, 18 (1997), pp. 1–22.
[8] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attrac-
tors, Springer-Verlag, New York, 1982.
[9] J. C. G. Walker, Numerical Adventures with Geochemical Cycles, Oxford
University Press, New York, 1991.
53
Chapter 8
Fourier Analysis
We all use Fourier analysis every day without even knowing it. Cell phones, disc
drives, DVDs, and JPEGs all involve fast finite Fourier transforms. This chapter
discusses both the computation and the interpretation of FFTs.
The acronym FFT is ambiguous. The first F stands for both “fast” and
“finite.” A more accurate abbreviation would be FFFT, but nobody wants to use
that. In Matlab the expression fft(x) computes the finite Fourier transform of
any vector x. The computation is fast if the integer n = length(x) is the product
of powers of small primes. We discuss this algorithm in section 8.6.
8.1 Touch-Tone Dialing
Touch-tone telephone dialing is an example of everyday use of Fourier analysis. The
basis for touch-tone dialing is the Dual Tone Multi-Frequency (DTMF) system. The
program touchtone demonstrates how DTMF tones are generated and decoded.
The telephone dialing pad acts as a 4-by-3 matrix (Figure 8.1). Associated with
each row and column is a frequency. These basic frequencies are
fr = [697 770 852 941];
fc = [1209 1336 1477];
If s is a character that labels one of the buttons on the keypad, the corre-
sponding row index k and column index j can be found with
switch s
case ’*’, k = 4; j = 1;
case ’0’, k = 4; j = 2;
case ’#’, k = 4; j = 3;
otherwise,
d = s-’0’; j = mod(d-1,3)+1; k = (d-j)/3+1;
end
February 15, 2008
1
2 Chapter 8. Fourier Analysis
1209 1336 1477
697
770
852
941
Figure 8.1. Telephone keypad.
A key parameter in digital sound is the sampling rate.
Fs = 32768
A vector of points in the time interval 0 ≤ t ≤ 0.25 at this sampling rate is
t = 0:1/Fs:0.25
The tone generated by the button in position (k,j) is obtained by superimposing
the two fundamental tones with frequencies fr(k) and fc(j).
y1 = sin(2*pi*fr(k)*t);
y2 = sin(2*pi*fc(j)*t);
y = (y1 + y2)/2;
If your computer is equipped with a sound card, the Matlab statement
sound(y,Fs)
plays the tone.
Figure 8.2 is the display produced by touchtone for the ’1’ button. The
top subplot depicts the two underlying frequencies and the bottom subplot shows a
portion of the signal obtained by averaging the sine waves with those frequencies.
The data file touchtone.mat contains a recording of a telephone being dialed.
Is it possible to determine the phone number by listening to the signal generated?
The statement
load touchtone
loads both a signal y and a sample rate Fs in the workspace. In order to reduce
file size, the vector y has been saved with 8-bit integer components in the range
−127 ≤ yk ≤ 127. The statement
8.1. Touch-Tone Dialing 3
400 600 800 1000 1200 1400 1600
0
0.5
1
f(Hz)
1
0 0.005 0.01 0.015
−1
−0.5
0
0.5
1
t(seconds)
Figure 8.2. The tone generated by the 1 button.
1 2 3 4 5 6 7 8 9
−1
0
1
Figure 8.3. Recording of an 11-digit telephone number.
y = double(y)/128;
rescales the vector and converts it to double precision for later use. The statements
n = length(y);
t = (0:n-1)/Fs
reproduce the sample times of the recording. The last component of t is 9.1309,
indicating that the recording lasts a little over 9 s. Figure 8.3 is a plot of the entire
signal.
This signal is noisy. You can even see small spikes on the graph at the times
the buttons were clicked. It is easy to see that 11 digits were dialed, but, on this
scale, it is impossible to determine the specific digits.
4 Chapter 8. Fourier Analysis
600 800 1000 1200 1400 1600
0
200
400
600
Figure 8.4. FFT of the recorded signal.
Figure 8.4 shows the magnitude of the FFT of the signal, which is the key to
determining the individual digits.
The plot was produced with
p = abs(fft(y));
f = (0:n-1)*(Fs/n);
plot(f,p);
axis([500 1700 0 600])
The x-axis corresponds to frequency. The axis settings limit the display to the
range of the DTMF frequencies. There are seven peaks, corresponding to the seven
basic frequencies. This overall FFT shows that all seven frequencies are present
someplace in the signal, but it does not help determine the individual digits.
The touchtone program also lets you break the signal into 11 equal segments
and analyze each segment separately. Figure 8.5 is the display from the first seg-
ment.
For this segment, there are only two peaks, indicating that only two of the
basic frequencies are present in this portion of the signal. These two frequencies
come from the ’1’ button. You can also see that the waveform of a short portion
of the first segment is similar to the waveform that our synthesizer produces for the
’1’ button. So we can conclude that the number being dialed in touchtones starts
with a 1. Exercise 8.1 asks you to continue the analysis and identify the complete
phone number.
8.2 Finite Fourier Transform
The finite, or discrete, Fourier transform of a complex vector y with n elements is
another complex vector Y with n elements
Yk =
n−1∑
j=0
ωjkyj ,
where ω is a complex nth root of unity:
ω = e−2πi/n.
8.2. Finite Fourier Transform 5
1 2 3 4 5 6 7 8 9
−1
−0.5
0
0.5
1
600 800 1000 1200 1400 1600
0
100
200
300
0.38 0.385 0.39 0.395 0.4 0.405 0.41 0.415 0.42
−1
−0.5
0
0.5
1
Figure 8.5. The first segment and its FFT.
In this chapter, the mathematical notation follows conventions common in signal
processing literature where i =
√
−1 is the complex unit and j and k are indices
that run from 0 to n− 1.
The Fourier transform can be expressed with matrix-vector notation:
Y = Fy,
where the Fourier matrix F has elements
fk,j = ω
jk.
It turns out that F is nearly its own inverse. More precisely, FH , the complex
conjugate transpose of F , satisfies
FHF = nI,
so
F−1 =
1
n
FH .
This allows us to invert the Fourier transform:
y =
1
n
FHY.
Hence
yj =
1
n
n−1∑
k=0
Ykω̄
jk,
6 Chapter 8. Fourier Analysis
where ω̄ is the complex conjugate of ω:
ω̄ = e2πi/n.
We should point out that this is not the only notation for the finite Fourier
transform in common use. The minus sign in the definition of ω after the first equa-
tion sometimes occurs instead in the definition of ω̄ used in the inverse transform.
The 1/n scaling factor in the inverse transform is sometimes replaced with 1/
√
n
scaling factors in both transforms.
In Matlab, the Fourier matrix F can be generated for any given n by
omega = exp(-2*pi*i/n);
j = 0:n-1;
k = j’
F = omega.^(k*j)
The quantity k*j is an outer product, an n-by-n matrix whose elements are the
products of the elements of two vectors. However, the built-in function fft takes
the finite Fourier transform of each column of a matrix argument, so an easier, and
quicker, way to generate F is
F = fft(eye(n))
8.3 fftgui
The GUI fftgui allows you to investigate properties of the finite Fourier transform.
If y is a vector containing a few dozen elements,
fftgui(y)
produces four plots.
real(y) imag(y)
real(fft(y)) imag(fft(y))
You can use the mouse to move any of the points in any of the plots, and the points
in the other plots respond.
Please run fftgui and try the following examples. Each illustrates some
property of the Fourier transform. If you start with no arguments,
fftgui
all four plots are initialized to zeros(1,32). Click your mouse in the upper left-
hand corner of the upper left-hand plot. You are taking the fft of the zeroth unit
vector, with one in the first component and zeros elsewhere. This should produce
Figure 8.6.
The real part of the result is constant and the imaginary part is zero. You
can also see this from the definition
Yk =
n−1∑
j=0
yje
−2ijkπ/n, k = 0, . . . , n− 1
8.3. fftgui 7
real(y) imag(y)
real(fft(y)) imag(fft(y))
Figure 8.6. FFT of the first unit vector is constant.
real(y) imag(y)
real(fft(y)) imag(fft(y))
Figure 8.7. FFT of the second unit vector is a pure sinusoid.
if y0 = 1 and y1 = · · · = yn−1 = 0. The result is
Yk = 1 · e0 + 0 + · · ·+ 0 = 1 for all k.
Click y0 again, hold the mouse down, and move the mouse vertically. The
amplitude of the constant result varies accordingly.
Next try the second unit vector. Use the mouse to set y0 = 0 and y1 = 1.
This should produce Figure 8.7. You are seeing the graph of
Yk = 0 + 1 · e−2ikπ/n + 0 + · · ·+ 0.
8 Chapter 8. Fourier Analysis
The nth root of unity can also be written
ω = cos δ − i sin δ, where δ = 2π/n.
Consequently, for k = 0, . . . , n− 1,
real(Yk) = cos kδ, imag(Yk) = − sin kδ.
We have sampled two trig functions at n equally spaced points in the interval
0 ≤ x < 2π. The first sample point is x = 0 and the last sample point is x = 2π−δ.
real(y) imag(y)
real(fft(y)) imag(fft(y))
Figure 8.8. FFT is the sum of two sinusoids.
Now set y2 = 1 and vary y4 with the mouse. One snapshot is Figure 8.8. We
have graphs of
cos 2kδ + η cos 4kδ and − sin 2kδ − η sin 4kδ
for various values of η = y4.
The point just to the right of the midpoint of the x-axis is particularly impor-
tant. It is known as the Nyquist point. With the points numbered from 0 to n− 1
for even n, it’s the point with index n
2
. If n = 32, it’s point number 16. Figure 8.9
shows that the fft of a unit vector at the Nyquist point is a sequence of alternating
+1’s and −1’s.
Now let’s look at some symmetries in the FFT. Make several random clicks on
the real(y) plot. Leave the imag(y) plot flat zero. Figure 8.10 shows an example.
Look carefully at the two fft plots. Ignoring the first point in each plot, the real
part is symmetric about the Nyquist point and the imaginary part is antisymmetric
about the Nyquist point. More precisely, if y is any real vector of length n and
Y = fft(y), then
real(Y0) =
∑
yj , imag(Y0) = 0,
real(Yj) = real(Yn−j), imag(Yj) = −imag(Yn−j), j = 1, . . . , n/2.
8.4. Sunspots 9
real(y) imag(y)
real(fft(y)) imag(fft(y))
Figure 8.9. The Nyquist point.
real(y) imag(y)
real(fft(y)) imag(fft(y))
Figure 8.10. Symmetry about the Nyquist point.
8.4 Sunspots
For centuries, people have noted that the face of the sun is not constant or uniform in
appearance, but that darker regions appear at random locations on a cyclical basis.
This activity is correlated with weather and other economically significant terrestrial
phenomena. In 1848, Rudolf Wolfer proposed a rule that combined the number and
size of these sunspots into a single index. Using archival records, astronomers
have applied Wolfer’s rule to determine sunspot activity back to the year 1700.
Today the sunspot index is measured by many astronomers, and the worldwide
distribution of the data is coordinated by the Solar Influences Data Center at the
10 Chapter 8. Fourier Analysis
Royal Observatory of Belgium [4].
The text file sunspot.dat in the Matlab demos directory has two columns
of numbers. The first column is the years from 1700 to 1987 and the second column
is the average Wolfer sunspot number for each year.
load sunspot.dat
t = sunspot(:,1)’;
wolfer = sunspot(:,2)’;
n = length(wolfer);
There is a slight upward trend to the data. A least squares fit gives the trend
line.
c = polyfit(t,wolfer,1);
trend = polyval(c,t);
plot(t,[wolfer; trend],’-’,t,wolfer,’k.’)
xlabel(’year’)
ylabel(’Wolfer index’)
title(’Sunspot index with linear trend’)
1700 1750 1800 1850 1900 1950
0
20
40
60
80
100
120
140
160
180
200
year
W
o
lfe
r
in
d
e
x
Sunspot index with linear trend
Figure 8.11. Sunspot index.
You can definitely see the cyclic nature of the phenomenon (Figure 8.11). The
peaks and valleys are a little more than 10 years apart.
Now subtract off the linear trend and take the FFT.
y = wolfer - trend;
Y = fft(y);
The vector |Y |2 is the power in the signal. A plot of power versus frequency
is a periodogram (Figure 8.12). We prefer to plot |Y |, rather than |Y |2, because the
8.4. Sunspots 11
scaling is not so exaggerated. The sample rate for these data is one observation per
year, so the frequency f has units of cycles per year.
Fs = 1; % Sample rate
f = (0:n/2)*Fs/n;
pow = abs(Y(1:n/2+1));
pmax = 5000;
plot([f; f],[0*pow; pow],’c-’, f,pow,’b.’, ...
’linewidth’,2,’markersize’,16)
axis([0 .5 0 pmax])
xlabel(’cycles/year’)
ylabel(’power’)
title(’Periodogram’)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
cycles/year
p
o
w
e
r
Periodogram
Figure 8.12. Periodogram of the sunspot index.
The maximum power occurs near frequency = 0.09 cycles/year. We would
like to know the corresponding period in years/cycle. Let’s zoom in on the plot and
use the reciprocal of frequency to label the x-axis.
k = 0:44;
f = k/n;
pow = pow(k+1);
plot([f; f],[0*pow; pow],’c-’,f,pow,’b.’, ...
’linewidth’,2,’markersize’,16)
axis([0 max(f) 0 pmax])
k = 2:3:41;
f = k/n;
period = 1./f;
periods = sprintf(’%5.1f|’,period);
12 Chapter 8. Fourier Analysis
set(gca,’xtick’,f)
set(gca,’xticklabel’,periods)
xlabel(’years/cycle’)
ylabel(’power’)
title(’Periodogram detail’)
144.0 57.6 36.0 26.2 20.6 16.9 14.4 12.5 11.1 9.9 9.0 8.2 7.6 7.0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
years/cycle
p
o
w
e
r
Periodogram detail
Figure 8.13. Detail of periodogram shows 11-year cycle.
As expected, there is a very prominent cycle with a length of about 11.1 years
(Figure 8.13). This shows that, over the last 300 years, the period of the sunspot
cycle has been slightly over 11 years.
The code for this section is in the NCM collection sunspotstx.m. The older
version in toolbox/matlab/demos/sunspots.m uses slightly different notation.
8.5 Periodic Time Series
The tones generated by a touch-tone telephone and the Wolfer sunspot index are
two examples of periodic time series, that is, functions of time that exhibit periodic
behavior, at least approximately. Fourier analysis allows us to estimate the period
from a discrete set of values sampled at a fixed rate. The following table shows the
relationship between the various quantities involved in this analysis.
8.6. Fast Finite Fourier Transform 13
y data
Fs samples/unit-time
n = length(y) number of samples
t = (0:n-1)/Fs total time
dt = 1/Fs time increment
Y = fft(y) finite Fourier transform
abs(Y) amplitude of FFT
abs(Y).^2 power
f = (0:n-1)*(Fs/n) frequency, cycles/unit-time
(n/2)*(Fs/n) = Fs/2 Nyquist frequency
p = 1./f period, unit-time/cycle
The periodogram is a plot of the FFT amplitude abs(Y), or power abs(Y).^2,
versus the frequency f. You only need to plot the first half because the second half
is a reflection of the first half about the Nyquist frequency.
8.6 Fast Finite Fourier Transform
One-dimensional FFTs with a million points and two-dimensional 1000-by-1000
transforms are common. The key to modern signal and image processing is the
ability to do these computations rapidly.
Direct application of the definition
Yk =
n−1∑
j=0
ωjkyj , k = 0, . . . , n− 1,
requires n multiplications and n additions for each of the n components of Y for
a total of 2n2 floating-point operations. This does not include the generation of
the powers of ω. A computer capable of doing one multiplication and addition
every microsecond would require a million seconds, or about 11.5 days, to do a
million-point FFT.
Several people discovered fast FFT algorithms independently and many people
have since contributed to their development, but it was a 1965 paper by John Tukey
of Princeton University and John Cooley of IBM Research that is generally credited
as the starting point for the modern usage of the FFT.
Modern fast FFT algorithms have computational complexity O(n log2n) in-
stead of O(n2). If n is a power of 2, a one-dimensional FFT of length n requires
fewer than 3n log2n floating-point operations. For n = 2
20, that’s a factor of almost
35,000 faster than 2n2. Even if n = 1024 = 210, the factor is about 70.
With Matlab 6.5 and a 700 MHz Pentium laptop, the time required for
fft(x) if length(x) is 220 = 1048576 is about 1 s. The built-in fft function is
based on FFTW, “The Fastest Fourier Transform in the West,” developed at M.I.T.
by Matteo Frigo and Steven G. Johnson [2].
The key to the fast FFT algorithms is the fact that the square of the 2nth
14 Chapter 8. Fourier Analysis
root of unity is the nth root of unity. Using complex notation,
ω = ωn = e
−2πi/n,
we have
ω22n = ωn.
The derivation of the fast algorithm starts with the definition of the finite
Fourier transform:
Yk =
n−1∑
j=0
ωjkyj , k = 0, . . . , n− 1.
Assume that n is even and that k ≤ n/2− 1. Divide the sum into terms with even
subscripts and terms with odd subscripts.
Yk =
∑
even j
ωjkyj +
∑
odd j
ωjkyj
=
n/2−1∑
j=0
ω2jky2j + ω
k
n/2−1∑
j=0
ω2jky2j+1.
The two sums on the right are components of the FFTs of length n/2 of the portions
of y with even and odd subscripts. In order to get the entire FFT of length n, we
have to do two FFTs of length n/2, multiply one of these by powers of ω, and
concatenate the results.
The relationship between an FFT of length n and two FFTs of length n/2 can
be expressed compactly in Matlab. If n = length(y) is even,
omega = exp(-2*pi*i/n);
k = (0:n/2-1)’;
w = omega .^ k;
u = fft(y(1:2:n-1));
v = w.*fft(y(2:2:n));
then
fft(y) = [u+v; u-v];
Now, if n is not only even but actually a power of 2, the process can be
repeated. The FFT of length n is expressed in terms of two FFTs of length n/2,
then four FFTs of length n/4, then eight FFTs of length n/8, and so on until we
reach n FFTs of length one. An FFT of length one is just the number itself. If
n = 2p, the number of steps in the recursion is p. There is O(n) work at each step,
so the total amount of work is
O(np) = O(n log2n).
If n is not a power of two, it is still possible to express the FFT of length n
in terms of several shorter FFTs. An FFT of length 100 is two FFTs of length 50,
8.7. ffttx 15
or four FFTs of length 25. An FFT of length 25 can be expressed in terms of five
FFTs of length 5. If n is not a prime number, an FFT of length n can be expressed
in terms of FFTs whose lengths divide n. Even if n is prime, it is possible to embed
the FFT in another whose length can be factored. We do not go into the details of
these algorithms here.
The fft function in older versions of Matlab used fast algorithms if the
length was a product of small primes. Beginning with Matlab 6, the fft function
uses fast algorithms even if the length is prime (see [2]).
8.7 ffttx
Our textbook function ffttx combines the two basic ideas of this chapter. If n is
a power of 2, it uses the O(n log2n) fast algorithm. If n has an odd factor, it uses
the fast recursion until it reaches an odd length, then sets up the discrete Fourier
matrix and uses matrix-vector multiplication.
function y = ffttx(x)
%FFTTX Textbook Fast Finite Fourier Transform.
% FFTTX(X) computes the same finite Fourier transform
% as FFT(X). The code uses a recursive divide and
% conquer algorithm for even order and matrix-vector
% multiplication for odd order. If length(X) is m*p
% where m is odd and p is a power of 2, the computational
% complexity of this approach is O(m^2)*O(p*log2(p)).
x = x(:);
n = length(x);
omega = exp(-2*pi*i/n);
if rem(n,2) == 0
% Recursive divide and conquer
k = (0:n/2-1)’;
w = omega .^ k;
u = ffttx(x(1:2:n-1));
v = w.*ffttx(x(2:2:n));
y = [u+v; u-v];
else
% The Fourier matrix
j = 0:n-1;
k = j’;
F = omega .^ (k*j);
y = F*x;
end
16 Chapter 8. Fourier Analysis
8.8 fftmatrix
The n-by-n matrix F generated by the Matlab statement
F = fft(eye(n,n))
is a complex matrix whose elements are powers of the nth root of unity
ω = e−2πi/n
The statement
plot(fft(eye(n,n)))
connects the elements of each column of F and thereby generates a subgraph of the
graph on n points. If n is prime, connecting the elements of all columns generates
the complete graph on n points. If n is not prime, the sparsity of the graph of all
columns is related to the speed of the FFT algorithm. The graphs for n = 8, 9, 10,
and 11 are shown in Figure 8.14.
8 9
10 11
Figure 8.14. Graphs of FFT matrix.
Because n = 11 is prime, the corresponding graph shows all possible connections.
But the other three values of n are not prime. Some of the links in their graphs
are missing, indicating that the FFT of a vector with that many points can be
computed more quickly.
The program fftmatrix allows you to investigate these graphs.
fftmatrix(n)
plots all the columns of the Fourier matrix of order n.
8.9. Other Fourier Transforms and Series 17
fftmatrix(n,j)
plots only the j+1st column.
fftmatrix
defaults to fftmatrix(10,4). In all cases, push buttons allow n, j and the choice
between one or all columns be changed.
8.9 Other Fourier Transforms and Series
We have been studying the finite Fourier transform, which converts one finite se-
quence of coefficients into another sequence of the same length, n. The transform
is
Yk =
n−1∑
j=0
yje
−2ijkπ/n, k = 0, . . . , n− 1.
The inverse transform is
yj =
1
n
n−1∑
k=0
Yke
2ijkπ/n, j = 0, . . . , n− 1.
The Fourier integral transform converts one complex function into another.
The transform is
F (µ) =
∫ ∞
−∞
f(t)e−2πiµtdt.
The inverse transform is
f(t) =
∫ ∞
−∞
F (µ)e2πiµtdµ.
The variables t and µ run over the entire real line. If t has units of seconds, then µ
has units of radians per second. Both of the functions f(t) and F (µ) are complex
valued, but in most applications the imaginary part of f(t) is zero.
Alternative units use ν = 2πµ, which has units of cycles or revolutions per
second. With this change of variable, there are no factors of 2π in the exponentials,
but there are factors of 1/
√
2π in front of the integrals, or a single factor of 1/(2π)
in the inverse transform. The Matlab Symbolic Toolbox uses this alternative
notation with the single factor in the inverse transform.
A Fourier series converts a periodic function into an infinite sequence of Fourier
coefficients. Let f(t) be the periodic function and let L be its period, so
f(t + L) = f(t) for all t.
The Fourier coefficients are given by integrals over the period
cj =
1
L
∫ L/2
−L/2
f(t)e−2πijtdt, j = . . . ,−1, 0, 1, . . . .
18 Chapter 8. Fourier Analysis
With these coefficients, the complex form of the Fourier series is
f(t) =
∞∑
j=−∞
cje
2πijt/L.
A discrete time Fourier transform converts an infinite sequence of data values
into a periodic function. Let xk be the sequence, with the index k taking on all
integer values, positive and negative.
The discrete time Fourier transform is the complex-valued periodic function
X(eiω) =
∞∑
k=−∞
xke
ikω.
The sequence can then be represented as
xk =
1
2π
∫ π
−π
X(eiω)e−ikωdω, k = . . . ,−1, 0, 1, . . . .
The Fourier integral transform involves only integrals. The finite Fourier trans-
form involves only finite sums of coefficients. Fourier series and the discrete time
Fourier transform involve both integrals and sequences. It is possible to “morph”
any of the transforms into any of the others by taking limits or restricting domains.
Start with a Fourier series. Let L, the length of the period, become infinite
and let j/L, the coefficient index scaled by the period length, become a continuous
variable, µ. Then the Fourier coefficients cj become the Fourier transform F (µ).
Again, start with a Fourier series. Interchanging the roles of the periodic
function and the infinite sequence of coefficients leads to the discrete time Fourier
transform.
Start with a Fourier series a third time. Now restrict t to a finite number
of integral values, k, and restrict j to the same finite number of values. Then the
Fourier coefficients become the finite Fourier transform.
In the Fourier integral transform context, Parseval’s theorem says
∫ +∞
−∞
|f(t)|2dt =
∫ +∞
−∞
|F (µ)|2dµ.
This quantity is known as the total power in a signal.
8.10 Further Reading
Van Loan [5] describes the computational framework for the fast transforms. A
page of links at the FFTW Web site [3] provides useful information.
Exercises
8.1. What is the telephone number recorded in touchtone.mat and analyzed by
touchtone.m?
Exercises 19
8.2. Modify touchtone.m so that it can dial a telephone number specified by an
input argument, such as touchtone(’1-800-555-1212’).
8.3. Our version of touchtone.m breaks the recording into a fixed number of
equally spaced segments, each corresponding to a single digit. Modify touchtone
so that it automatically determines the number and the possibly disparate
lengths of the segments.
8.4. Investigate the use of the Matlab functions audiorecorder andaudioplayer,
or some other system for making digital recordings. Make a recording of a
phone number and analyze it with your modified version of touchtone.m.
8.5. Recall that the Fourier matrix F is the n-by-n complex matrix with elements
fk,j = ω
jk,
where
ω = e−2πi/n.
Show that 1√
n
F is unitary. In other words, show that FH , the complex
conjugate transpose of F , satisfies
FHF = nI.
The notation here is a little unusual for matrices because the subscripts j
and k run from 0 to n− 1, instead of from 1 to n.
8.6. What relationship between n and j causes fftmatrix(n,j) to produce a
five-point star? What relationship produces a regular pentagon?
8.7. El Niño. The climatological phenomenon el Niño results from changes in
atmospheric pressure in the southern Pacific ocean. The ‘Southern Oscillation
Index’ is the difference in atmospheric pressure between Easter Island and
Darwin, Australia, measured at sea level at the same moment. The text file
elnino.dat contains values of this index measured on a monthly basis over
the 14-year period 1962 through 1975.
Your assignment is to carry out an analysis similar to the sunspot example
on the el Niño data. The unit of time is one month instead of one year.
You should find there is a prominent cycle with a period of 12 months and a
second, less prominent, cycle with a longer period. This second cycle shows
up in about three of the Fourier coefficients, so it is hard to measure its
length, but see if you can make an estimate.
8.8. Train whistle. The Matlab demos directory contains several sound samples.
One of them is a train whistle. The statement
load train
gives you a long vector y and a scalar Fs whose value is the number of samples
per second. The time increment is 1/Fs seconds.
If your computer has sound capabilities, the statement
sound(y,Fs)
20 Chapter 8. Fourier Analysis
plays the signal, but you don’t need that for this problem.
The data do not have a significant linear trend. There are two pulses of the
whistle, but the harmonic content of both pulses is the same.
(a) Plot the data with time in seconds as the independent variable.
(b) Produce a periodogram with frequency in cycles/second as the indepen-
dent variable.
(c) Identify the frequencies of the six peaks in the periodogram. You should
find that ratios between these six frequencies are close to ratios between
small integers. For example, one of the frequencies is 5/3 times another. The
frequencies that are integer multiples of other frequencies are overtones. How
many of the peaks are fundamental frequencies and how many are overtones?
8.9. Bird chirps. Analyze the chirp sound sample from the Matlab demos di-
rectory. By ignoring a short portion at the end, it is possible to segment the
signal into eight pieces of equal length, each containing one chirp. Plot the
magnitude of the FFT of each segment. Use subplot(4,2,k) for k = 1:8
and the same axis scaling for all subplots. Frequencies in the range from
roughly 400 Hz to 800 Hz are appropriate. You should notice that one or two
of the chirps have distinctive plots. If you listen carefully, you should be able
to hear the different sounds.
Bibliography
[1] J. W. Cooley and J. W. Tukey, An algorithm for the machine calculation of
complex Fourier series, Mathematics of Computation, 19 (1965), pp. 297–301.
[2] M. Frigo and S. G. Johnson, FFTW: An adaptive software architecture for
the FFT, in Proceedings of the 1998 IEEE International Conference on Acoustics
Speech and Signal Processing, 3 (1998), pp. 1381–1384.
http://www.fftw.org
[3] M. Frigo and S. G. Johnson, Links to FFT-related resources.
http://www.fftw.org/links.html
[4] Solar Influences Data Center, Sunspot archive and graphics.
http://sidc.oma.be
[5] C. Van Loan, Computational Frameworks for the Fast Fourier Transform,
SIAM, Philadelphia, 1992.
21
Chapter 9
Random Numbers
This chapter describes algorithms for the generation of pseudorandom numbers with
both uniform and normal distributions.
9.1 Pseudorandom Numbers
Here is an interesting number:
0.95012928514718
This is the first number produced by the Matlab random number generator with
its default settings. Start up a fresh Matlab, set format long, type rand, and it’s
the number you get.
If all Matlab users, all around the world, all on different computers, keep
getting this same number, is it really “random”? No, it isn’t. Computers are (in
principle) deterministic machines and should not exhibit random behavior. If your
computer doesn’t access some external device, like a gamma ray counter or a clock,
then it must really be computing pseudorandom numbers. Our favorite definition
was given in 1951 by Berkeley professor D. H. Lehmer, a pioneer in computing and,
especially, computational number theory:
A random sequence is a vague notion . . . in which each term is unpre-
dictable to the uninitiated and whose digits pass a certain number of
tests traditional with statisticians . . .
9.2 Uniform Distribution
Lehmer also invented the multiplicative congruential algorithm, which is the basis for
many of the random number generators in use today. Lehmer’s generators involve
three integer parameters, a, c, and m, and an initial value, x0, called the seed. A
February 15, 2008
1
2 Chapter 9. Random Numbers
sequence of integers is defined by
xk+1 = axk + c mod m.
The operation “mod m” means take the remainder after division by m. For example,
with a = 13, c = 0, m = 31, and x0 = 1, the sequence begins with
1, 13, 14, 27, 10, 6, 16, 22, 7, 29, 5, 3, . . . .
What’s the next value? Well, it looks pretty unpredictable, but you’ve been ini-
tiated. So you can compute (13 · 3) mod 31, which is 8. The first 30 terms in
the sequence are a permutation of the integers from 1 to 30 and then the sequence
repeats itself. It has a period equal to m− 1.
If a pseudorandom integer sequence with values between 0 and m is scaled
by dividing by m, the result is floating-point numbers uniformly distributed in the
interval [0, 1]. Our simple example begins with
0.0323, 0.4194, 0.4516, 0.8710, 0.3226, 0.1935, 0.5161, . . . .
There is only a finite number of values, 30 in this case. The smallest value is 1/31;
the largest is 30/31. Each one is equally probable in a long run of the sequence.
In the 1960s, the Scientific Subroutine Package (SSP) on IBM mainframe
computers included a random number generator named RND or RANDU. It was a
multiplicative congruential with parameters a = 65539, c = 0, and m = 231. With
a 32-bit integer word size, arithmetic mod 231 can be done quickly. Furthermore,
because a = 216+3, the multiplication by a can be done with a shift and an addition.
Such considerations were important on the computers of that era, but they gave
the resulting sequence a very undesirable property. The following relations are all
taken mod 231:
xk+2 = (2
16 + 3)xk+1 = (2
16 + 3)2xk
= (232 + 6 · 216 + 9)xk
= [6 · (216 + 3)− 9]xk.
Hence
xk+2 = 6xk+1 − 9xk for all k.
As a result, there is an extremely high correlation among three successive random
integers of the sequence generated by RANDU.
We have implemented this defective generator in an M-file randssp. A demon-
stration program randgui tries to compute π by generating random points in a cube
and counting the fraction that actually lie within the inscribed sphere. With these
M-files on your path, the statement
randgui randssp
will show the consequences of the correlation of three successive terms. The resulting
pattern is far from random, but it can still be used to compute π from the ratio of
the volumes of the cube and sphere.
9.2. Uniform Distribution 3
For many years, the Matlab uniform random number function, rand, was
also a multiplicative congruential generator. The parameters were
a = 75 = 16807,
c = 0,
m = 231 − 1 = 2147483647.
These values are recommended in a 1988 paper by Park and Miller [11].
This old Matlab multiplicative congruential generator is available in the M-
file randmcg. The statement
randgui randmcg
shows that the points do not suffer the correlation of the SSP generator. They
generate a much better “random” cloud within the cube.
Like our toy generator, randmcg and the old version of the Matlab function
rand generate all real numbers of the form k/m for k = 1, . . . , m− 1. The smallest
and largest are 0.00000000046566 and 0.99999999953434. The sequence repeats
itself after m − 1 values, which is a little over 2 billion numbers. A few years ago,
that was regarded as plenty. But today, an 800 MHz Pentium laptop can exhaust
the period in less than half an hour. Of course, to do anything useful with 2 billion
numbers takes more time, but we would still like to have a longer period.
In 1995, version 5 of Matlab introduced a completely different kind of random
number generator. The algorithm is based on work of George Marsaglia, a professor
at Florida State University and author of the classic analysis of random number
generators, “Random numbers fall mainly in the planes” [6].
Marsaglia’s generator [9] does not use Lehmer’s congruential algorithm. In
fact, there are no multiplications or divisions at all. It is specifically designed to
produce floating-point values. The results are not just scaled integers. In place of a
single seed, the new generator has 35 words of internal memory or state. Thirty-two
of these words form a cache of floating-point numbers, z, between 0 and 1. The
remaining three words contain an integer index i, which varies between 0 and 31, a
single random integer j, and a “borrow” flag b. This entire state vector is built up
a bit at a time during an initialization phase. Different values of j yield different
initial states.
The generation of the ith floating-point number in the sequence involves a
“subtract-with-borrow” step, where one number in the cache is replaced with the
difference of two others:
zi = zi+20 − zi+5 − b.
The three indices, i, i+20, and i+5, are all interpreted mod 32 (by using just their
last five bits). The quantity b is left over from the previous step; it is either zero or
a small positive value. If the computed zi is positive, b is set to zero for the next
step. But if the computed zi is negative, it is made positive by adding 1.0 before it
is saved and b is set to 2−53 for the next step. The quantity 2−53, which is half of
the Matlab constant eps, is called one ulp because it is one unit in the last place
for floating-point numbers slightly less than 1.
4 Chapter 9. Random Numbers
By itself, this generator would be almost completely satisfactory. Marsaglia
has shown that it has a huge period—almost 21430 values would be generated before
it repeated itself. But it has one slight defect. All the numbers are the results of
floating-point additions and subtractions of numbers in the initial cache, so they
are all integer multiples of 2−53. Consequently, many of the floating-point numbers
in the interval [0, 1] are not represented.
The floating-point numbers between 1/2 and 1 are equally spaced with a
spacing of one ulp, and our subtract-with-borrow generator will eventually generate
all of them. But numbers less than 1/2 are more closely spaced and the generator
would miss most of them. It would generate only half of the possible numbers in
the interval [1/4, 1/2], only a quarter of the numbers in [1/8, 1/4], and so on. This
is where the quantity j in the state vector comes in. It is the result of a separate,
independent, random number generator based on bitwise logical operations. The
floating-point fraction of each zi is XORed with j to produce the result returned by
the generator. This breaks up the even spacing of the numbers less than 1/2. It is
now theoretically possible to generate all the floating-point numbers between 2−53
and 1− 2−53. We’re not sure if they are all actually generated, but we don’t know
of any that can’t be.
Figure 9.1 shows what the new generator is trying to accomplish. For this
graph, one ulp is equal to 2−4 instead of 2−53.
1/16 1/8 1/4 1/2
1/8
1/4
1/2
1
Figure 9.1. Uniform distribution of floating-point numbers.
The graph depicts the relative frequency of each of the floating-point numbers.
A total of 32 floating-point numbers are shown. Eight of them are between 1/2 and
1, and they are all equally like to occur. There are also eight numbers between
1/4 and 1/2, but, because this interval is only half as wide, each of them should
occur only half as often. As we move to the left, each subinterval is half as wide as
the previous one, but it still contains the same number of floating-point numbers,
so their relative frequencies must be cut in half. Imagine this picture with 253
numbers in each of 232 smaller intervals and you will see what the new random
9.3. Normal Distribution 5
number generator is doing.
With the additional bit fiddling, the period of the new generator becomes
something like 21492. Maybe we should call it the Christopher Columbus generator.
In any case, it will run for a very long time before it repeats itself.
9.3 Normal Distribution
Almost all algorithms for generating normally distributed random numbers are
based on transformations of uniform distributions. The simplest way to gener-
ate an m-by-n matrix with approximately normally distributed elements is to use
the expression
sum(rand(m,n,12),3) - 6
This works because R = rand(m,n,p) generates a three-dimensional uniformly dis-
tributed array and sum(R,3) sums along the third dimension. The result is a
two-dimensional array with elements drawn from a distribution with mean p/2 and
variance p/12 that approaches a normal distribution as p increases. If we take
p = 12, we get a pretty good approximation to the normal distribution and we
get the variance to be equal to one without any additional scaling. There are two
difficulties with this approach. It requires twelve uniforms to generate one normal,
so it is slow. And the finite p approximation causes it to have poor behavior in the
tails of the distribution.
Older versions of Matlab—before Matlab 5—used the polar algorithm.
This generates two values at a time. It involves finding a random point in the
unit circle by generating uniformly distributed points in the [−1, 1]× [−1, 1] square
and rejecting any outside the circle. Points in the square are represented by vectors
with two components. The rejection portion of the code is
r = Inf;
while r > 1
u = 2*rand(2,1)-1
r = u’*u
end
For each point accepted, the polar transformation
v = sqrt(-2*log(r)/r)*u
produces a vector with two independent normally distributed elements. This algo-
rithm does not involve any approximations, so it has the proper behavior in the tails
of the distribution. But it is moderately expensive. Over 21% of the uniform num-
bers are rejected if they fall outside of the circle, and the square root and logarithm
calculations contribute significantly to the cost.
Beginning with Matlab 5, the normal random number generator randn uses a
sophisticated table lookup algorithm, also developed by George Marsaglia. Marsaglia
calls his approach the ziggurat algorithm. Ziggurats are ancient Mesopotamian ter-
raced temple mounds that, mathematically, are two-dimensional step functions. A
one-dimensional ziggurat underlies Marsaglia’s algorithm.
6 Chapter 9. Random Numbers
Marsaglia has refined his ziggurat algorithm over the years. An early version
is described in Knuth’s classic The Art of Computer Programming [5]. The version
used in Matlab is described by Marsaglia and W. W. Tsang in [7]. A Fortran
version is described in [2, sect. 10.7]. A more recent version is available in the
online electronic Journal of Statistical Software [8]. We describe this recent version
here because it is the most elegant. The version actually used in Matlab is more
complicated, but is based on the same ideas and is just as effective.
The probability density function, or pdf, of the normal distribution is the
bell-shaped curve
f(x) = ce−x
2/2,
where c = 1/(2π)1/2 is a normalizing constant that we can ignore. If we generate
random points (x, y), uniformly distributed in the plane, and reject any of them that
do not fall under this curve, the remaining x’s form our desired normal distribution.
The ziggurat algorithm covers the area under the pdf by a slightly larger area with
n sections. Figure 9.2 has n = 8; actual code might use n = 128. The top n − 1
sections are rectangles. The bottom section is a rectangle together with an infinite
tail under the graph of f(x). The right-hand edges of the rectangles are at the
points zk, k = 2, . . . , n, shown with circles in the picture. With f(z1) = 1 and
f(zn+1) = 0, the height of the kth section is f(zk) − f(zk+1). The key idea is to
choose the zk’s so that all n sections, including the unbounded one on the bottom,
have the same area. There are other algorithms that approximate the area under
the pdf with rectangles. The distinguishing features of Marsaglia’s algorithm are
the facts that the rectangles are horizontal and have equal areas.
0.00 0.74 1.03 1.26 1.49 1.72 1.98 2.34
0.06
0.14
0.23
0.33
0.45
0.59
0.76
1.00
Figure 9.2. The ziggurat algorithm.
For a specified number, n, of sections, it is possible to solve a transcendental
equation to find zn, the point where the infinite tail meets the first rectangular
section. In our picture with n = 8, it turns out that zn = 2.34. In an actual code
with n = 128, zn = 3.4426. Once zn is known, it is easy to compute the common
9.4. randtx, randntx 7
area of the sections and the other right-hand endpoints zk. It is also possible to
compute σk = zk−1/zk, which is the fraction of each section that lies underneath
the section above it. Let’s call these fractional sections the core of the ziggurat.
The right-hand edge of the core is the dotted line in our picture. The computation
of these zk’s and σk’s is done in initialization code that is run only once.
After the initialization, normally distributed random numbers can be com-
puted very quickly. The key portion of the code computes a single random integer,
j, between 1 and n and a single uniformly distributed random number, u, between
−1 and 1. A check is then made to see if u falls in the core of the jth section. If it
does, then we know that uzj is the x-coordinate of a point under the pdf and this
value can be returned as one sample from the normal distribution. The code looks
something like this.
j = ceil(128*rand);
u = 2*rand-1;
if abs(u) < sigma(j)
r = u*z(j);
return
end
Most of the σj ’s are greater than 0.98, and the test is true over 97% of the time.
One normal random number can usually be computed from one random integer, one
random uniform, an if-test, and a multiplication. No square roots or logarithms are
required. The point determined by j and u will fall outside the core less than 3% of
the time. This happens if j = 1 because the top section has no core, if j is between
2 and n−1 and the random point is in one of the little rectangles covering the graph
of f(x), or if j = n and the point is in the infinite tail. In these cases, additional
computations involving logarithms, exponentials, and more uniform samples are
required.
It is important to realize that, even though the ziggurat step function only
approximates the probability density function, the resulting distribution is exactly
normal. Decreasing n decreases the amount of storage required for the tables and
increases the fraction of time that extra computation is required, but does not affect
the accuracy. Even with n = 8, we would have to do the more costly corrections
almost 23% of the time, instead of less than 3%, but we would still get an exact
normal distribution.
With this algorithm, Matlab 6 can generate normally distributed random
numbers as fast as it can generate uniformly distributed ones. In fact, Matlab on
an 800 MHz Pentium laptop can generate over 10 million random numbers from
either distribution in less than one second.
9.4 randtx, randntx
Our NCM M-file collection includes textbook functions randtx and randntx. For
these two functions, we have chosen to fully reproduce the behavior of the corre-
sponding Matlab built-in functions rand and randn. The two textbook functions
use the same algorithms and produce the same results (to within roundoff error) as
8 Chapter 9. Random Numbers
the two built-in functions. All four functions—rand with or without an n and with
or without a tx—have the same usage. With no arguments, the expression randtx
or randntx generates a single uniformly or normally distributed pseudorandom
value. With one argument, the expression randtx(n) or randntx(n) generates an
n-by-n matrix. With two arguments, the expression randtx(m,n) or randntx(m,n)
generates an m-by-n matrix.
It is usually not necessary to access or set the internal state of any of the gener-
ators. But if you want to repeat a computation using the same sequence of pseudo-
random numbers, you can reset the generator state. By default, a generator starts
at the state set by randtx(’state’,0) or randntx(’state’,0). At any point dur-
ing a computation, you can access the current state with s = randtx(’state’) or
s = randntx(’state’). You can later restore that state with randtx(’state’,s)
or randntx(’state’,s). You can also set the state with randtx(’state’,j) or
randntx(’state’,j), where j is a single integer in the range 0 ≤ j ≤ 231− 1. The
number of states that can be set by a single 32-bit integer is only a tiny fraction of
the total number of states.
For the uniform generator randtx, the state s is a vector with 35 elements.
Thirty-two of the elements are floating-point numbers between 2−53 and 1− 2−53.
The other three elements in s are small integer multiples of eps. Although they
cannot all be reached from default initial settings, the total number of possible bit
patterns in the randtx state is 2 · 32 · 232 · 232·52, which is 21702.
For the normal generator randntx, the state s is a vector with two 32-bit
integer elements, so the total number of possible states is 264.
Both generators have setup calculations that are done only when the generator
is first used or reset. For randtx, the setup generates the initial floating-point
numbers in the state vector one bit at a time. For randntx, the setup computes
the breakpoints in the ziggurat step function.
After the setup, the principal portion of the uniform generator randtx is
U = zeros(m,n);
for k = 1:m*n
x = z(mod(i+20,32)+1) - z(mod(i+5,32)+1) - b;
if x < 0
x = x + 1;
b = ulp;
else
b = 0;
end
z(i+1) = x;
i = i+1;
if i == 32, i = 0; end
[x,j] = randbits(x,j);
U(k) = x;
end
This takes the difference between two elements in the state, subtracts any carry bit
b from the previous calculation, adjusts the result if it is negative, and inserts it
9.5. Twister 9
into the state. The auxiliary function randbits does an XOR operation between
the fraction of the floating-point number x and the random integer j.
After the setup, the principal portion of the normal generator randntx is
R = zeros(m,n);
for k = 1:m*n
[u,j] = randuni;
rk = u*z(j+1);
if abs(rk) < z(j)
R(k) = rk;
else
R(k) = randntips(rk,j,z);
end
end
This uses a subfunction randuni to generate a random uniform u and a random
integer j. A single multiplication generates a candidate result rk and checks to see
if it is within the “core” of the ziggurat. Almost all of the time it is in the core and
so becomes an element of the final result. If rk is outside the core, then additional
computation must be done by the auxiliary subfunction randtips.
9.5 Twister
0 10 20 30 40 50 60 70 80 90 100
0
200
400
600
800
1000
1200
1400
1600
1800
Figure 9.3. Histogram of run lengths.
Let’s do a run length test using the uniform random number generator that
we have described in this chapter and that is implemented in randtx.
rand(’state’,0)
x = rand(1,2^24);
delta = .01;
10 Chapter 9. Random Numbers
k = diff(find(x
10.10. Experiment with eigsvdgui on random symmetric and nonsymmetric ma-
trices, randn(n). Choose values of n appropriate for the speed of your com-
puter and investigate the three variants eig, symm, and svd. The title in the
eigsvdgui shows the number of iterations required. Roughly, how does the
number of iterations for the three different variants depend upon the order
of the matrix?
10.11. Pick a value of n and generate a matrix with
A = diag(ones(n-1,1),-1) + diag(1,n-1);
Explain any atypical behavior you observe with each of the following.
eigsvdgui(A,’eig’)
eigsvdgui(A,’symm’)
eigsvdgui(A,’svd’)
10.12. The NCM file imagesvd.m helps you investigate the use of PCA in digital
image processing. If you have them available, use your own photographs. If
you have access to the Matlab Image Processing Toolbox, you may want to
use its advanced features. However, it is possible to do basic image processing
without the toolbox.
For an m-by-n color image in JPEG format, the statement
X = imread(’myphoto.jpg’);
produces a three-dimensional m-by-n-by-3 array X with m-by-n integer subar-
rays for the red, green, and blue intensities. It would be possible to compute
three separate m-by-n SVDs of the three colors. An alternative that requires
less work involves altering the dimensions of X with
X = reshape(X,m,3*n)
and then computing one m-by-3n SVD.
(a) The primary computation in imagesvd is done by
[V,S,U] = svd(X’,0)
34 Chapter 10. Eigenvalues and Singular Values
How does this compare with
[U,S,V] = svd(X,0)
(b) How does the choice of approximating rank affect the visual qualities of
the images? There are no precise answers here. Your results will depend
upon the images you choose and the judgments you make.
10.13. This exercise investigates a model of the human gait developed by Nikolaus
Troje at the Bio Motion Lab of Ruhr University in Bochum, Germany. Their
Web page provides an interactive demo [3]. Two papers describing the work
are also available on the Web [4, 5]. Troje’s data result from motion capture
experiments involving subjects wearing reflective markers walking on a tread-
mill. His model is a five-term Fourier series with vector-valued coefficients
obtained by principal component analysis of the experimental data. The
components, which are also known as postures or eigenpostures, correspond
to static position, forward motion, sideways sway, and two hopping/bouncing
movements that differ in the phase relationship between the upper and lower
portions of the body. The model is purely descriptive; it does not make any
direct use of physical laws of motion.
8
7
6
14
13
15
1
12
5
10
11
9
4
2
3
Figure 10.13. Walker at rest.
The moving position v(t) of the human body is described by 45 functions
of time, which correspond to the location of 15 points in three-dimensional
space. Figure 10.13 is a static snapshot. The model is
v(t) = v1 + v2 sin ωt + v3 cos ωt + v4 sin 2ωt + v5 cos 2ωt.
If the postures v1, . . . , v5 are regarded as the columns of a single 45-by-5 ma-
trix V , the calculation of v(t) for any t involves a matrix-vector multiplication.
The resulting vector can then be reshaped into a 15-by-3 array that exposes
Exercises 35
the spatial coordinates. For example, at t = 0, the time-varying coefficients
form the vector w = [1 0 1 0 1]’. Consequently, reshape(V*w,15,3) pro-
duces the coordinates of the initial position. The five postures for an indi-
vidual subject are obtained by a combination of principal component and
Fourier analysis. The individual characteristic frequency ω is an independent
speed parameter. If the postures are averaged over the subjects with a par-
ticular characteristic, the result is a model for the “typical” walker with that
characteristic. The characteristics available in the demo on the Web page in-
clude male/female, heavy/light, nervous/relaxed, and happy/sad. Our M-file
walker.m is based on the postures for a typical female walker, f1,. . . ,f5, and
a typical male walker, m1, . . . ,m5. Slider s1 varies the time increment and
hence the apparent walking speed. Sliders s2, . . . , s5 vary the amount that
each component contributes to the overall motion. Slider s6 varies a linear
combination of the female and male walkers. A slider setting greater than
1.0 overemphasizes the characteristic. Here is the complete model, including
the sliders:
f(t) = f1 + s2f2 sinωt + s3f3 cos ωt + s4f4 sin 2ωt + s5f5 cos 2ωt,
m(t) = m1 + s2m2 sin ωt + s3m3 cosωt + s4m4 sin 2ωt + s5m5 cos 2ωt,
v(t) = (f(t) + m(t))/2 + s6(f(t)−m(t))/2.
(a) Describe the visual differences between the gaits of the typical female and
male walkers.
(b) File walkers.mat contains four data sets. F and M are the postures of the
typical female and typical male obtained by analyzing all the subjects. A and
B are the postures of two individual subjects. Are A and B male or female?
(c) Modify walker.m to add a waving hand as an additional, artificial, pos-
ture.
(d) What does this program do?
load walkers
F = reshape(F,15,3,5);
M = reshape(M,15,3,5);
for k = 1:5
for j = 1:3
subplot(5,3,j+3*(k-1))
plot([F(:,j,k) M(:,j,k)])
ax = axis;
axis([1 15 ax(3:4)])
end
end
(e) Change walker.m to use a Fourier model parametrized by amplitude and
phase. The female walker is
f(t) = f1 + s2a1 sin (ωt + s3φ1) + s4a2 sin (2ωt + s5φ2).
36 Chapter 10. Eigenvalues and Singular Values
A similar formulation is used for the male walker. The linear combination of
the two walkers using s6 is unchanged. The amplitude and phase are
a1 =
√
f22 + f
2
3 ,
a2 =
√
f24 + f
2
5 ,
φ1 = tan
−1(f3/f2),
φ2 = tan
−1(f5/f4).
10.14. In English, and in many other languages, vowels are usually followed by con-
sonants and consonants are usually followed by vowels. This fact is revealed
by a principal component analysis of the digraph frequency matrix for a sam-
ple of text. English text uses 26 letters, so the digraph frequency matrix is
a 26-by-26 matrix, A, with counts of pairs of letters. Blanks and all other
punctuation are removed from the text and the entire sample is thought of
as circular or periodic, so the first letter follows the last letter. The matrix
entry ai,j is the number of times the ith letter is followed by the jth letter
in the text. The row and column sums of A are the same; they count the
number of times individual letters occur in the sample. So the fifth row and
fifth column usually have the largest sums because the fifth letter, which is
“E,” is usually the most frequent.
A principal component analysis of A produces a first component,
A ≈ σ1u1vT1 ,
that reflects the individual letter frequencies. The first right- and left-singular
vectors, u1 and v1, have elements that are all of the same sign and that are
roughly proportional to the corresponding frequencies. We are primarily
interested in the second principal component,
A ≈ σ1u1vT1 + σ2u2vT2 .
The second term has positive entries in vowel-consonant and consonant-vowel
positions and negative entries in vowel-vowel and consonant-consonant posi-
tions. The NCM collection contains a function digraph.m that carries out
this analysis. Figure 10.14 shows the output produced by analyzing Lincoln’s
Gettysburg Address with
digraph(’gettysburg.txt’)
The ith letter of the alphabet is plotted at coordinates (ui,2, vi,2). The dis-
tance of each letter from the origin is roughly proportional to its frequency,
and the sign patterns cause the vowels to be plotted in one quadrant and
the consonants to be plotted in the opposite quadrant. There is more detail.
The letter “N” is usually preceded by a vowel and often followed by another
consonant, like “D” or “G,” and so it shows up in a quadrant pretty much
by itself. On the other hand, “H” is often preceded by another consonant,
namely “T,” and followed by a vowel, “E,” so it also gets its own quadrant.
Exercises 37
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
A
B
C
D
E
F
G
H
I
JK
LM
N
O
P
Q
R
S
T
U
V
WXYZ
1149 characters
Figure 10.14. The second principal component of a digraph matrix.
(a) Explain how digraph uses sparse to count letter pairs and create the
matrix. help sparse should be useful.
(b) Try digraph on other text samples. Roughly how many characters are
needed to see the vowel-consonant frequency behavior?
(c) Can you find any text with at least several hundred characters that does
not show the typical behavior?
(d) Try digraph on M-files or other source code. Do computer programs typ-
ically have the same vowel-consonant behavior as prose?
(e) Try digraph on samples from other languages. Hawaiian and Finnish
are particularly interesting. You may need to modify digraph to accom-
modate more or fewer than 26 letters. Do other languages show the same
vowel-consonant behavior as English?
10.15. Explain the behavior of circlegen for each of the following values of the
step size h. What, if anything, is special about these particular values? Is
the orbit a discrete set of points? Does the orbit stay bounded, grow linearly,
or grow exponentially? If necessary, increase the axis limits in circlegen so
that it shows the entire orbit. Recall that φ = (1+
√
5)/2 is the golden ratio:
h =
√
2− 2 cos (2π/30) (the default),
h = 1/φ,
h = φ,
h = 1.4140,
h =
√
2,
h = 1.4144,
h < 2,
h = 2,
h > 2.
38 Chapter 10. Eigenvalues and Singular Values
10.16. (a) Modify circlegen so that both components of the new point are deter-
mined from the old point, that is,
xn+1 = xn + hyn,
yn+1 = yn − hxn.
(This is the explicit Euler’s method for solving the circle ordinary differential
equation.) What happens to the “circles”? What is the iteration matrix?
What are its eigenvalues?
(b) Modify circlegen so that the new point is determined by solving a 2-by-2
system of simultaneous equations:
xn+1 − hyn+1 = xn,
yn+1 + hxn+1 = yn.
(This is the implicit Euler’s method for solving the circle ordinary differential
equation.) What happens to the “circles”? What is the iteration matrix?
What are its eigenvalues?
10.17. Modify circlegen so that it keeps track of the maximum and minimum
radius during the iteration and returns the ratio of these two radii as the value
of the function. Compare this computed aspect ratio with the eigenvector
condition number, cond(X), for various values of h.
Bibliography
[1] J. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford,
1965.
[2] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide:
Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi
Methods, SIAM, Philadelphia, 1998.
http://www.caam.rice.edu/software/ARPACK
[3] Bio Motion Labr
[4] N. Troje.
http://journalofvision.org/2/5/2
[5] N. Troje.
[6] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J.
Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney,
and D. Sorensen, LAPACK Users’ Guide, Third Edition, SIAM, Philadelphia,
1999.
http://www.netlib.org/lapack
[7] J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
[8] G. H. Golub and C. F. Van Loan, Matrix Computations, Third Edition,
The Johns Hopkins University Press, Baltimore, 1996.
[9] G. W. Stewart, Introduction to Matrix Computations, Academic Press, New
York, 1973.
[10] G. W. Stewart, Matrix Algorithms: Basic Decompositions, SIAM, Philadel-
phia, 1998.
[11] L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM,
Philadelphia, 1997.
39
Chapter 11
Partial Differential
Equations
A wide variety of partial differential equations occurs in technical computing. We
cannot begin to cover them all in this book. In this chapter, we limit ourselves to
three model problems for second-order partial differential equations in one or two
space dimensions.
11.1 Model Problems
All the problems we consider involve the Laplacian operator, which is
4 = ∂
2
∂x2
in one space dimension and
4 = ∂
2
∂x2
+
∂2
∂y2
in two space dimensions. We let ~x denote the single variable x in one dimension
and the pair of variables (x, y) in two dimensions.
The first model problem is the Poisson equation. This elliptic equation does
not involve a time variable, and so describes the steady state, quiescent behavior of
a model variable:
4u = f(~x).
There are no initial conditions.
The second model problem is the heat equation. This parabolic equation occurs
in models involving diffusion and decay:
∂u
∂t
= 4u− f(~x).
The initial condition is
u(~x, 0) = u0(~x).
February 15, 2008
1
2 Chapter 11. Partial Differential Equations
The third model problem is the wave equation. This hyperbolic equation de-
scribes how a disturbance travels through matter. If the units are chosen so that
the wave propagation speed is equal to one, the amplitude of a wave satisfies
∂2u
∂t2
= 4u.
Typical initial conditions specify the initial amplitude and take the initial velocity
to be zero:
u(~x, 0) = u0(~x),
∂u
∂t
(~x, 0) = 0.
In one dimension, all the problems take place on a finite interval on the x-axis. In
more than one space dimension, geometry plays a vital role. In two dimensions, all
the problems take place in a bounded region Ω in the (x, y) plane. In all cases, f(~x)
and u0(~x) are given functions of ~x. All the problems involve boundary conditions
where the value of u or some partial derivative of u is specified on the boundary of
Ω. Unless otherwise specified, we will take the boundary values to be zero.
11.2 Finite Difference Methods
Basic finite difference methods for approximating solutions to these problems use
a uniform mesh with spacing h. In one dimension, for the interval a ≤ x ≤ b, the
spacing is h = (b− a)/(m + 1) and the mesh points are
xi = a + ih, i = 0, . . . ,m + 1.
The second derivative with respect to x is approximated by the 3-point centered
second difference:
4hu(x) =
u(x + h)− 2u(x) + u(x− h)
h2
.
In two dimensions, the mesh is the set of points
(xi, yj) = (ih, jh)
that lie within the region Ω. Approximating the partial derivatives with centered
second differences gives the 5-point discrete Laplacian
4hu(x, y) =
u(x + h, y)− 2u(x, y) + u(x− h, y)
h2
+
u(x, y + h)− 2u(x, y) + u(x, y − h)
h2
.
Alternative notation uses P = (x, y) for a mesh point and N = (x, y + h), E =
(x + h, y), S = (x, y − h), and W = (x − h, y) for its four neighbors in the four
compass directions. The discrete Laplacian is
4hu(P ) =
u(N) + u(W ) + u(E) + u(S)− 4u(P )
h2
.
11.2. Finite Difference Methods 3
The finite difference Poisson problem involves finding values of u so that
4hu(~x) = f(~x)
for each point ~x on the mesh.
If the source term f(~x) is zero, Poisson’s equation is called Laplace’s equation:
4hu(x) = 0.
In one dimension, Laplace’s equation has only trivial solutions. The value of u at a
mesh point x is the average of the values of u at its left and right neighbors, so u(x)
must be a linear function of x. Taking the boundary conditions into consideration
implies that u(x) is the linear function connecting the two boundary values. If the
boundary values are zero, then u(x) is identically zero. In more than one dimension,
solutions to Laplace’s equation are called harmonic functions and are not simply
linear functions of ~x.
The finite difference heat and wave equations also make use of first and second
differences in the t direction. Let δ denote the length of a time step. For the heat
equation, we use a difference scheme that corresponds to Euler’s method for ordinary
differential equations:
u(~x, t + δ)− u(~x, t)
δ
= 4hu(~x).
Starting with the initial conditions u(~x, 0) = u0(~x), we can step from any value of
t to t + δ with
u(~x, t + δ) = u(~x, t) + δ4hu(~x, t)
for all of the mesh points ~x in the region. The boundary conditions supply values on
the boundary or outside the region. This method is explicit because each new value
of u can be computed directly from values of u at the previous time step. More
complicated methods are implicit because they involve the solution of systems of
equations at each step.
For the wave equation, we can use a centered second difference in t:
u(~x, t + δ)− 2u(~x, t) + u(~x, t− δ)
δ2
= 4hu(~x, t).
This requires two “layers” of values of the solution, one at t − δ and one at t. In
our simple model problem, the initial condition
∂u
∂t
(~x, 0) = 0
allows us to start with both u(~x, 0) = u0(~x) and u(~x, δ) = u0(~x). We compute
subsequent layers with
u(~x, t + δ) = 2u(~x, t)− u(~x, t− δ) + δ24hu(~x, t)
for all of the mesh points ~x in the region. The boundary conditions supply values
on the boundary or outside the region. Like our scheme for the heat equation, this
method for solving the wave equation is explicit.
4 Chapter 11. Partial Differential Equations
11.3 Matrix Representation
If a one-dimensional mesh function is represented as a vector, the one-dimensional
difference operator 4h becomes the tridiagonal matrix
1
h2
−2 1
1 −2 1
1 −2 1
. . .
. . .
. . .
1 −2 1
1 −2
.
This matrix is symmetric. (It is also negative definite.) Most importantly, even if
there are thousands of interior mesh points, there are at most three nonzero elements
in each row and column. Such matrices are the prime examples of sparse matrices.
When computing with sparse matrices, it is important to use data structures that
store only the locations and values of the nonzero elements.
With u represented as a vector and h24h as a matrix A, the Poisson problem
becomes
Au = b,
where b is a vector (the same size as u) containing the values of h2f(x) at the
interior mesh points. The first and last components of b would also include any
nonzero boundary values.
In Matlab, the solution to the discrete Poisson problem is computed using
sparse backslash, which takes advantage of the sparsity in A:
u = A
The situation for meshes in two dimensions is more complicated. Let’s number
the interior mesh points in Ω from top to bottom and from left to right. For example,
the numbering of an L-shaped region would be
L =
0 0 0 0 0 0 0 0 0 0 0
0 1 5 9 13 17 21 30 39 48 0
0 2 6 10 14 18 22 31 40 49 0
0 3 7 11 15 19 23 32 41 50 0
0 4 8 12 16 20 24 33 42 51 0
0 0 0 0 0 0 25 34 43 52 0
0 0 0 0 0 0 26 35 44 53 0
0 0 0 0 0 0 27 36 45 54 0
0 0 0 0 0 0 28 37 46 55 0
0 0 0 0 0 0 29 38 47 56 0
0 0 0 0 0 0 0 0 0 0 0
The zeros are points on the boundary or outside the region. With this numbering,
the values of any function defined on the interior of the region can be reshaped into
a long column vector. In this example, the length of the vector is 56.
11.3. Matrix Representation 5
If a two-dimensional mesh function is represented as a vector, the finite dif-
ference Laplacian becomes a matrix. For example, at point number 43,
h24hu(43) = u(34) + u(42) + u(44) + u(52)− 4u(43).
If A is the corresponding matrix, then its 43rd row would have five nonzero elements:
a43,34 = a43,42 = a43,44 = a43,52 = 1, and a43,43 = −4.
A mesh point near the boundary has only two or three interior neighbors, so the
corresponding row of A has only three or four nonzero entries.
The complete matrix A has −4’s on its diagonal, four 1’s off the diagonal in
most of its rows, two or three 1’s off the diagonal in some of its rows, and zeros
elsewhere. For the example region above, A would be 56 by 56. Here is A if there
are only 16 interior points.
A =
-4 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
1 -4 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 -4 1 1 0 0 0 0 0 0 0 0 0 0 0
0 1 1 -4 0 1 0 0 0 0 0 0 0 0 0 0
0 0 1 0 -4 1 1 0 0 0 0 0 0 0 0 0
0 0 0 1 1 -4 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 -4 1 0 0 0 1 0 0 0 0
0 0 0 0 0 1 1 -4 1 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1 -4 1 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1 -4 1 0 0 0 1 0
0 0 0 0 0 0 0 0 0 1 -4 0 0 0 0 1
0 0 0 0 0 0 1 0 0 0 0 -4 1 0 0 0
0 0 0 0 0 0 0 1 0 0 0 1 -4 1 0 0
0 0 0 0 0 0 0 0 1 0 0 0 1 -4 1 0
0 0 0 0 0 0 0 0 0 1 0 0 0 1 -4 1
0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 -4
This matrix is symmetric, negative definite, and sparse. There are at most five
nonzero elements in each row and column.
Matlab has two functions that involve the discrete Laplacian, del2 and
delsq. If u is a two-dimensional array representing a function u(x, y), then del2(u)
computes 4hu, scaled by h2/4, at interior points, and uses one-sided formulas at
points near the boundary. For example, the function u(x, y) = x2 + y2 has 4u = 4.
The statements
h = 1/20;
[x,y] = meshgrid(-1:h:1);
u = x.^2 + y.^2;
d = (4/h^2) * del2(u);
produce an array d, the same size as x and y, with all the elements equal to 4.
6 Chapter 11. Partial Differential Equations
If G is a two-dimensional array specifying the numbering of a mesh, then
A = -delsq(G) is the matrix representation of the operator h24h on that mesh.
The mesh numbering for several specific regions is generated by numgrid. For
example,
m = 5
L = numgrid(’L’,2*m+1)
generates the L-shaped mesh with 56 interior points shown above. And
m = 3
A = -delsq(numgrid(’L’,2*m+1))
generates the 16-by-16 matrix A shown above.
The function inregion can also generate mesh numberings. For example, the
coordinates of the vertices of the L-shaped domain are
xv = [0 0 1 1 -1 -1 0];
yv = [0 -1 -1 1 1 0 0];
The statement
[x,y] = meshgrid(-1:h:1);
generates a square grid of width h. The statement
[in,on] = inregion(x,y,xv,yv);
generates arrays of zeros and ones that mark the points that are contained in the
domain, including the boundary, as well as those that are strictly on the boundary.
The statements
p = find(in-on);
n = length(p);
L = zeros(size(x));
L(p) = 1:n;
number the n interior points from top to bottom and left to right. The statement
A = -delsq(L);
generates the n-by-n sparse matrix representation of the discrete Laplacian on the
mesh.
With u represented as a vector with n components, the Poisson problem be-
comes
Au = b,
where b is a vector (the same size as u) containing the values of h2f(x, y) at the
interior mesh points. The components of b that correspond to mesh points with
neighbors on the boundary or outside the region also include any nonzero boundary
values.
As in one dimension, the solution to the discrete Poisson problem is computed
using sparse backslash.
u = A
11.4. Numerical Stability 7
11.4 Numerical Stability
The time-dependent heat and wave equations generate a sequence of vectors, u(k),
where the k denotes the kth time step. For the heat equation, the recurrence is
u(k+1) = u(k) + σAu(k),
where
σ =
δ
h2
.
This can be written
u(k+1) = Mu(k),
where
M = I + σA.
In one dimension, the iteration matrix M has 1− 2σ on the diagonal and one
or two σ’s off the diagonal in each row. In two dimensions, M has 1 − 4σ on the
diagonal and two, three, or four σ’s off the diagonal in each row. Most of the row
sums in M are equal to 1; a few are less than 1. Each element of u(k+1) is a linear
combination of elements of u(k) with coefficients that add up to 1 or less. Now here
is the key observation. If the elements of M are nonnegative, then the recurrence is
stable. In fact, it is dissipative. Any error or noise in u(k) is not magnified in u(k+1).
But if the diagonal elements of M are negative, then the recurrence can be unstable.
Error and noise, including roundoff error and noise in the initial conditions, can be
magnified with each time step. Requiring 1 − 2σ or 1 − 4σ to be positive leads to
a very important stability condition for this explicit method for the heat equation.
In one dimension,
σ ≤ 1
2
.
And, in two dimensions,
σ ≤ 1
4
.
If this condition is satisfied, the iteration matrix has positive diagonal elements and
the method is stable.
Analysis of the wave equation is a little more complicated because it involves
three levels, u(k+1), u(k), and u(k−1). The recurrence is
u(k+1) = 2u(k) − u(k−1) + σAu(k),
where
σ =
δ2
h2
.
The diagonal elements of the iteration matrix are now 2 − 2σ, or 2 − 4σ. In
one dimension, the stability condition is
σ ≤ 1.
8 Chapter 11. Partial Differential Equations
And, in two dimensions,
σ ≤ 1
2
.
These stability conditions are known as the CFL conditions, after Courant, Friedrichs
and Lewy, who wrote a paper in 1928 that used finite difference methods to prove
existence of solutions to the partial differential equations of mathematical physics.
Stability conditions are restrictions on the size of the time step, δ. Any attempt to
speed up the computation by taking larger time steps is likely to be disastrous. For
the heat equation, the stability condition is particularly severe—the time step must
be smaller than the square of the space mesh width. More sophisticated methods,
often involving some implicit equation solving at each step, have less restrictive or
unlimited stability conditions.
The M-file pdegui illustrates the concepts discussed in this chapter by of-
fering the choice among several domains and among the model partial differential
equations. For Poisson’s equation, pdegui uses sparse backslash to solve
4hu = 1
in the chosen domain. For the heat and wave equations, the stability parameter
σ can be varied. If the critical value, 0.25 for the heat equation and 0.50 for the
wave equation, is exceeded by even a small amount, the instability rapidly becomes
apparent.
You will find much more powerful capabilities in the Matlab Partial Differ-
ential Equation Toolbox.
11.5 The L-Shaped Membrane
Separating out periodic time behavior in the wave equation leads to solutions of the
form
u(~x, t) = cos (
√
λ t) v(~x).
The functions v(~x) depend upon λ. They satisfy
4v + λv = 0
and are zero on the boundary. The quantities λ that lead to nonzero solutions
are the eigenvalues, and the corresponding functions v(~x) are the eigenfunctions or
modes. They are determined by the physical properties and the geometry of each
particular situation. The square roots of the eigenvalues are resonant frequencies. A
periodic external driving force at one of these frequencies generates an unboundedly
strong response in the medium.
Any solution of the wave equation can be expressed as a linear combination of
these eigenfunctions. The coefficients in the linear combination are obtained from
the initial conditions.
In one dimension, the eigenvalues and eigenfunctions are easily determined.
The simplest example is a violin string, held fixed at the ends of the interval of
length π. The eigenfunctions are
vk(x) = sin (kx).
11.5. The L-Shaped Membrane 9
The eigenvalues are determined by the boundary condition, vk(π) = 0. Hence, k
must be an integer and
λk = k
2.
If the initial condition, u0(x), is expanded in a Fourier sine series,
u0(x) =
∑
k
ak sin (kx),
then the solution to the wave equation is
u(x, t) =
∑
k
ak cos (kt) sin (kx)
=
∑
k
ak cos (
√
λk t) vk(x).
In two dimensions, an L-shaped region formed from three unit squares is in-
teresting for several reasons. It is one of the simplest geometries for which solutions
to the wave equation cannot be expressed analytically, so numerical computation
is necessary. Furthermore, the 270◦ nonconvex corner causes a singularity in the
solution. Mathematically, the gradient of the first eigenfunction is unbounded near
the corner. Physically, a membrane stretched over such a region would rip at the
corner. This singularity limits the accuracy of finite difference methods with uni-
form grids. The MathWorks has adopted a surface plot of the first eigenfunction of
the L-shaped region as the company logo. The computation of this eigenfunction
involves several of the numerical techniques we have described in this book.
Simple model problems involving waves on an L-shaped region include an
L-shaped membrane, or L-shaped tambourine, and a beach towel blowing in the
wind, constrained by a picnic basket on one fourth of the towel. A more practical
example involves ridged microwave waveguides. One such device, shown in Figure
11.1, is a waveguide-to-coax adapter. The active region is the channel with the
H-shaped cross section visible at the end of the adapter. The ridges increase the
bandwidth of the guide at the expense of higher attenuation and lower power-
handling capability. Symmetry of the H about the dotted lines shown in the contour
plot of the electric field implies that only one quarter of the domain needs to be
considered and that the resulting geometry is our L-shaped region. The boundary
conditions are different than our membrane problem, but the differential equation
and the solution techniques are the same.
Eigenvalues and eigenfunctions of the L-shaped domain can be computed by
finite difference methods. The Matlab statements
m = 200
h = 1/m
A = delsq(numgrid(’L’,2*m+1))/h^2
set up the 5-point finite difference approximation to the Laplacian on an 200-by-200
mesh in each of the three squares that make up the domain. The resulting sparse
matrix A has order 119201 and 594409 nonzero entries. The statement
10 Chapter 11. Partial Differential Equations
Figure 11.1. A double-ridge microwave-to-coax adapter and its H-shaped
region. Photo courtesy Advanced Technical Materials, Inc. [1].
lambda = eigs(A,6,0)
uses Arnoldi’s method from the Matlab implementation of ARPACK to compute
the first six eigenvalues. It takes less than 2 min on a 1.4 GHz Pentium laptop to
produce
lambda =
9.64147
15.19694
19.73880
29.52033
31.91583
41.47510
The exact values are
9.63972
15.19725
19.73921
29.52148
31.91264
41.47451
You can see that even with this fine mesh and large matrix calculation, the computed
eigenvalues are accurate to only three or four significant digits. If you try to get
more accuracy by using a finer mesh and hence a larger matrix, the computation
requires so much memory that the total execution time is excessive.
For the L-shaped domain and similar problems, a technique using analytic
solutions to the underlying differential equation is much more efficient and accu-
rate than finite difference methods. The technique involves polar coordinates and
11.5. The L-Shaped Membrane 11
fractional-order Bessel functions. With parameters α and λ, the functions
v(r, θ) = Jα(
√
λ r) sin (α θ)
are exact solutions to the polar coordinate version of the eigenfunction equation
∂2v
∂r2
+
1
r
∂v
∂r
+
1
r2
∂2v
∂θ2
+ λv = 0.
For any value of λ, the functions v(r, θ) satisfy the boundary conditions
v(r, 0) = 0 and v(r, π/α) = 0
on the two straight edges of a circular sector with angle π/α. If
√
λ is chosen to be a
zero of the Bessel function, Jα(
√
λ) = 0, then v(r, θ) is also zero on the circle, r = 1.
Figure 11.2 shows a few of the eigenfunctions of the circular sector with angle 3π/2.
The eigenfunctions have been chosen to illustrate symmetry about 3π/4 and π/2.
8.9494
33.4927
14.3559
44.0711
20.7146
55.6455
Figure 11.2. Eigenfunctions of the three-quarter disc.
We approximate the eigenfunctions of the L-shaped domain and other regions
with corners by linear combinations of the circular sector solutions:
v(r, θ) =
∑
j
cjJαj (
√
λ r) sin (αj θ).
The angle of the reentrant 270◦ corner in the L-shaped region is 3π/2, or π/(2/3),
so the values of α are integer multiples of 2/3:
αj =
2j
3
.
These functions v(r, θ) are exact solutions to the eigenfunction differential equation.
There is no finite difference mesh involved. The functions also satisfy the boundary
12 Chapter 11. Partial Differential Equations
conditions on the two edges that meet at the reentrant corner. All that remains is
to pick the parameter λ and the coefficients cj so that the boundary conditions on
the remaining edges are satisfied.
A least squares approach involving the SVD is used to determine λ and the
cj . Pick m points, (ri, θi), on the remaining edges of the boundary. Let n be the
number of fundamental solutions to be used. Generate an m-by-n matrix A with
elements that depend upon λ:
Ai,j(λ) = Jαj (
√
λ ri) sin (αj θi), i = 1, . . . , m, j = 1, . . . , n.
Then, for any vector c, the vector Ac is the vector of boundary values, v(ri, θi).
We want to make ||Ac|| small without taking ||c|| small. The SVD provides the
solution.
Let σn(A(λ)) denote the smallest singular value of the matrix A(λ). and let
λk denote a value of λ that produces a local minimum of the smallest singular value:
λk = kth minimizer(σn(A(λ))).
Each λk approximates an eigenvalue of the region. The corresponding right singular
vector provides the coefficients for the linear combination c = V(:,n).
9.6397 15.1973 19.7392
31.9126 44.9485 49.3480
Figure 11.3. Eigenfunctions of the L-shaped region.
It is worthwhile to take advantage of symmetries. It turns out that the eigen-
functions fall into three symmetry classes:
• symmetric about the center line at θ = 3π/4, so v(r, θ) = v(r, 3π/2− θ);
• antisymmetric about the center line at θ = 3π/4, so v(r, θ) = −v(r, 3π/2− θ);
• eigenfunction of the square, so v(r, π/2) = 0 and v(r, π) = 0.
These symmetries allow us to restrict the values of αj used in each expansion:
Exercises 13
• αj = 2j3 , j odd and not a multiple of 3;
• αj = 2j3 , j even and not a multiple of 3;
• αj = 2j3 , j a multiple of 3.
The M-file membranetx in the NCM directory computes eigenvalues and eigenfunc-
tions of the L-membrane using these symmetries and a search for local minima of
σn(A(λ)). The M-file membrane, distributed with Matlab in the demos directory,
uses an older version of the algorithm based on the QR decomposition instead of the
SVD. Figure 11.3 shows six eigenfunctions of the L-shaped region, with two from
each of the three symmetry classes. They can be compared with the eigenfunctions
of the sector shown in Figure 11.2. By taking the radius of the sector to be 2/
√
π,
both regions have the same area and the eigenvalues are comparable.
The demo M-file logo makes a surf plot of the first eigenfunction, then adds
lighting and shading to create The MathWorks logo. After being so careful to satisfy
the boundary conditions, the logo uses only the first two terms in the circular sector
expansion. This artistic license gives the edge of the logo a more interesting curved
shape.
Exercises
11.1. Let n be an integer and generate n-by-n matrices A, D, and I with the
statements
e = ones(n,1);
I = spdiags(e,0,n,n);
D = spdiags([-e e],[0 1],n,n);
A = spdiags([e -2*e e],[-1 0 1],n,n);
(a) For an appropriate value of h, the matrix (1/h2)A approximates 4h for
the interval 0 ≤ x ≤ 1. Is that value of h equal to 1/(n−1), 1/n, or 1/(n+1)?
(b) What does (1/h)D approximate?
(c) What are DT D and DDT ?
(d) What is A2?
(e) What is kron(A,I)+kron(I,A)?
(f) Describe the output produced by plot(inv(full(-A))).
11.2. (a) Use finite differences to compute a numerical approximation to the solu-
tion u(x) to the one-dimensional Poisson problem
d2u
dx2
= exp (−x2)
on the interval −1 ≤ x ≤ 1. The boundary conditions are u(−1) = 0 and
u(1) = 0. Plot your solution.
(b) If you have access to dsolve in the Symbolic Toolbox, or if you are very
good at calculus, find the analytic solution of the same problem and compare
it with your numerical approximation.
14 Chapter 11. Partial Differential Equations
11.3. Reproduce the contour plot in Figure 11.1 of the first eigenfunction of the
H-shaped ridge waveguide formed from four L-shaped regions.
11.4. Let h(x) be the function defined by the M-file humps(x). Solve four different
problems involving h(x) on the interval 0 ≤ x ≤ 1.
(a) One-dimensional Poisson problem with humps as the source term:
d2u
dx2
= −h(x),
with boundary conditions
u(0) = 0, u(1) = 0.
Make plots, similar to Figure 11.4, of h(x) and u(x). Compare diff(u,2)
with humps(x).
0 0.5 1
0
20
40
60
80
100
0 0.5 1
−1
0
1
2
3
4
5
Figure 11.4. h(x) and u(x).
(b) One-dimensional heat equation with humps as the source term:
∂u
∂t
=
∂2u
∂x2
+ h(x),
with initial value
u(0, x) = 0
and boundary conditions
u(0, t) = 0, u(1, t) = 0.
Create an animated plot of the solution as a function of time. What is the
limit as t →∞ of u(x, t)?
(c) One-dimensional heat equation with humps as the initial value:
∂u
∂t
=
∂2u
∂x2
,
with initial value
u(x, 0) = h(x)
Exercises 15
and boundary conditions
u(0, t) = h(0), u(1, t) = h(1).
Create an animated plot of the solution as a function of time. What is the
limit as t →∞ of u(x, t)?
(d) One-dimensional wave equation with humps as the initial value:
∂2u
∂t2
=
∂2u
∂x2
,
with initial values
u(x, 0) = h(x),
∂u
∂t
(x, 0) = 0,
and boundary conditions
u(0, t) = h(0), u(1, t) = h(1).
Create an animated plot of the solution as a function of time. For what values
of t does u(x, t) return to its initial value h(x)?
11.5. Let p(x, y) be the function defined by the M-file peaks(x,y). Solve four
different problems involving p(x, y) on the square −3 ≤ x ≤ 3, −3 ≤ y ≤ 3.
−2 0 2
−2
0
2
−2 0 2
−2
0
2
Figure 11.5. p(x, y) and u(x, y).
(a) Two-dimensional Poisson problem with peaks as the source term:
∂2u
∂x2
+
∂2u
∂y2
= p(x, y),
with boundary conditions
u(x, y) = 0 if |x| = 3 or |y| = 3.
Make contour plots, similar to Figure 11.5, of p(x, y) and u(x, y).
16 Chapter 11. Partial Differential Equations
(b) Two-dimensional heat equation with peaks as the source term:
∂u
∂t
=
∂2u
∂x2
+
∂2u
∂y2
− p(x, y),
with initial value
u(x, y, 0) = 0
and boundary conditions
u(x, y, t) = 0 if |x| = 3 or |y| = 3.
Create an animated contour plot of the solution as a function of time. What
is the limit as t →∞ of u(x, t)?
(c) Two-dimensional heat equation with peaks as the initial value:
∂u
∂t
=
∂2u
∂x2
,
with initial value
u(x, y, 0) = p(x, y)
and boundary conditions
u(x, y, t) = p(x, y) if |x| = 3 or |y| = 3.
Create an animated contour plot of the solution as a function of time. What
is the limit as t →∞ of u(x, t)?
(d) Two-dimensional wave equation with peaks as the initial value:
∂2u
∂t2
=
∂2u
∂x2
,
with initial values
u(x, y, 0) = p(x, y),
∂u
∂t
(x, y, 0) = 0
and boundary conditions
u(x, y, t) = p(x, y) if |x| = 3 or |y| = 3.
Create an animated contour plot of the solution as a function of time. Does
the limit as t →∞ of u(x, t) exist?
11.6. The method of lines is a convenient technique for solving time-dependent
partial differential equations. Replace all the spatial derivatives with finite
differences, but leave the time derivatives intact. Then use a stiff ordinary
differential equation solver on the resulting system. In effect, this is an im-
plicit time-stepping finite difference algorithm with the time step determined
Exercises 17
automatically and adaptively by the ODE solver. For our model heat and
wave equations, the ODE systems are simply
u̇ = (1/h2)Au
and
ü = (1/h2)Au.
The matrix (1/h2)A represents 4h, and u is the vector-valued function of t
formed from all the elements u(xi, t) or u(xi, yj , t) at the mesh points.
(a) The Matlab function pdepe implements the method of lines in a general
setting. Investigate its use for our one- and two-dimensional model heat
equations.
(b) If you have access to the Partial Differential Equation Toolbox, investigate
its use for our two-dimensional model heat and wave equations.
(c) Implement your own method of lines solutions for our model equations.
11.7. Answer the following questions about pdegui.
(a) How does the number of points n in the grid depend upon the grid size
h for the various regions?
(b) How does the time step for the heat equation and for the wave equation
depend upon the grid size h?
(c) Why are the contour plots of the solution to the Poisson problem and the
eigenvalue problem with index = 1 similar?
(d) How do the contour plots produced by pdegui of the eigenfunctions of
the L-shaped domain compare with those produced by
contourf(membranetx(index))
(e) Why are the regions Drum1 and Drum2 interesting? Search the Web for
“isospectral” and “Can you hear the shape of a drum?” You should find
many articles and papers, including ones by Gordon, Webb, and Wolpert [3]
and Driscoll [2].
11.8. Add the outline of your hand that you obtained in exercise 3.4 as another
region to pdegui. Figure 11.6 shows one of the eigenfunctions of my hand.
11.9. The electrostatic capacity of a region Ω is the quantity
∫ ∫
Ω
u(x, y)dxdy,
where u(x, y) is the solution to the Poisson problem
4u = −1 in Ω
and u(x, y) = 0 on the boundary of Ω.
(a) What is the capacity of the unit square?
(b) What is the capacity of the L-shaped domain?
(c) What is the capacity of your hand?
18 Chapter 11. Partial Differential Equations
Figure 11.6. An eigenfunction of a hand.
11.10. The statements
load penny
P = flipud(P)
contour(P,1:12:255)
colormap(copper)
axis square
access a file in the Matlab demos directory and produce Figure 11.7. The
data were obtained in 1984 at what was then the National Bureau of Stan-
dards by an instrument that makes precise measurements of the depth of a
mold used to mint the U.S. one cent coin.
Figure 11.7. The depth of a mold used to mint the U.S. one cent coin.
The NCM function pennymelt uses this penny data as the initial condition,
u(x, y, 0), for the heat equation and produces an animated, evolving plot
of the solution, u(x, y, t). You can choose either a lighted, surface plot or
Exercises 19
a contour plot. You can choose the time step δ with uicontrols or with
pennymelt(delta). You can also choose a time-stepping algorithm know as
the ADI or alternating direction implicit method. Each time step involves
two half-steps, one implicit in the x direction and the other implicit in the y
direction.
−σu(k+1/2)(N) + (1 + 2σ)u(k+1/2)(P )− σu(k+1/2)(S)
= σu(k)(E) + (1− 2σ)u(k)(P ) + σu(k)(W ),
−σu(k+1)(E) + (1 + 2σ)u(k+1)(P )− σu(k+1)(W )
= σu(k+1/2)(N) + (1− 2σ)u(k+1/2)(P ) + σu(k+1/2)(S).
Solving these implicit equations on an m-by-n grid requires the solution of m
tridiagonal systems of order n for the first half-step and then n tridiagonal
systems of order m for the second half-step.
Answer these questions about pennymelt.
(a) What is the limiting behavior of u(x, y, t) as t →∞?
(b) For what values of δ is the explicit time-stepping algorithm stable?
(c) Demonstrate that the ADI method is stable for any value of δ.
11.11. Let p(x, y) be the function defined on a 128-by-128 square by the penny data
described in the previous exercise.
(a) Make a contour plot of p(x, y) and make a lighted surface plot using the
section of code in pennymelt.m.
(b) Solve the discrete Poisson problem
4hu = p
with u(x, y) = 0 outside the square, and plot the solution u(x, y).
(c) Use del2 to compute
f = 4hu,
and compare f(x, y) with p(x, y).
11.12. Modify pennymelt.m to solve the wave equation instead of the heat equation.
11.13. Modify waves.m to use nine eigenfunctions instead of four.
11.14. The eigenvalues and eigenfunctions of the unit square are
λm,n = (m
2 + n2)π2,
um,n = sin mx sin ny.
If the λm,n are indexed with one subscript and listed in increasing order, we
have
λk = (2, 5, 5, 8, 10, 10, 13, 13, 17, 17, 18, 20, 20, . . .)π
2.
We see that λ1, λ4, and λ11 are simple eigenvalues, but that most of the
eigenvalues are double.
(a) What is the smallest triple eigenvalue of the unit square and what is its
index? In other words, what is the smallest integer that can be written as
the sum of two squares in three different ways?
(b) What is the smallest quadruple eigenvalue of the unit square?
20 Chapter 11. Partial Differential Equations
11.15. By reflecting the eigenfunctions of the unit square twice, we obtain some of
the eigenfunctions of the L-shaped domain. The indexing is different because
the L also has eigenfunctions that are not derived from the square. For ex-
ample, λ3 of the L is 2π2 because it is equal to λ1 of the square. And λ8 = λ9
of the L is a double eigenvalue, 5π2, corresponding to λ2 = λ3 of the square.
(a) Roughly what fraction of the eigenvalues of the L-shaped region are also
eigenvalues of the square?
(b) What is the smallest triple eigenvalue of the L-shaped region and what
is its index?
(c) What is the smallest quadruple eigenvalue of the L-shaped region?
(d) Neither membranetx nor pdegui uses the sin mx sin ny representation of
eigenfunctions of the square. This is OK because these eigenfunctions are not
unique and can have other representations. How do membranetx and pdegui
compute eigenfunctions? How do they get a set of linearly independent eigen-
functions for eigenvalues with multiplicity greater than one?
11.16. Enter the commands
ncmlogo
cameratoolbar
Or, just enter the command ncmlogo and then select Camera Toolbar
from the View tab on the figure window. Experiment with the various icons
available on the new toolbar. What do they all do?
11.17. Make your own copy of ncmlogo and modify it to create a logo for your own
book or company.
Bibliography
[1] Advanced Technical Materials, Inc.
http://www.atmmicrowave.com
[2] T. A. Driscoll, Eigenmodes of isospectral drums, SIAM Review, 39 (1997),
pp. 1–17.
[3] C. Gordon, D. Webb, and S. Wolpert, Isospectral plane domains and
surfaces via Riemannian orbifolds, Inventiones Mathematicae, 110 (1992), pp.
1–22.
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