Table Of ContentL
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MATLAB Numerical Calculations
MATLAB is a high-level language and environment for numerical computation, visualization, M
and programming. Using MATLAB, you can analyze data, develop algorithms, and create
models and applications. The language, tools, and built-in math functions enable you to A
explore multiple approaches and reach a solution faster than with spreadsheets or traditional T
programming languages, such as C/C++ or Java. This book is designed for use as a
L
scientific/business calculator so that you can get numerical solutions to problems involving
a wide array of mathematics using MATLAB. Just look up the function you want in the book A
and you are ready to use it in MATLAB or use the book to learn about the enormous range of
B
options that MATLAB offers.
MATLAB Numerical Calculations focuses on MATLAB capabilities to give you numerical
N
solutions to problems you are likely to encounter in your professional or scholastic life. It
introduces you to the MATLAB language with practical hands-on instructions and results, u
allowing you to quickly achieve your goals. Starting with a look at basic MATLAB functionality
m
with integers, rational numbers and real and complex numbers, and MATLAB’s relationship
with Maple, you will learn how to solve equations in MATLAB, and how to simplify the results.
e
You will see how MATLAB incorporates vector, matrix and character variables, and functions
thereof. MATLAB is a powerful tool used to define, manipulate and simplify complex r
i
algebraic expressions. With MATLAB you can also work with ease in matrix algebra, making
c
use of commands which allow you to find eigenvalues, eigenvectors, determinants, norms
a
and various matrix decompositions, among many other features. Lastly, you will see how you
can write scripts and use MATLAB to explore numerical analysis, finding approximations of l
integrals, derivatives and numerical solutions of differential equations. C
• How to use MATLAB to work with numeric, vector, matrix and character a
variables
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• How to use MATLAB to work with matrix and vector functions, including c
all the standard matrix operations and decompositions
u
• How to define and work with functions of one or more variables, recursive
l
functions, and use loops and scripts in MATLAB
a
• How MATLAB can be used to explore numerical analysis, finding
t
approximations of integrals and derivatives, and numerical solutions
i
of differential equations o
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s
ISBN 978-1-4842-0347-7 Shelve in
55999
Applications/General
User level
Intermediate–Advanced
9781484203477
For your convenience Apress has placed some of the front
matter material after the index. Please use the Bookmarks
and Contents at a Glance links to access them.
Contents at a Glance
About the Author �����������������������������������������������������������������������������������������������������������������xi
Introduction ������������������������������������������������������������������������������������������������������������������������xv
■ Chapter 1: Introduction to MATLAB �����������������������������������������������������������������������������������1
■ Chapter 2: Integers, Divisibility and Number Systems ����������������������������������������������������11
■ Chapter 3: Real and Complex Numbers ���������������������������������������������������������������������������43
■ Chapter 4: Numerical Variables, Vectors and Matrices ���������������������������������������������������95
■ Chapter 5: Vectors and Matrices �����������������������������������������������������������������������������������139
■ Chapter 6: Functions �����������������������������������������������������������������������������������������������������179
■ Chapter 7: Programming and Numerical Analysis ��������������������������������������������������������211
■ Chapter 8: Numerical Algorithms: Equations, Derivatives, Integrals and
Differential Equations ����������������������������������������������������������������������������������������������������279
iii
Introduction
MATLAB is the tool of the modern day mathematician or engineer. The incredible deep functionality of MATLAB
makes what would have taken hours 40 years ago, often less than a minute with MATLAB built in functions. MATLAB
enables you to explore multiple approaches and reach a solution faster often more accurately than with other tools
or traditional programming languages, such as C/C++ or Java. More importantly, it has changed the way we learn
and made getting solutions immensely simpler. So, you can now focus on the application, instead of the math. And
with that easily available power, you can explore and find more and more functions, test hypotheses and become a
significantly more powerful worker.
This book is designed for use, in part, as a tool to enable you to use MATLAB as a scientific/business calculator
so that you can get numerical solutions to problems involving a wide array of mathematics using MATLAB. But it
is broader in scope than that. The book can be used as an independent resource for MATLAB. A background in the
necessary mathematics is assumed, so this book shows you how to interpret your problems to get MATLAB to do what
you want it to do. Just look up the function you want in the book and you are ready to use it in MATLAB or use the
book to learn about the enormous range of options that MATLAB offers. The book is topical, picking examples to show
not only general methods in using MATLAB, but specifics to use MATLAB for advanced mathematical computations
while giving a glimpse at their application.
MATLAB Numerical Calculations focuses on MATLAB capabilities to give you numerical solutions to problems
you are likely to encounter in your professional or scholastic life. It introduces you to the MATLAB language with
practical hands-on instructions and results, allowing you to quickly achieve your goals. Starting with a look at basic
MATLAB functionality with integers, rational numbers and real and complex numbers, and MATLAB’s relationship
with Maple, you will learn how to solve equations in MATLAB, and how to simplify the results. You will see how
MATLAB incorporates vector, matrix and character variables, and functions thereof. MATLAB is a powerful tool used
to defined, manipulate and simplify complex algebraic expressions. With MATLAB you can also work with ease in
matrix algebra, making use of commands which allow you to find eigenvalues, eigenvectors, determinants, norms and
various matrix decompositions, among many other features. Lastly, you will see how you can write scripts and use
MATLAB to explore numerical analysis, finding approximations of integrals, derivatives and numerical solutions of
differential equations.
xv
Chapter 1
Introduction to MATLAB
1.1 Numerical Calculations with MATLAB
You can use MATLAB as a powerful numerical calculator. While most calculators handle numbers only to a preset
degree of accuracy, MATLAB works to whichever precision is necessary for any given calculation. In addition, unlike
calculators, we can perform operations not only with individual numbers, but also with objects such as matrices.
Most classical numerical analysis topics are treated by MATLAB. It supports matrix algebra, statistics,
interpolation, fit by least squares, numerical integration, minimization of functions, linear programming, numerical
solutions of algebraic equations and differential equations and a long list of further techniques.
Here are some examples of numerical calculations with MATLAB (once the commands have been entered to the
right of the input prompt “>>” simply hit Enter to obtain the result):
1. We can calculate 4 + 3 and get 7 as a result. To do this, just type 4 + 3 and then Enter:
>> 4 + 3
ans =
7
2. We can also find the value of 3 raised to the power 100, without previously fixing the
precision. For this it is enough to simply type 3 ^ 100:
>> 3 ^ 100
ans =
5. 1538e + 047
3. We can also use the command format long e to obtain results in scientific notation with
16 more exponential digits of precision:
>> format long e;
>> 3 ^ 100
ans =
5.153775207320115e+047
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Chapter 1 ■ IntroduCtIon to MatLaB
4. We can also work with complex numbers. We find the result of the operation
raising (2 + 3i) to the power 10 by typing the expression (2 + 3i) ^ 10:
>> (2 + 3i) ^ 10
ans =
-3 415250000000001e + 005 - 1. 456680000000001e + 005i
The command format long g will optimize the output of future calculations.
>> format long g
>> (2 + 3i) ^ 10
ans =
-341525 - 145668i
5. The previous result can also be obtained in short format using the command format short:
>> format short;
>> (2 + 3i) ^ 10
ans =
-3.4152e+005- 1.4567e+005i
6. We can calculate the value of the Bessel function at the point 13.5. To do this, we type
Besselj(0,13.5):
>> Besselj(0,13.5)
ans =
0.2150
7. We can also perform numerical integration. To calculate the integral between 0 and p
of sin(x), we type the expression int(sin(x), 0, pi) after having declared the variable x as
symbolic with the command syms:
>> syms x
>> int(sin(x), 0, pi)
ans =
2
These themes will be treated more thoroughly later.
2
Chapter 1 ■ IntroduCtIon to MatLaB
1.2 Symbolic Calculations with MATLAB
MATLAB handles symbolic mathematical computation perfectly, manipulating formulae and algebraic expressions
easily and efficiently. You can expand, factor and simplify polynomials, rational functions and trigonometric
expressions; you can find algebraic solutions of polynomial equations and systems of equations; you can evaluate
derivatives and integrals symbolically and find solutions of differential equations; you can manipulate power series,
find limits and explore many other facets of algebraic series.
To perform this task, MATLAB requires that all variables (or algebraic expressions) are previously declared as
symbolic using the command syms.
Here are some examples of symbolic computations with MATLAB:
1. We find the cube of the algebraic expression: (x + 1)(x + 2) - (x + 2)^2. This is done by
typing the expression: expand((x + 1)*(x + 2) - (x + 2)^2)^3). The result will be another
algebraic expression:
>> syms x
>> expand (((x + 1)*(x + 2)-(x + 2)^2)^3).
ans =
-x ^ 3-6 * x ^ 2-12 * x-8
2. We can factor the result of the above calculation by typing
factor((x + 1) *(x + 2) - (x + 2)^2)^3):
>> syms x
>> factor(((x + 1)*(x + 2)-(x + 2)^2)^3)
ans =
-(x+2) ^ 3
3. We can find the indefinite integral of the function
(x ^ 2)sin(x)^2 by typing int(x^2 * sin(x)^2, x):
>> syms x
>> int(x^2*sin(x)^2, x)
ans =
x ^ 2 * (-1/2 * cos(x) * sin(x) + 1/2 * x)-1/2 * x * cos(x) ^ 2 + 1/4 *
cos(x) * sin(x) + 1/4 * 1/x-3 * x ^ 3
4. We can simplify the previous result:
>> syms x
>> simplify(int(x^2*sin(x)^2, x))
ans =
-1/2 * x ^ 2 * cos(x) * sin(x) + 1/6 * x ^ 3-1/2 * x * cos(x) ^ 2 + 1/4 *
cos(x) * sin(x) + 1/4 * x
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Chapter 1 ■ IntroduCtIon to MatLaB
5. We can display the previous result in standard mathematical notation:
>> syms x
>> pretty(simplify(int('x^2*sin(x)^2', 'x')))
2 3 2
-1/2 x cos(x) sin(x) + 1/6 x - 1/2 x cos(x) + 1/4 cos(x) sin(x) + 1/4 x
6. We can expand the function x^2 * sin(x)^2 as a power series up to order 12, presenting the
result in standard form:
>> syms x
>> pretty(taylor(x^2*sin(x)^2,12))
4 6 8 10 12
x 1/3 x + 2/45 x - 1/315 x + 0 (x)
7. We can solve the equation 3ax 7x^2 + x^3 = 0 (where a is a parameter):
>> syms x a
>> solve('3*a*x-7*x^2 + x^3 = 0', x)
ans =
[ 0 ]
[7/2 + 1/2 *(49-12*a) ^(1/2) ]
[7/2-1/2 *(49-12*a) ^(1/2) ]
8. We can find the five solutions of the equation x^5 + 2x + 1 = 0 :
>> syms x
>> solve('x^5+2*x+1','x')
ans =
[-.7018735688558619 -.8796971979298240 * i]
[-.7018735688558619 +.8796971979298240 * i]
[-.4863890359345430 ]
[ .9450680868231334 -.8545175144390459 * i]
[ .9450680868231334 +.8545175144390459 * i]
On the other hand, MATLAB can be used together with the Maple program libraries to work with symbolic
mathematics, thus extending its field of action. In this way, MATLAB can be used to work on topics such as differential
forms, Euclidean geometry, projective geometry, statistics, etc.
At the same time, we can extend MATLAB’s numerical calculation cababilities by using the Maple libraries
(combinatorics, optimization, number theory, statistics, etc.)
4
Chapter 1 ■ IntroduCtIon to MatLaB
1.3 MATLAB and Maple
Whenever it is necessary to use a Maple command or function from MATLAB, use the command maple followed
by the corresponding Maple syntax. This functionality is only available if you have installed the symbolic
computation Toolbox “Extended Symbolic Math Toolbox”. This is the tool one uses to work with linear algebra and
mathematical analysis.
To use a Maple command from MATLAB, the syntax is as follows:
maple(‘Maple_command_syntax’)
or alternatively:
maple ‘Maple_command_syntax’
To use a Maple command with N arguments from MATLAB, the syntax is as follows:
maple(‘Maple_command_syntax’, argument1, argument2, . . . , argumentN)
Let’s see some examples:
1. We can calculate the limit of the function (x^3 -1) / (x-1) as x - > 1:
>> maple('limit((x^3-1)/(x-1),x=1)')
ans =
3
We could also have used the following syntax:
>> maple 'limit((x^3-1)/(x-1),x=1)'
ans =
3
2. We can calculate the greatest common divisor of 10,000 and 5,000:
>> maple('gcd', 10000, 5000)
ans =
5000
1.4 General Notation. The Command Window
Whenever a program is used, it is necessary to become familiar with the general characteristics of its notation. Like
any program, the best way to learn MATLAB is to use it. Each example consists of the user input prompt “>>” followed
by the command and the MATLAB response on the next line. See Figure 1-1.
5
Chapter 1 ■ IntroduCtIon to MatLaB
Figure 1-1.
Sometimes, depending on the type of command (user input) given to MATLAB in the Command Window, the
response will begin with the expression ans =. See Figure 1-2.
Figure 1-2.
It is important to pay attention to uppercase and lowercase characters, parentheses and square brackets, and the
use of spaces and punctuation (in particular commas and semicolons).
Like the C programming language, MATLAB is case sensitive; for example, Sin(x) is different to sin(x). The names
of all built-in functions begin with lowercase letters. Each bracket has its own meaning, as we will see later.
To indicate that two variables must be multiplied you put the symbol * between them, and there cannot be spaces
in the names of commands, variables or functions. In other cases, spaces are ignored, but they can be included to
make the input more readable.
6
