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MATLAB Numerical Calculations
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Cesar Perez Lopez
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2014
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MATLAB is a high-level language and environment for numerical computation, visualization, 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 explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. This book is designed for use as a 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 and you are ready to use it in MATLAB or use the book to learn about the enormous range of options that MATLAB offers.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 MATLABs 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.What youll learn How to use MATLAB to work with numeric, vector, matrix and character variables. How to use MATLAB to work with matrix and vector functions, including all the standard matrix operations and decompositions. How to define and work with functions of one or more variables, recursive functions, and use loops and scripts in MATLAB. How MATLAB can be used to explore numerical analysis, finding approximations of integrals and derivatives, and numerical solutions of differential equations.Who this book is for This book is for anyone who wants to work in a practical, hands-on manner on numeric calculations in algebra or calculus problems with MATLAB. Youll already have a core understanding of undergraduate level calculus, algebra and linear algebra, and have access to an installed version of MATLAB, but no previous experience of MATLAB is assumed. Table of ContentsChapter 1 Introduction to MATLABChapter 2 Integers, Divisibility and Numbering SystemsChapter 3 Real and Complex ExpressionsChapter 4: Numeric VariablesChapter 5: Vectors and MatricesChapter 6: FunctionsChapter 7: Programming and Numerical Analysis MethodsChapter 8: Numerical Algorithms: Equations, Derivatives, Integrals and Differential Equations

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Csar Prez Lpez 2014

Csar Prez Lpez MATLAB Numerical Calculations 10.1007/978-1-4842-0346-0_1

1. Introduction to MATLAB

Csar Prez Lpez 1

(1)

Madrid, Spain

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):

We can calculate 4 + 3 and get 7 as a result. To do this, just type 4 + 3 and then Enter :

>> 4 + 3

ans =

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

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

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

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

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

We can also perform numerical integration. To calculate the integral between 0 and 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 =

These themes will be treated more thoroughly later.

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:

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

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

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

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

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

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)

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) ]

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 MATLABs numerical calculation cababilities by using the Maple libraries (combinatorics, optimization, number theory, statistics, etc.)

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)

Lets see some examples:

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 =

We could also have used the following syntax:

>> maple 'limit((x^3-1)/(x-1),x=1)'

ans =

We can calculate the greatest common divisor of 10,000 and 5,000 :

>> maple('gcd', 10000, 5000)

ans =

5000

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