chapter 1an engineer’s guide to matlab copyright © edward b. magrab 20091 an engineer’s guide...

Copyright © Edward B. Magrab 2009 1

Chapter 1An Engineer’s Guide to MATLAB

AN ENGINEER’S GUIDE TO MATLAB

3rd Edition

CHAPTER 1

INTRODUCTION



Goal of Course

For you to be able to generate readable, compact, and verifiably correct MATLAB programs that obtain numerical solutions to a wide range of physical and empirical models, and to display the results with fully annotated graphics.



MATLAB founded in 1984 by Jack Little, Steve Bangert, and Clive Moler

1985 - MIT bought 10 copies

1986 - MATLAB 2

1987 - MATLAB 3

1990 - Simulink 1

1994 - MATLAB 4

1996 - MATLAB 5

2000 - MATLAB 6

2004 - MATLAB 7

2009 – MATLAB 7.8 (2009a)



Chapter 1 – Objective

To introduce the fundamental characteristics of the MATLAB environment and the language’s basic syntax



MATLAB

• A computing language devoted to processing data in the form of arrays of numbers (called matrices).

• Integrates computation and visualization into a flexible computer environment, and provides a diverse family of built-in functions that can be used to obtain numerical solutions to a wide range of engineering problems.

• Derives its name from MATrix LABoratory.



Some Suggestions on How to Use MATLAB

Use the Help files extensively.

This will minimize errors caused by incorrect syntax and by incorrect or inappropriate application of a MATLAB function.

Write scripts and functions in a text editor and save them as M files.

This will save time, save the code, and greatly facilitate the debugging process, especially if the MATLAB editor/debugger is used.




Attempt to minimize the number of expressions comprising scripts and functions.

This usually leads to a tradeoff between readability and compactness, but it can encourage the search for MATLAB functions and procedures that can perform some of the steps faster and more directly.

When practical, use graphical output as the script or function is being developed.

This usually shortens the code development process by identifying potential coding errors and can facilitate the understanding of the physical process being modeled or analyzed.




Most importantly, verify by independent means that the outputs from the scripts and functions are correct.



Notation Conventions

Variable/Function Name Font Example

User-created variable Times Roman ExitPressurea2, sig

MATLAB function Courier cosh(x), pi

User-created function Times Roman BeamRoots(a, x, k)Bold

Numerical Value Font Example

Provided in program Times Roman 5.672

Output to command window Helvetica 5.672or to a graphical display



The MATLAB Environment

Preliminaries: Command Window Management

Executing Expressions from the MATLAB Command Window: Basic MATLAB Syntax

Clarification and Exceptions to MATLAB Syntax

MATLAB Functions

Creating Scripts and Executing Them from the MATLAB Editor

Online Help

Symbolic Toolbox

What We Will Cover in Chapter 1



Command Window, Command History, Workspace



Command Window, Command History, Workspace, and Editor



Command Window and Editor

MATLAB command window (left) and editor (right) after closing the command history and workspace windows



Clearing of Command Window and Workspace

MATLAB function Description

clc Clear the command window

clear Removes variables from the workspace (computer memory)

close all Closes (deletes) all graphic windows

format Formats the display of numerical output to the command window

format compact Removes empty (blank) lines



These operation can also be done with



and with

format compact

format long e

format short



Option Display (number > 0) Display (number < 0)

shortlongshort elong eshort glong gshort englong engrationalhexbank

444.44444.444444444444445e+0024.4444e+0024.444444444444445e+002444.44444.444444444444444.4444e+000444.444444444444e+0004000/9407bc71c71c71c72444.44

0.00440.0044444444444444.4444e-0034.444444444444444e-0030.00444440.004444444444444444.4444e-0034.44444444444444e-0031/2253f723456789abcdf0.00

Results from Different Format Selections



Preferences Menu Selections: Font Size



MATLAB Variable Names

• 63 alphanumeric characters

• Start with uppercase or lowercase letter

Followed by any combination of uppercase and lowercase letters, numbers, and the underscore character (_)

• Case sensitive - junk different from junK

• Example –

exit_pressure or ExitPressure

• Length of variable names -

Tradeoff between easily recognizable identifiers and readability of the resulting expressions



Keywords Reserved Explicitly for the MATLAB Programming Language

breakcasecatchcontinueelseelseifendforfunction

globalifotherwisepersistentreturnswitchtrywhile



» p = 7.1p = 7.1000» x = 4.92x = 4.9200» k = -1.7k = -1.7000

User types and hits Enter

System response


System response


System response

Command Window Interaction



» p = 7.1;» x = 4.92;» k = -1.7;»

Suppression of System Response - Semicolon

Several Expressions Placed on One Line

p = 7.1, x = 4.92, k = 1.7

System response –p = 7.1000x = 4.9200k = -1.7000»

Using semicolon instead of commas

suppresses this output



Arithmetic Operators

Hierarchy Level

( ) Parentheses 1

´ Prime (Apostrophe) 1

^ Exponentiation 2

* Multiplication 3

/ Division 3

+ Addition 4

Subtraction 4



Parentheses

• Needed so that the mathematical operations are performed on the proper collections of quantities and in their proper order.

• Within each set of parentheses the mathematical operation are performed from left to right on quantities at the same hierarchical level.

These rules can help minimize the number of parentheses.



Some Examples

Mathematical expression

MATLAB expression

1 dcx+2 dcx + 2 (2/d)cx+2 (dcx + 2)/g2.7

2xdc

1-d*c^(x+2) d*c^x+2 or 2+d*c^x (2/d)*c^(x+2) or 2/d*c^(x+2) or 2*c^(x+2)/d (d*c^x+2)/g^2.7 sqrt(d*c^x+2) or (d*c^x+2)^0.5



Another Example

Consider

1

1+

k

tpx

The MATLAB script is

p = 7.1; x = 4.92; k = 1.7; % Numerical values % must be assigned

firstt = (1/(1+p*x))^k

which results in

t = 440.8779



Syntax Clarification and Exceptions

Blanks

In an arithmetic expression, blanks can be employed with no consequence – except when

expression appears in an array specification; that is, between two brackets [ ]

Excluding blanks in assignment statements, variable names on the right hand side of the equal sign must be separated by

one of the arithmetic operators

a comma (,)

a semicolon (;)



Two Exceptions

(1) Complex Numbers: z = a +1jb or z = a +1ib (i = j = )

Example

a = 2; b = 3;z = (a+1j*b)*(4-7j)

-1

Note:

Real and complex numbers can be mixed without any special concerns.

Example 1

z = 4 + sqrt(-4)

System displaysz = 4.0000 + 2.0000i

Example 2

z = 1i^1i

System displaysz = 0.2079



(2) Exponential Form: x = 4.56102

x = 4.56e-2 or

x = 0.0456 or

x = 4.56*10^-2

Note:

Maximum number of digits that can follow the ‘e’ is 3.



System Assignment of Variable Names

Type in the command window

cos(pi/3)

The system responds with

ans =

0.5000

The variable ans can now be used as one would any other variable. Then, typing in the command window

ans+2

the system responds with ans = 2.5000



Scalars versus Arrays

MATLAB considers all variables as arrays of numbers –

• When using the five arithmetic operators, +, , *, /, and ^, these operations have to obey the rules of linear (matrix) algebra

• When the variables are scalar quantities [arrays of one element (one row and one column)] the usual rules of algebra apply

• One can operate on arrays of numbers and suspend the rules of linear algebra by using dot operations



Mathematical function MATLAB expression

ex ex 1 x << 1

x ln(x) or loge(x) log10(x) |x| signum(x) loge(1+x) x << 1 n! All prime numbers n

exp(x) expm1(x) sqrt(x)

log(x)

log10(x) abs(x) sign(x) log1p(x) factorial(n) primes(n)

Some Elementary MATLAB Functions



Some MATLAB Constants and Special Quantities

Mathematical quantity or operation

MATLAB expression

Comments

1

Floating point relative accuracy 0/0, 0×, / Largest floating-point number before overflow Smallest floating-point number before underflow

pi i or j eps inf NaN realmax realmin

3.141592653589793 Indicates complex quantity. 2.2210-16 Infinity Undefined mathematical operation 1.7977e+308

2.2251e-308



MATLAB Trigonometric and Hyperbolic Functions

Trigonometric Hyperbolic

Function (radians) (degrees) Inverse Inverse

sine cosine tangent secant cosecant cotangent

sin(x) cos(x) tan(x) sec(x) csc(x) cot(x)

sind(x) cosd(x) tand(x) secd(x) cscd(x) cotd(x)

asin(x) acos(x) atan(x)† asec(x) acsc(x) acot(x)

sinh(x) cosh(x) tanh(x) sech(x) csch(x) coth(x)

asinh(x) acosh(x) atanh(x) asech(x) acsch(x) acoth(x)

† atan2(y, x) is the four quadrant version.



Several Specialized Mathematical Functions

Mathematical function

MATLAB Expression

Description

Ai(x) Bi(x) I(x) J(x) K(x) Y(x) B(x,w) K(m), E(m) erf(x), erfc(x) E1(z) (a)

( )mnP x

airy(0,x) airy(2,x) besseli(nu, x) besselj(nu, x) besselk(nu, x) bessely(nu, x) beta(x, w) ellipke(m) erf(x), erfc(x) expint(x) gamma(a) legendre(n, x)

Airy function Airy function Modified Bessel function of first kind Bessel function of first kind Modified Bessel function of second kind Bessel function of second kind Beta function Complete elliptic integrals of 1st & 2nd kind Error and complementary error function Exponential integral Gamma function Associated Legendre function



Several Specialized Statistical Functions

Mathematical function MATLAB Expression

Description

maximum value of x median minimum value of x mode or s 2 or s2

max(x) mean(x) median(x) min(x) mode(x) std(x) var(x)

Largest element(s) in an array Average or mean value of an array Median value of an array Smallest element(s) in an array Most frequent values in an array Standard deviation of an array Variance of an array of values



MATLAB Relational Operators

Conditional Mathematical symbol

MATLAB symbol

equal not equal less than greater than less than or equal greater than or equal

= < >

== ~= < > <= >=



Example

For x = 0.1 and a = 0.5, determine the value of y when

-= - sin( )/cosh( ) - ln ( )πxey e x a x + a

The script is

x = 0.1; a = 0.5;y = sqrt(abs(exp(-pi*x)-sin(x)/cosh(a)-log(x+a)))

When executed, the system returnsy = 1.0736



Decimal-to-Integer Conversion Functions

MATLAB function

x y Description

y = fix(x)

2.7 1.9 2.49-2.51j

2.0000 1.0000 2.0000 2.0000i

Round toward

zero

y = round(x)

2.7 1.9 2.49 2.51j

3.0000 2.0000 2.0000 3.0000i

Round to nearest

integer

y = ceil(x)

2.7 1.9 2.49 2.51j

3.0000 1.0000 3.0000 - 2.0000i

Round toward

infinity

y = floor(x)

2.7 1.9 2.49 2.51j

2.0000 2.0000 2.0000 3.0000i

Round toward

minus infinity



Complex Number Manipulation Functions

MATLAB function z y Description

z = complex(a, b) a + b*j - Form complex number; a and b real

y = abs(z) 3 + 4j 5 Absolute value: 2 2a b

y = conj(z) 3 + 4j 3 4j Complex conjugate

y = real(z) 3 + 4j 3 Real part

y = imag(z) 3 + 4j 4 Imaginary part

y = angle(z) a +b*j atan2(b, a) Phase angle in radians: y



Additional Special Characters and a Summary of Their Usage



Creating Programs and Executing Them from the MATLAB Editor

When to use the editor -

1. The program will contain more than a few lines of code.

2. The program will be used again.

3. A permanent record is desired.

4. It is expected that occasional upgrading will be required.

5. Substantial debugging is required.

6. One wants to transfer the listing to another person or organization.



Additional Reasons -• Required when creating functions

• Editor has many features

Commenting/Un-commenting lines

Smart Indenting

Parentheses checking

Keywords in blue

Comments in green

Strings in violet

Smart indent



Additional Reasons -

• Click one icon to save and run a program (Must use Save As first time)

Save and Run icon



Additional Reasons -

Enabling M-Lint from the Preferences menu



Illustration of M-Lint

Orange bar

Red bar

Red square

With cursor placed on red bar, this error message is displayed.



Additional Reasons –

Enabling the cell feature of the Editor



Illustration of Editing Cells

Run the highlighted

cell

Run the highlighted

cell and advance to

the next cell

Selected cell is highlighted

Second cell

Third cell



A Script or a Function Typically Has the Following Attributes -

1. Documentation:

Purpose and operations performed

Programmer's Name

Date originated

Date(s) revised

Description of the input(s): number, meaning, and type

Description of the output(s): number, meaning, and type



2. Input -

which includes numerous checks to ensure that all input values have the qualities required for the script/function to work properly.

3. Initialization -

where the appropriate variables are assigned their numerical values.

4. Computation -

where the numerical evaluations are performed.

5. Output -

where the results are presented as graphical and/or annotated numerical quantities.



Naming of Programs and Functions

File name follows that for variable names.

Must start with an upper or lower case letter followed by up to 62 contiguous alphanumeric characters and the underscore character.

No blank spaces are allowed in file names.

[This is different from what is allowed by the Windows operating system]

A ‘.m’ suffix must be affixed to the file name –

Consequently, these files are called ‘M’ files



Saving/Executing (Running) M Files

Set Path window (File)

Change current path to file being executed.



Accessing the Browser Window to Change Current Path (Directory)

Clicking on this icon brings up the

Browser



Example - Flow in a circular channel

3/23/2 5/2c

5/2

2 - 0.5sin 2=

8 sin 1- cos

D g θ θQ

θ θ

12

cosc

dD

Let d = 2 m, g = 9.8 m/s2, and = 60 = /3. Then the script is

g = 9.8; d = 2; th = pi/3; % InputDc = d/2*(1-cos(th));Qnum = 2^(3/2)*Dc^(5/2)*sqrt(g)*(th-0.5*sin(2*th))^(3/2);Qden = 8*sqrt(sin(th))*(1-cos(th))^(5/2);Q = Qnum/Qden % m^3/s

Dc

d/2 d/2 2



The various MATLAB windows after executing the

program



On-Line Help: Getting Started



On-Line Help: Desktop



On-Line Help: Command Window



On-Line Help: When Function Name is Known

Type in function name



On-Line Help: When Function Name Is Not Known

Type search entry and press Enter



Symbolic Toolbox

The Symbolic Math Toolbox provides the capability of manipulating symbols to perform algebraic, matrix, and calculus operations symbolically

When one couples the results obtained from symbolic operations with MATLAB’s ability to create functions, one has a very effective means of numerically evaluating symbolically obtained expressions. This is introduced in Chapter 5.



We will illustrate by example –

• Syntax

• Variable precision arithmetic

• Taylor series expansions

• Differentiation and integration

• Limits

• Substitution

• Inverse Laplace transform



The shorthand way to create symbolic variables is with

syms a b c …

where a, b, and c are now symbolic variables.

The blanks between each variable are required

If the variables are restricted to being real variables, then we modify this statement as

syms a b c real

These symbols can be intermixed with non-symbolic variable names, numbers, and MATLAB functions, with the result being a symbolic expression.



Consider the expression 2-= 11.92 +af e b/d

Assuming that d = 4.2 (1/d = 0.238095), the script to represent this expression symbolically is

syms a bd = 4.2;f = 11.92*exp(-a^2)+b/d

which, upon execution, displays

f =298/25*exp(-a^2)+5/21*b

where f is a symbolic object.

Note that:

21/5 = 4.2

298/25 = 11.92



Numbers in a symbolic expression are converted to the ratio of two integers, when possible.

If the decimal representation of numbers is desired, then one uses

vpa(f, n)

where f is the symbolic expression and n is the number of digits.

Thus, to revert to the decimal notation with five decimal digits, the script becomes

syms a bd = 4.2;f = vpa(11.92*exp(-a^2)+b/d, 5)

2-= 11.92 +af e b/d



which yields

f =11.920*exp(-1.*a^2)+.23810*b

Note:

1/d = 1/4.2 = 0.2381

Variable Precision Arithmetic

One can also use vpa to calculate quantities with more than 15 digits of accuracy as follows

vpa('Expression', n)

where Expression is a valid MATLAB symbolic relation and n is the desired number of digits of precision.



Consider the evaluation of the following expression 10032!y e

The script to evaluate this relation with 50 digits of precision is

y = vpa('factorial(32)-exp(100)', 50)

Its execution gives

y =

-26881171155030517550432725348582123713611118.773742

If variable precision arithmetic had not been used, then

y = 2.688117115503052×1043



Symbolic Differentiation and Integration

Differentiation is performed with the function

diff(f, x, n)

where

f = f(x) is a symbolic expression

x = variable with which differentiation is performed

n = number of differentiations to be performed

For example, when n = 2 the second derivative is taken



Example

We shall take the derivative of bcos(bt), first with respect to t and then with respect to b.

The script is

syms b tdt = diff(b*cos(b*t), t, 1)db = diff(b*cos(b*t), b, 1)

Upon execution, we obtain

dt =-b^2*sin(b*t)db =cos(b*t)-b*sin(b*t)*t



Integration is performed with the function

int(f, x, c, d)

where

f = f(x) is a symbolic expressionx = variable of integrationc = lower limit of integration d = upper limit of integration

When c and d are omitted, the application of int results in the indefinite integral of f(x)



Example

We shall integrate the results of the differentiation performed previously. Thus,

syms b tf = b*cos(b*t);dt = diff(f, t, 1);db = diff(f, b, 1);it = int(dt, t)ib = int(db, b)

The execution results in

it =b*cos(b*t)ib =b*t*cos(b*t))



Limits

One can take the limit of a symbolic expression using

limit(f, x, z)

where f = f(x) is the symbolic function x = symbolic variable that is to assume the limiting value z

Example

Determine the limit of2

Lim3 4a

a b

a



The script is

syms a bLim = limit((2*a+b)/(3*a-4), a, inf)

where inf =

Example

Determine the limit of

2Lim

3 4a

a b

a

xLim 1

xy

x



The script is

syms y xLim = limit((1+y/x)^x, x, inf)


Lim =exp(y)

In other words, the limit is ey.

xLim 1

xy

x



Taylor Series Expansion

An n-term Taylor series expansion of a function f(x) about the point a is given by

( )1

0

( )( )

!

knk

k

f ax a

k

The function that performs this operation is

taylor(f, n, a, x)



Example

Obtain a four-term expansion of cos about o. The script is

syms x thoTay = taylor(cos(x), 4, tho, x)


Tay =cos(tho)-sin(tho)*(x-tho)-1/2*cos(tho)*(x-tho)^2 +1/6*sin(tho)*(x-tho)^3

That is,

2 31 1cos sin cos sin

2 6o o o o o o ox x x



Substitution

To substitute one expression b for another expression a, the following function is used

subs(f, a, b)

where f = f(a)

Inverse Laplace Transform

If the Laplace transform of a function f(t) is F(s), where s is the Laplace transform parameter, then the inverse Laplace transform is obtained from

ilaplace(F, s, t)



2

1( )

2 1F s

s s

Example

Consider the expression

where 0 < < 1.

The inverse Laplace transform is obtained from

syms s t z f = ilaplace(1/(s^2+2*z*s+1), s, t)

The execution of this script gives

f =

exp(-t*z)*sinh(t*(z^2-1)^(1/2))/(z^2-1)^(1/2)



To simplify this expression, we make use of the following change of variables

2 2 2z^2-1 1 1 r

Then, a simplified expression can be obtained by modifying the original script as follows

syms s t z r f = ilaplace(1/(s^2+2*z*s+1), s, t)f = simple(subs(f,(z^2-1), -r^2))


f =exp(-t*z)*sin(t*r)/r 2

2sin 1

1

tet



Example

Determine the curvature of the parametric curves2 cos cos 2

2 sin sin 2

x b t b t

y b t b t

The curvature is determined from

3/ 22 2

x y y x

x y

where the prime denoted the derivative with respect to t. The script is



syms t a b x = 2*b*cos(t) +b*cos(2*t);y = 2*b*sin(t)-b*sin(2*t);xp = diff(x, t, 1);xpp = diff(x, t, 2);yp = diff(y, t, 1);ypp = diff(y, t, 2);kn = xp*ypp-yp*xpp;kd = xp^2+yp^2;kn = factor(simple(kn));kd = factor(simple(kd));k = simple(kn/kd^(3/2))

The execution of this script gives

k =-1/4/(2-2*cos(3*t))^(1/2)/b

1

4 2 2cos3b t



This result can be simplified with the following trigonometric identity

21 cos 2sin ( / 2)a a

Then

22-2*cos(3*t) 2 1 cos3 4sin 3 / 2t t

To make this final change, we employ subs as follows

k = simple(subs(k, 2-2*cos(3*t), 4*sin(3*t/2)^2))

to obtaink =-1/8/sin(3/2*t)/b

1

8 sin(3 / 2)b t

chapter 1an engineer’s guide to matlab copyright © edward b. magrab 20091 an engineer’s guide...

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