Script to Plot tan(θ) ( 1)

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Matlab Programming

Gerald W. Recktenwald

Department of Mechanical Engineering Portland State University

gerry@me.pdx.edu

These slides are a supplement to the book Numerical Methods with Matlab: Implementations and Applications, by Gerald W. Recktenwald,

c 2001, Prentice-Hall, Upper Saddle River, NJ. These slides are c 2001 Gerald W. Recktenwald. The PDF version of these slides may be downloaded or stored or printed only for noncommercial, educational use. The repackaging or sale of these slides in any form, without written consent of the author, is prohibited.

The latest version of this PDF ﬁle, along with other supplemental material for the book, can be found at www.prenhall.com/recktenwald.

Version 0.97 August 28, 2001

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Overview

• Script m-ﬁles

Creating

Side eﬀects

• Function m-ﬁles

Syntax of I/O parameters

Text output

Primary and secondary functions

• Flow control

Relational operators

Conditional execution of blocks

Loops

• Vectorization

Using vector operations instead of loops

Preallocation of vectors and matrices

Logical and array indexing

• Programming tricks

Variable number of I/O parameters

Indirect function evaluation

Inline function objects

Global variables

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Preliminaries

• Programs are contained in m-ﬁles

Plain text ﬁles – not binary ﬁles produced by word processors

File must have “.m” extension

• m-ﬁle must be in the path

Matlab maintains its own internal path

The path is the list of directories that Matlab will search when looking for an m-ﬁle to execute.

A program can exist, and be free of errors, but it will not run if Matlab cannot ﬁnd it.

Manually modify the path with the path, addpath, and rmpath built-in functions, or with addpwd NMM toolbox function

. . . or use interactive Path Browser

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Script Files

• Not really programs

No input/output parameters

Script variables are part of workspace

• Useful for tasks that never change

• Useful as a tool for documenting homework:

Write a function that solves the problem for arbitrary parameters

Use a script to run function for speciﬁc parameters required by the assignment

Free Advice: Scripts oﬀer no advantage over functions.

Functions have many advantages over scripts.

Always use functions instead of scripts.

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Script to Plot tan(θ) ( 1)

Enter statements in ﬁle called tanplot.m 1. Choose New. . . from File menu 2. Enter lines listed below

Contents of tanplot.m:

theta = linspace(1.6,4.6);

tandata = tan(theta);

plot(theta,tandata);

xlabel(’\theta (radians)’);

ylabel(’tan(\theta)’);

grid on;

axis([min(theta) max(theta) -5 5]);

3. Choose Save. . . from File menu Save as tanplot.m

4. Run it

>> tanplot

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Script to Plot tan(θ) ( 2)

Running tanplot produces the following plot:

2 2.5 3 3.5 4 4.5

-5 -4 -3 -2 -1 0 1 2 3 4 5

θ (radians)

tan(θ)

If the plot needs to be changed, edit the tanplot script and rerun it. This saves the eﬀort of typing in the commands. The tanplot script also provides written documentation of how to create the plot.

Example: Put a % character at beginning of the line

containing the axis command, then rerun the script

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Script Side-Eﬀects ( 1)

All variables created in a script ﬁle are added to the workplace.

This may have undesirable eﬀects because

• Variables already existing in the workspace may be overwritten

• The execution of the script can be aﬀected by the state variables in the workspace.

Example: The easyplot script

% easyplot: Script to plot data in file xy.dat

% Load the data

D = load(’xy.dat’); % D is a matrix with two columns x = D(:,1); y = D(:,2); % x in 1st column, y in 2nd column plot(x,y) % Generate the plot and label it xlabel(’x axis, unknown units’)

ylabel(’y axis, unknown units’)

title(’Plot of generic x-y data set’)

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Script Side-Eﬀects ( 2)

The easyplot script aﬀects the workspace by creating three variables:

>> clear

>> who

(no variables show)

>> easyplot

>> who

Your variables are:

D x y

The D, x, and y variables are left in the workspace. These generic variable names might be used in another sequence of calculations in the same Matlab session. See Exercise 10 in Chapter 4.

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Script Side-Eﬀects ( 3)

Side Eﬀects, in general:

• Occur when a module changes variables other than its input and output parameters

• Can cause bugs that are hard to track down

• Cannot always be avoided

Side Eﬀects, from scripts

• Create and change variables in the workspace

• Give no warning that workspace variables have changed

Because scripts have side eﬀects, it is better to encapsulate any mildly complicated numerical in a function m-ﬁle

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Function m-ﬁles ( 1)

• Functions are subprograms:

Functions use input and output parameters to

communicate with other functions and the command window

Functions use local variables that exist only while the function is executing. Local variables are distinct from variables of the same name in the workspace or in other functions.

• Input parameters allow the same calculation procedure (same algorithm) to be applied to diﬀerent data. Thus, function m-ﬁles are reusable.

• Functions can call other functions.

• Speciﬁc tasks can be encapsulated into functions. This modular approach enables development of structured solutions to complex problems.

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Function m-ﬁles ( 2)

Syntax:

The ﬁrst line of a function m-ﬁle has the form:

function [outArgs] = funName(inArgs)

outArgs are enclosed in [ ]

• outArgs is a comma-separated list of variable names

• [ ] is optional if there is only one parameter

• functions with no outArgs are legal

inArgs are enclosed in ( )

• inArgs is a comma-separated list of variable names

• functions with no inArgs are legal

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Function Input and Output ( 1)

Examples: Demonstrate use of I/O arguments

• twosum.m — two inputs, no output

• threesum.m — three inputs, one output

• addmult.m — two inputs, two outputs

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Function Input and Output ( 2)

twosum.m

function twosum(x,y)

% twosum Add two matrices

% and print the result x+y

threesum.m

function s = threesum(x,y,z)

% threesum Add three variables

% and return the result

s = x+y+z;

addmult.m

function [s,p] = addmult(x,y)

% addmult Compute sum and product

% of two matrices

s = x+y;

p = x*y;

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Function Input and Output Examples ( 3)

Example: Experiments with twosum:

>> twosum(2,2) ans =

4

>> x = [1 2]; y = [3 4];

>> twosum(x,y) ans =

4 6

>> A = [1 2; 3 4]; B = [5 6; 7 8];

>> twosum(A,B);

ans =

6 8

10 12

>> twosum(’one’,’two’) ans =

227 229 212

Notes: 1. The result of the addition inside twosum is exposed because the x+y expression does not end in a semicolon. (What if it did?)

2. The strange results produced by twosum(’one’,’two’) are obtained by adding the numbers associated with the ASCII character codes for each of the letters in ‘one’ and ‘two’.

Try double(’one’) and double(’one’) + double(’two’).

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Function Input and Output Examples ( 4)

Example: Experiments with twosum:

>> clear

>> x = 4; y = -2;

>> twosum(1,2) ans =

3

>> x+y ans =

2

>> disp([x y])

4 -2

>> who

Your variables are:

ans x y

In this example, the x and y variables deﬁned in the workspace are distinct from the x and y variables deﬁned in twosum. The x and y in twosum are local to twosum.

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Function Input and Output Examples ( 5)

Example: Experiments with threesum:

>> a = threesum(1,2,3) a =

6

>> threesum(4,5,6) ans =

15

>> b= threesum(7,8,9);

Note: The last statement produces no output because the assignment expression ends with a semicolon. The value of 24 is stored in b.

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Function Input and Output Examples ( 6)

Example: Experiments with addmult:

>> [a,b] = addmult(3,2) a =

5 b =

6

>> addmult(3,2) ans =

5

>> v = addmult(3,2) v =

5

Note: addmult requires two return variables. Calling

addmult with no return variables or with one return variable causes undesired behavior.

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Summary of Input and Output Parameters

• Values are communicated through input arguments and output arguments.

• Variables deﬁned inside a function are local to that function.

Local variables are invisible to other functions and to the command environment.

• The number of return variables should match the number of output variables provided by the function. This can be

relaxed by testing for the number of return variables with nargout (See § 3.6.1.).

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Text Input and Output

It is usually desirable to print results to the screen or to a ﬁle.

On rare occasions it may be helpful to prompt the user for information not already provided by the input parameters to a function.

Inputs to functions:

• input function can be used (and abused!).

• Input parameters to functions are preferred.

Text output from functions:

• disp function for simple output

• fprintf function for formatted output.

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Prompting for User Input

The input function can be used to prompt the user for numeric or string input.

>> x = input(’Enter a value for x’);

>> yourName = input(’Enter your name’,’s’);

Prompting for input betrays the Matlab novice. It is a nuisance to competent users, and makes automation of computing tasks impossible.

Free Advice: Avoid using the input function. Rarely is it necessary. All inputs to a function should be provided via the input parameter list. Refer to the demonstration of the inputAbuse function in § 3.3.1.

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Text Output with disp and fprintf

Output to the command window is achieved with either the disp function or the fprintf function. Output to a ﬁle requires the fprintf function.

disp Simple to use. Provides limited control over appearance of output.

fprintf Slightly more complicated than disp.

Provides total control over appearance of output.

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The disp function ( 1)

Syntax:

disp(outMatrix)

where outMatrix is either a string matrix or a numeric matrix.

Examples: Numeric output

>> disp(5) 5

>> x = 1:3; disp(x)

1 2 3

>> y = 3-x; disp([x; y])

1 2 3

2 1 0

>> disp([x y])

1 2 3 2 1 0

>> disp([x’ y])

??? All matrices on a row in the bracketed expression must have the same number of rows.

Note: The last statement shows that the input to disp must be a legal matrix.

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The disp function ( 2)

Examples: String output

>> disp(’Hello, world!) Hello, world!

>> s = ’MATLAB 6 is built with LAPACK’; disp(s) MATLAB 6 is built with LAPACK

>> t = ’Earlier versions used LINPACK and EISPACK’;

>> disp([s; t])

??? All rows in the bracketed expression must have the same number of columns.

>> disp(char(s,t))

MATLAB 6 is built with LAPACK

Earlier versions used LINPACK and EISPACK

The disp[s; t] expression causes an error because s has fewer elements than t. The built-in char function constructs a string matrix by putting each input on a separate row and padding the rows with blanks as necessary.

>> S = char(s,t);

>> length(s), length(t), length(S(1,:)) ans =

29 ans =

41 ans =

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The num2str function ( 1)

The num2str function is often used to with the disp function to create a labeled output of a numeric value.

Syntax:

stringValue = num2str(numericValue)

converts numericValue to a string representation of that numeric value.

Examples:

>> num2str(pi) ans =

3.1416

>> A = eye(3) A =

1 0 0

0 1 0

0 0 1

>> S = num2str(A) S =

1 0 0 0 1 0

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The num2str function ( 2)

Although A and S appear to contain the same values, they are not equivalent. A is a numeric matrix, and S is a string matrix.

>> clear

>> A = eye(3); S = num2str(A); B = str2num(S);

>> A-S

??? Error using ==> -

Matrix dimensions must agree.

>> A-B ans =

0 0 0

>> whos

Name Size Bytes Class

A 3x3 72 double array

B 3x3 72 double array

S 3x7 42 char array

ans 3x3 72 double array

Grand total is 48 elements using 258 bytes

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Using num2str with disp ( 1)

Combine num2str and disp to print a labeled output of a numeric value

>> x = sqrt(2);

>> outString = [’x = ’,num2str(x)];

>> disp(outString) x = 1.4142

or, build the input to disp on the ﬂy

>> disp([’x = ’,num2str(x)]);

x = 1.4142

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Using num2str with disp ( 2)

The

disp([’x = ’,num2str(x)]);

construct works when x is a row vector, but not when x is a column vector or matrix

>> z = y’;

>> disp([’z = ’,num2str(z)])

??? All matrices on a row in the bracketed expression must have the same number of rows.

Instead, use two disp statements to display column of vectors or matrices

>> disp(’z = ’); disp(z) z =

1 2 3 4

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Using num2str with disp ( 3)

The same eﬀect is obtained by simply entering the name of the variable with no semicolon at the end of the line.

>> z (enter z and press return) z =

1 2 3 4

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The format function

The format function controls the precision of disp output.

>> format short

>> disp(pi) 3.1416

>> format long

>> disp(pi)

3.14159265358979

Alternatively, a second parameter can be used to control the precision of the output of num2str

>> disp([’pi = ’,num2str(pi,2)]) pi = 3.1

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The fprintf function ( 1)

Syntax:

fprintf(outFormat,outVariables)

fprintf(fileHandle,outFormat,outVariables)

uses the outFormat string to convert outVariables to strings that are printed. In the ﬁrst form (no fileHandle) the output is displayed in the command window. In the second form, the

output is written to a ﬁle referred to by the fileHandle (more on this later).

Notes to C programmers:

1. The Matlab fprintf function uses single quotes to deﬁne the format string.

2. The fprintf function is vectorized. (See examples below.)

Example:

>> x = 3;

>> fprintf(’Square root of %g is %8.6f\n’,x,sqrt(x));

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The fprintf function ( 2)

The outFormat string speciﬁes how the outVariables are

converted and displayed. The outFormat string can contain any text characters. It also must contain a conversion code for each of the outVariables. The following table shows the basic

conversion codes.

Code Conversion instruction

%s format as a string

%d format with no fractional part (integer format)

%f format as a ﬂoating-point value

%e format as a ﬂoating-point value in scientiﬁc notation

%g format in the most compact form of either %f or %e

\n insert newline in output string

\t insert tab in output string

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The fprintf function ( 3)

In addition to specifying the type of conversion (e.g. %d, %f, %e) one can also specify the width and precision of the result of the conversion.

Syntax:

%wd

%w.pf

%w.pe

where w is the number of characters in the width of the ﬁnal result, and p is the number of digits to the right of the decimal point to be displayed.

Examples:

Format String Meaning

%14.5f use ﬂoating point format to convert a numerical value to a string 14 characters wide with 5 digits after the decimal point

%12.3e use scientiﬁc notation format to convert numerical value to a string 12 characters wide with 3 digits after the decimal point. The 12 characters for the string include the e+00 or e-00 (or e+000 or e-000

TM

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The fprintf function ( 4)

More examples of conversion codes

Value %8.4f %12.3e %10g %8d

2 2.0000 2.000e+00 2 2

sqrt(2) 1.4142 1.414e+00 1.41421 1.414214e+00

sqrt(2e-11) 0.0000 4.472e-06 4.47214e-06 4.472136e-06 sqrt(2e11) 447213.5955 4.472e+05 447214 4.472136e+05

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The fprintf function ( 5)

The fprintf function is vectorized. This enables printing of

vectors and matrices with compact expressions. It can also lead to some undesired results.

Examples:

>> x = 1:4; y = sqrt(x);

>> fprintf(’%9.4f\n’,y) 1.0000

1.4142 1.7321 2.0000

The %9.4f format string is reused for each element of y. The recycling of a format string may not always give the intended result.

>> x = 1:4; y = sqrt(x);

>> fprintf(’y = %9.4f\n’,y) y = 1.0000

y = 1.4142 y = 1.7321 y = 2.0000

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The fprintf function ( 6)

Vectorized fprintf cycles through the outVariables by columns. This can also lead to unintended results

>> A = [1 2 3; 4 5 6; 7 8 9]

A =

1 2 3

4 5 6

7 8 9

>> fprintf(’%8.2f %8.2f %8.2f\n’,A)

1.00 4.00 7.00

2.00 5.00 8.00

3.00 6.00 9.00

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How to print a table with fprintf ( 1)

Many times a tabular display of results is desired.

The boxSizeTable function listed below, shows how the fprintf function creates column labels and formats numeric data into a tidy tabular display. The for loop construct is discussed later in these slides.

function boxSizeTable

% boxSizeTable Demonstrate tabular output with fprintf

% --- labels and sizes for shiping containers label = char(’small’,’medium’,’large’,’jumbo’);

width = [5; 5; 10; 15];

height = [5; 8; 15; 25];

depth = [15; 15; 20; 35];

vol = width.*height.*depth/10000; % volume in cubic meters fprintf(’\nSizes of boxes used by ACME Delivery Service\n\n’);

fprintf(’size width height depth volume\n’);

fprintf(’ (cm) (cm) (cm) (m^3)\n’);

for i=1:length(width)

fprintf(’%-8s %8d %8d %8d %9.5f\n’,...

label(i,:),width(i),height(i),depth(i),vol(i)) end

Note: length is a built-in function that returns the number of elements in a vector. width, height, and depth

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How to print a table with fprintf ( 2)

Example: Running boxSizeTable gives

>> boxSizeTable

Sizes of boxes used by ACME Delivery Service

size width height depth volume

(cm) (cm) (cm) (m^3)

small 5 5 15 0.03750

medium 5 8 15 0.06000

large 10 15 20 0.30000

jumbo 15 25 35 1.31250

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The fprintf function ( 3)

File Output with fprintf requires creating a ﬁle handle with the fopen function. All aspects of formatting and vectorization

discussed for screen output still apply.

Example: Writing contents of a vector to a ﬁle.

x = ... % content of x

fout = fopen(’myfile.dat’,’wt’); % open myfile.dat fprintf(fout,’ k x(k)\n’);

for k=1:length(x)

fprintf(fout,’%4d %5.2f\n’,k,x(k));

end

fclose(fout) % close myfile.dat

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Flow Control ( 1)

To enable the implementation of computer algorithms, a computer language needs control structures for

• Repetition: looping or iteration

• Conditional execution: branching

• Comparison

We will consider these in reverse order.

Comparison

Comparison is achieved with relational operators. Relational operators are used to test whether two values are equal, or whether one value is greater than or less than another. The

result of a comparison may also be modiﬁed by logical operators.

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Relational Operators ( 1)

Relational operators are used in comparing two values.

Operator Meaning

< less than

<= less than or equal to

> greater than

>= greater than or equal to

~= not equal to

The result of applying a relational operator is a logical value, i.e.

the result is either true or false.

In Matlab any nonzero value, including a non-empty string, is equivalent to true. Only zero is equivalent to false.

Note: The <=, >=, and ~= operators have “=” as the second character. =<, => and =~ are not valid operators.

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Relational Operators ( 2)

The result of a relational operation is a true or false value.

Examples:

>> a = 2; b = 4;

>> aIsSmaller = a < b aIsSmaller =

1

>> bIsSmaller = b < a bIsSmaller =

0

Relational operations can also be performed on matrices of the same shape, e.g.,

>> x = 1:5; y = 5:-1:1;

>> z = x>y z =

0 0 0 1 1

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Logical Operators

Logical operators are used to combine logical expressions (with

“and” or “or”), or to change a logical value with “not”

Operator Meaning

& and

| or

~ not

Examples:

>> a = 2; b = 4;

>> aIsSmaller = a < b;

>> bIsSmaller = b < a;

>> bothTrue = aIsSmaller & bIsSmaller bothTrue =

0

>> eitherTrue = aIsSmaller | bIsSmaller eitherTrue =

1

>> ~eitherTrue ans =

0

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Logical and Relational Operators

Summary:

• Relational operators involve comparison of two values.

• The result of a relational operation is a logical (True/False) value.

• Logical operators combine (or negate) logical values to produce another logical value.

• There is always more than one way to express the same comparison

Free Advice:

• To get started, focus on simple comparison. Do not be afraid to spread the logic over multiple lines (multiple comparisons) if necessary.

• Try reading the test out loud.

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Conditional Execution

Conditional Execution or Branching:

As the result of a comparison, or another logical (true/false) test, selected blocks of program code are executed or skipped.

Conditional execution is implemented with if, if...else, and if...elseif constructs, or with a switch construct.

There are three types of if constructs 1. Plain if

2. if...else 3. if...elseif

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if Constructs

Syntax:

if expression

block of statements end

The block of statements is executed only if the expression is true.

Example:

if a < 0

disp(’a is negative’);

end

One line format uses comma after if expression

if a < 0, disp(’a is negative’); end

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if. . . else

Multiple choices are allowed with if. . . else and if. . . elseif constructs

if x < 0

error(’x is negative; sqrt(x) is imaginary’);

else

r = sqrt(x);

end

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if. . . elseif

It’s a good idea to include a default else to catch cases that don’t match preceding if and elseif blocks

if x > 0

disp(’x is positive’);

elseif x < 0

disp(’x is negative’);

else

disp(’x is exactly zero’);

end

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The switch Construct

A switch construct is useful when a test value can take on discrete values that are either integers or strings.

Syntax:

switch expression case value1,

block of statements case value2,

block of statements ..

.

otherwise,

Example:

color = ’...’; % color is a string switch color

case ’red’

disp(’Color is red’);

case ’blue’

disp(’Color is blue’);

case ’green’

disp(’Color is green’);

otherwise

disp(’Color is not red, blue, or green’);

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Flow Control ( 3)

Repetition or Looping

A sequence of calculations is repeated until either

1. All elements in a vector or matrix have been processed or

2. The calculations have produced a result that meets a predetermined termination criterion

Looping is achieved with for loops and while loops.

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for loops

for loops are most often used when each element in a vector or matrix is to be processed.

Syntax:

for index = expression block of statements end

Example: Sum of elements in a vector

x = 1:5; % create a row vector sumx = 0; % initialize the sum for k = 1:length(x)

sumx = sumx + x(k);

end

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for loop variations

Example: A loop with an index incremented by two

for k = 1:2:n ...

end

Example: A loop with an index that counts down

for k = n:-1:1 ...

end

Example: A loop with non-integer increments

for x = 0:pi/15:pi

fprintf(’%8.2f %8.5f\n’,x,sin(x));

end

Note: In the last example, x is a scalar inside the loop. Each time through the loop, x is set equal to one of the columns of 0:pi/15:pi.

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while loops ( 1)

while loops are most often used when an iteration is repeated until some termination criterion is met.

Syntax:

while expression

The block of statements is executed as long as expression is true.

Example: Newton’s method for evaluating √ x r_k = 1

2

r_k−1 + x rk−1

r = ... % initialize

rold = ...

while abs(rold-r) > delta rold = r;

r = 0.5*(rold + x/rold);

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while loops ( 2)

It is (almost) always a good idea to put a limit on the number of iterations to be performed by a while loop.

An improvement on the preceding loop,

maxit = 25;

it = 0;

while abs(rold-r) > delta & it<maxit rold = r;

r = 0.5*(rold + x/rold);

it = it + 1;

end

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while loops ( 3)

The break and return statements provide an alternative way to exit from a loop construct. break and return may be applied to for loops or while loops.

break is used to escape from an enclosing while or for loop.

Execution continues at the end of the enclosing loop construct.

return is used to force an exit from a function. This can have the eﬀect of escaping from a loop. Any statements following the loop that are in the function body are skipped.

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The break command

Example: Escape from a while loop

function k = breakDemo(n)

% breakDemo Show how the "break" command causes

% exit from a while loop.

% Search a random vector to find index

% of first element greater than 0.8.

%

% Synopsis: k = breakDemo(n)

%

% Input: n = size of random vector to be generated

%

% Output: k = first (smallest) index in x such that x(k)>0.8 x = rand(1,n);

k = 1;

while k<=n if x(k)>0.8

break end

k = k + 1;

end

fprintf(’x(k)=%f for k = %d n = %d\n’,x(k),k,n);

% What happens if loop terminates without finding x(k)>0.8 ?

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The return command

Example: Return from within the body of a function

function k = returnDemo(n)

% returnDemo Show how the "return" command

% causes exit from a function.

% Search a random vector to find

% index of first element greater than 0.8.

%

% Synopsis: k = returnDemo(n)

%

% Input: n = size of random vector to be generated

%

% Output: k = first (smallest) index in x

% such that x(k)>0.8

x = rand(1,n);

k = 1;

while k<=n if x(k)>0.8

return end

k = k + 1;

end

% What happens if loop terminates without finding x(k)>0.8 ?

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Comparison of break and return

break is used to escape the current while or for loop.

return is used to escape the current function.

function k = demoBreak(n) ...

while k<=n if x(k)>0.8 break;

end

k = k + 1;

end

function k = demoReturn(n) ...

while k<=n if x(k)>0.8 return;

end

k = k + 1;

end

jump to end of enclosing

“while ... end” block

return to calling function

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Vectorization

Vectorization is the use of vector operations (Matlab expressions) to process all elements of a vector or matrix.

Properly vectorized expressions are equivalent to looping over the elements of the vectors or matrices being operated upon. A vectorized expression is more compact and results in code that executes faster than a non-vectorized expression.

To write vectorized code:

• Use vector operations instead of loops, where applicable

• Pre-allocate memory for vectors and matrices

• Use vectorized indexing and logical functions

Non-vectorized code is sometimes called “scalar code” because the operations are performed on scalar elements of a vector or matrix instead of the vector as a whole.

Free Advice: Code that is slow and correct is always better than code that is fast and incorrect. Start with scalar code, then vectorize as needed.

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Replace Loops with Vector Operations

Scalar Code

for k=1:length(x) y(k) = sin(x(k)) end

Vectorized equivalent

y = sin(x)

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Preallocate Memory

The following loop increases the size of s on each pass.

y = ... % some computation to define y for j=1:length(y)

if y(j)>0

s(j) = sqrt(y(j));

else

s(j) = 0;

end end

Preallocate s before assigning values to elements.

y = ... % some computation to define y s = zeros(size(y));

for j=1:length(y) if y(j)>0

s(j) = sqrt(y(j));

end end

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Vectorized Indexing and Logical Functions ( 1)

Thorough vectorization of code requires use of array indexing and logical indexing.

Array Indexing:

Use a vector or matrix as the “subscript” of another matrix:

>> x = sqrt(0:4:20) x =

0 2.0000 2.8284 3.4641 4.0000 4.47210

>> i = [1 2 5];

>> y = x(i) y =

0 2 4

The x(i) expression selects the elements of x having the indices in i. The expression y = x(i) is equivalent to

k = 0;

for i = [1 2 5]

k = k + 1;

y(k) = x(i);

end

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Vectorized Indexing and Logical Functions ( 2)

Logical Indexing:

Use a vector or matrix as the mask to select elements from another matrix:

>> x = sqrt(0:4:20) x =

0 2.0000 2.8284 3.4641 4.0000 4.47210

>> j = find(rem(x,2)==0) j =

1 2 5

>> z = x(j) z =

0 2 4

The j vector contains the indices in x that correspond to elements in x that are integers.

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Vectorized Indexing and Logical Functions ( 3)

Example: Vectorization of Scalar Code

We just showed how to pre-allocate memory in the code snippet:

for j=1:length(y) if y(j)>0

s(j) = sqrt(y(j));

end end

In fact, the loop can be replaced entirely by using logical and array indexing

y = ... % some computation to define y

s = zeros(size(y));

i = find(y>0); % indices such that y(i)>0 s(y>0) = sqrt(y(y>0))

If we don’t mind redundant computation, the preceding expressions can be further contracted:

s(y>0) = sqrt(y(y>0))

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Vectorized Copy Operations ( 1)

Example: Copy entire columns (or rows)

Scalar Code

[m,n] = size(A); % assume A and B have

% same number of rows for i=1:m

B(i,1) = A(i,1);

end

Vectorized Code

B(:,1) = A(:,1);

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Vectorized Copy Operations ( 2)

Example: Copy and transform submatrices

Scalar Code

for j=2:3

B(1,j) = A(j,3);

end

Vectorized Code

B(1,2:3) = A(2:3,3)’

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Deus ex Machina

Matlab has features to solve some recurring programming problems:

• Variable number of I/O parameters

• Indirect function evaluation with feval

• In-line function objects (Matlab version 5.x)

• Global Variables

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Variable Input and Output Arguments ( 1)

Each function has internal variables, nargin and nargout.

Use the value of nargin at the beginning of a function to ﬁnd out how many input arguments were supplied.

Use the value of nargout at the end of a function to ﬁnd out how many input arguments are expected.

Usefulness:

• Allows a single function to perform multiple related tasks.

• Allows functions to assume default values for some inputs, thereby simplifying the use of the function for some tasks.

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Variable Input and Output Arguments ( 2)

Consider the built-in plot function

Inside the plot function nargin nargout

plot(x,y) 2 0

plot(x,y,’s’) 3 0

plot(x,y,’s--’) 3 0

plot(x1,y1,’s’,x2,y2,’o’) 6 0

h = plot(x,y) 2 1

The values of nargin and nargout are determined when the plot function is invoked.

Refer to the demoArgs function in Example 3.13

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Indirect Function Evaluation ( 1)

The feval function allows a function to be evaluated indirectly.

Usefulness:

• Allows routines to be written to process an arbitrary f(x).

• Separates the reusable algorithm from the problem-speciﬁc code.

feval is used extensively for root-ﬁnding (Chapter 6),

curve-ﬁtting (Chapter 9), numerical quadrature (Chapter 11) and numerical solution of initial value problems (Chapter 12).

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Indirect Function Evaluation ( 2)

>> fsum(’sin’,0,pi,5) ans =

2.4142

>> fsum(’cos’,0,pi,5) ans =

0

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Use of feval

function s = fsum(fun,a,b,n)

% FSUM Computes the sum of function values, f(x), at n equally

% distributed points in an interval a <= x <= b

%

% Synopsis: s = fsum(fun,a,b,n)

%

% Input: fun = (string) name of the function to be evaluated

% a,b= endpoints of the interval

% n = number of points in the interval

x = linspace(a,b,n); % create points in the interval y = feval(fun,x); % evaluate function at sample points s = sum(y); % compute the sum

function y = sincos(x)

% SINCOS Evaluates sin(x)*cos(x) for any input x

%

% Synopsis: y = sincos(x)

%

% Input: x = angle in radians, or vector of angles in radians

%

% Output: y = value of product sin(x)*cos(x) for each element in x y = sin(x).*cos(x);

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Inline Function Objects

Matlab version 5.x introduced object-oriented programming extensions. Though OOP is an advanced and somewhat subtle way of programming, in-line function objects are simple to use and oﬀer great program ﬂexibility.

Instead of

function y = myFun(x) y = x.^2 - log(x);

Use

myFun = inline( ’x.^2 - log(x)’ );

Both deﬁnitions of myFun allow expressions like

z = myFun(3);

s = linspace(1,5);

t = myFun(s);

Usefulness:

• Eliminates need to write separate m-ﬁles for functions that evaluate a simple formula.

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Global Variables

workspace

» x = 1

» y = 2

» s = 1.2

» z = localFun(x,y,s)

function d = localFun(a,b,c) ...

d = a + b^c

localFun.m

(x,y,s) (a,b,c)

z d

Communication of values via input and output variables

workspace

» x = 1;

» y = 2;

» global ALPHA;

» ...

» ALPHA = 1.2;

» z = globalFun(x,y)

function d = globalFun(a,b) ...

global ALPHA ...

d = a + b^ALPHA;

globalFun.m

(x,y) (a,b)

z d

Communication of values via input and output variables and global variables shared by the workspace and function

Usefulness:

• Allows bypassing of input parameters if no other mechanism (such as pass-through parameters) is available.

• Provides a mechanism for maintaining program state (GUI applications)