King M.R., Mody N.A. Numerical and Statistical Methods for Bioengineering: Applications in MATLAB

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0, which represents false. The elements of a logical array are all of the logical

data type and have values equal to either 1 or 0.

Short-circuit logical operators operate on scalar logical values (see Table A.2).

The && operator returns 1 (true) if both logical expressions evaluate as true, and

returns 0 (false) if any one or both of the logical expressions evaluate as false. The ||

operator returns true if any one or both of the logical expressions are true.

These operators are called short-circuit operators because the result of the logical

operation is determ ined by evaluating only the ﬁrst logical expression. If the result of

the logical operation still remains unclear, then the second logical expression is

evaluated.

Element-wise logical operators compare logical arrays element-wise and return a

logical array that contains the result (see Table A.3).

A1.2.2 Program control statements

Programs contain not only operators that operate on data, but also control

statements. Two categories of control that we discuss in this section are

conditional control and loop control.

Conditional control

if-else-end

The most basic form of the if statement is

if logical expression

statement 1

statement 2

statement n

end

Table A.2. Short-circuit logical operators

Operator Deﬁnition

&& AND

|| OR

Table A.3. Element-wise logical

operators

Operator Deﬁnition

& AND

| OR

~ NOT

567

Introduction to MATLAB

The if statement evaluates a logical expression. If the expression is true, the set

of statements contained within the if-e nd block are executed. If the logical

expression is found to be false, control of the program transfers directly to the

line immediately after the end statement. The statements contained within the if-

end block are not executed. Good coding practice includes indenting statements

located within a block as shown above. Indentation improves readability of the

code.

Another syntax for the if control block is as follows:

if logical expression #1

set of statements #1

elseif logical expression #2

set of statements #2

else

set of statements #3

end

In this control statement, if logical expression #1 turns out to be false, logical

expression #2 is evaluated. If this evaluates to true, then statement set #2 is executed,

otherwise statement set #3 is executed. Note the following points.

(1) The elseif sub-block as well as the else sub-block are option al.

(2) There can be more than one elseif statement speciﬁed within the if-end block.

(3) If none of the logical expressions hold true, the statements speciﬁed between the

else line and the end line are executed.

switch-case

When the decision to ex ecute a set of statements rests on the value of an expression,

the switch-case control block should be used. The syntax is

switch expression

case value_1

set of statements #1

case value_2

set of statements #2

case value_n

set of statements #n

otherwise

set of statements #(n +1)

end

The value of the expression is evaluated. If its value matches the value(s) given in

the ﬁrst case statement, then statement set #1 is executed. If not, the next case

value(s) is checked. Once a matching value is found and the corresponding

statements for that case are executed, control of the pro gram is immediately

transferred to the line following the end statement, i.e. the values listed in

subsequent case statements are not checked. Optionally, one can add an

otherwise statement to the switch-case block. If none of the case values

match the expression value, the statements in the otherwise sub-block are

executed.

568

Appendix A

Loop control

for loop

The for loop is used to execute a set of statements a speciﬁed number of times. The

syntax is

for index = ﬁrst:step:last

statements

end

If the value of index is inadvertently changed within the loop, this will not

modify the number of loops made.

while loop

The while loop executes a series of statements over and over again as long as a

speciﬁed condition holds true. Once the condition becomes false, control of the

program is transferred to the line following the while-end block. The syntax is

while logical expression

statements

end

The break statement is used to “break” out of a for-end or while-end loop.

Usually, the break statement is contained within an if-end block placed within

the loop. A particular condition is tested by the if-end statement. If the logical

expression evaluated by the if statement is found to be true, the break command is

encountered and looping terminates. Control of the program is trans ferred to the

next line after the end statement of the loop .

Example A1.3

This example is adapted from Program 5.5. The following function m-file uses the fixed-point iterative

method to solve a nonlinear equation in a single variable:

x1 = function ﬁxedpointmethod(gfunc, x0, tolx)

% Fixed-Point Iteration used to solve a nonlinear equation in x

% Input variables

% gfunc : nonlinear function g(x) whose ﬁxed-point we seek

% x0 : initial guess value

% tolx : tolerance for error in estimating root

maxloops = 50;

for i = 1:maxloops

x1 = feval(gfunc, x0);

if (abs(x1 – x0) <= tolx)

break % Jump out of the for loop

end

x0 = x1;

end

To perform fixed-point iteration we must first convert the nonlinear equation f (x) = 0 whose root we

seek into the form x = g(x), where g(x) is evaluated iteratively to obtain a sufficiently accurate value of x that

satisfies x = g(x). Once the difference in two consecutive values of x falls below a threshold called the

tolerance limit, the program exits out of the loop.

569

Introduction to MATLAB

MATLAB has deﬁned certain parameters that return special values. Some

important functions that are discussed in this book are:

ans: Any time you evaluate an expression at the command line, but have not

assigned a variable to store the value of the expression, MATLAB places the value

into a function named ans. The most recent answer not assigned to a particular

variable is stored in ans.

pi: This function stores the value of the irrational number π = 3.141592653589793.

inf: This function represents inﬁnity. Numerical calculations that involve division

by zero generate the result inf.

eps: Gap in the ﬂoating-point number line. This function is explained in Chapt er 1.

realmax: The largest ﬂoating-point number that is recognized by the machine.

This function is explained in Chapter 1.

realmin: The smallest ﬂoating point number that is recognized by the machine.

This function is explained in Chapter 1.

A1.3 Two-dimensional plotting

MATLAB provides a variety of plotting commands and tools to visualize data. The

plotting function of particular interest in this book is plot, which draws a two-

dimensional graph with linear-scale x- and y-axes. Before plotting any data, it is

recommended that you ﬁrst create a ﬁgure window in which the graph axes and line

plot will appear.

This is done by typing the statement

ﬁgure

This precedes the plot statement. The plot function has many syntax forms. We

start with a basic example,

plot(x, y)

where x and y are two vectors that must be of the same length. This statement plots a

line using data points (x, y) such that the elements of y are paired with the

corresponding elements of x. You can also specify the line style and color of the

plot, e.g.

plot(x,y,’ -k’ )

plots the points as a continuous black line.

The color codes and line style codes are given in Table A.4. You can also mark the

location of data points with a marker shape of any color (see Table A.5); e.g.

plot(x,y,’ sb’ )

plots a blue square where each data point is located.

You can combine line styles and marker styles to distinguish one line from another

when producing black and white plo ts. Suppose we plot two sets of data on the same

axes. The following statement is an example of how you can differentiate the two

lines simply using different markers:

plot(x1, y1, ’-xk’ , x2, y2, ’-ok’ )

Note that a plot function can accept multiple data sets.

If you do not do this, you run the risk of generating a new plot in the current ﬁgure window, thereby

erasing or modifying a previously created ﬁgure.

570

Appendix A

You can specify the width of the line by providing a value for the LineWidth

property, equal to the number of points:

plot (x, y, ’-k ’, ’LineWidth’ ,2)

You can set a range of properties of the plot axes using the set function. To set

the properties of an existing ﬁgure axes, you will need to obtain a handle of the axes

object. The handle of the current axis is returned by the function gca, which stands

for “get current axes.”

Suppose you wish to change the width of the axes lines and the size of the tick label font:

set(gca, ’LineWidth’ ,2, ’FontSize’ , 16)

You can specify the axes limits using the axis function, whose syntax is

axis([xmin xmax ymin ymax])

If the data units must be equal in length for both the x- and y-axis, for instance to

make a plot of a circle appear as a circle rather than an ellipse, the command

Table A.4. Color and line style codes for MATLAB

plotting function

Letter Color Character Line style

k black - continuous line

b blue -- dashed line

y yellow : dotted line

m magenta -. dash–dot line

g green

w white

r red

c cyan

Table A.5. Marker style codes for MATLAB

plotting functions

Character Marker

^ upward-pointing

triangle

v downward-pointing

triangle

d diamond

s square

o circle

* star

x ×

+ plus sign

571

Introduction to MATLAB

axis equal

will equalize the unit length on both axes.

You can provide labels

for the x-axis and y-axis using the xlabel and ylabel

functions, respectively. The title function is used to give the ﬁgure a title.

Example A1.4

The following script plots the sin x function for a range of values of x:

x = -5:0.01:5;

y = sin(x);

plot (x, y, ’-k ’,’ LineWidth ’,2)

set(gca, ’LineWidth’ ,2, ’FontSize’ ,16)

xlabel( ’x’ )

ylabel( ’sin(x)’ )

axis equal

axis([-5, 5, -1.2, 1.2])

The output of this script is shown in Figure A1.1.

If multiple plots are to be displayed within a single ﬁgure window, use the subplot

function. The syntax is

subplot(no_of_rows, no_of_columns, ﬁgure_ position)

Suppose there are six plots in a ﬁgure arranged in three rows and two columns.

Each plot is positioned in the ﬁgure using the following statements:

ﬁgure

% ﬁrst plot

subplot(3, 2, 1)

plot(x1, y1)

% second plot

subplot(3,2,2)

plot(x2, y2)

% sixth plot

subplot(3,2,6)

plot(x6, y6)

Figure A1.1

Plot of sin x.

−5 0 5

−1

sin(x)

Formatting of text such as italics, superscripts, subscripts, etc. can be done using TeX commands. See

MATLAB help for details.

572

Appendix A

A1.4 Data import and export

A1.4.1 Importing data

There are two ways to load text data (ASCII format) from a ﬁle into the workspace.

One method is to use the Import Wizard that is accessible from the ﬁle menu (File 

Import Data). When the Import Wizar d starts, it will ask you to select the ﬁle that

contains the da ta you wish to import. Next, you will be asked to specify the

delimiting character that separates adjacent data values. Often, the delimiter will

be the space character, although a comma, semicolon, or tab character can also be

used as a delimiter. The data extracted from the ﬁle are displayed in rows and

columns in the same window for your veriﬁcation. The Wizard creates a single

variable named after the ﬁlename and stores all of the numerical data in that

variable. It also creates another variable for alphabetical data. If row headers or

column headers are present in the data, the Wizard extracts the headers and creat es a

separate variable for them. In the next window, you are asked to choose which

variables to save. When row (or column) headers are available, MATLAB will ask if

you want to save each row (or column) as a separate variable. In this case, each row

variable is named after its corresponding header.

Import of numerical data from a ﬁle can be automated within an m-ﬁle using the

load function. Consider the following statement:

load a.dat

This is the command syntax for load. The load function will treat the ﬁle a.dat as

a text (ASCII) ﬁle and will save the numerical data contained in the ﬁle to a single

variable of type double. The variable saved in the workspace is a two-dimensional

array and is named a, after the ﬁlename. For the load function to execute without

error, the data stored in the ﬁle must be in rectangular form, i.e. the number of

elements in each row must be the same. Also, within each row the row elements

should be separated by one or more spaces. The functional form of the syntax for

load is

b = load(a.dat);

Using the functional form you can specify the name of the variable that will

contain the data extracted from a.dat.

A1.4.2 Exporting data

Data generated within a function or script can be output to the screen or to a ﬁle

using the fprintf function. This function outputs the data as a character string.

You can specify how the numerical data should appear in the output using

conversion speciﬁers. The syntax for this function is

fprintf(ﬁd, format, a, b, ...)

where ﬁd is a ﬁle identi ﬁer that speciﬁes which ﬁle the data should be written to; ﬁd is

an integer and is obtained from the function fopen, e.g.

ﬁd = fopen(’ myﬁle.txt’ , ’w’ )

573

Introduction to MATLAB

To specify, output to screen ﬁd =1.Ifﬁd is not speciﬁed, output is sent to screen.

Note that format is a string consisting of text and conversion speciﬁers. The

numerical data (a, b, ...) to be written is provided after the format string in

sequential order of appearance.

A conversion speciﬁer always begins with a percent sign (%). Several conversion

speciﬁers are presented in Table A.6. Note that %d can be used with integers and with

non-integral values that have been converted to integers using the MATLAB

functions ﬁx, ﬂoor , ceil,orround. The minimum width of numeric data

(number of signiﬁcant digits) and the precision (number of signiﬁcant digits to the

right of the decimal point) can be optionally speciﬁed with the conversion speciﬁer.

See MATLAB help for more information on fprintf and fopen.

Example A1.5

This example illustrates the use of fprintf. The function squareroot calculates the square root of

each element in vector x. It then prints to the screen the formatted result:

function squareroot(x)

y = sqrt(x);

z = [x; y]; % x and y are row vectors

fprintf( ’Square root of %6.2f is %6.2f.\n’ ,z)

Note that ‘\n’ creates a new line. At the command prompt, we type the following:

x = 0:0.1:1;

squareroot(x)

Square root of 0.00 is 0.00.

Square root of 0.10 is 0.32.

Square root of 0.20 is 0.45.

Square root of 0.30 is 0.55.

Square root of 0.40 is 0.63.

Square root of 0.50 is 0.71.

Square root of 0.60 is 0.77.

Square root of 0.70 is 0.84.

Square root of 0.80 is 0.89.

Square root of 0.90 is 0.95.

Square root of 1.00 is 1.00.

To save data that does not require special formatting, use the save command. You

can save the entire workspace or speciﬁc workspace variables to a ﬁle using save.By

itself, the save command creates a binary ﬁle with a .mat extension called

matlab.mat which contains all the workspace varia bles. You can specify the

name of the ﬁle using the following syntax:

Table A.6. Conversion speciﬁers for the

fprintf function

Speciﬁer Purpose

%e exponential notation

%f ﬁxed-point notation

%d integer notation

574

Appendix A

save ﬁlename.mat

The variables from this ﬁle can be loaded into the memor y using the load

command. You can choose which variables to save by specifying them in the save

statement:

save ﬁlename.mat x y z

You can also save the ﬁle as an ASCII text ﬁle (8 digit form) using the -ascii

option. This creates a text ﬁle that can be input into other programs such as Excel

and Word. To save the numeric data in 16 digit form, include the -double option:

save ﬁlename.dat x y z –ascii – double

575

Introduction to MATLAB

Appendix B Location of nodes for

Gauss–Legendre quadrature

Here we derive the optimal location of the nodes for an n-point Gaussian quadrature.

The truncation error of the Gaussian quadrature formula is the integral of Equation

(6.17), i.e. the truncation error of the polynomial interpolant,

ExðÞ¼

nþ1

ξ xðÞðÞ

n þ 1ðÞ!

x  x

ðÞx  x

ð Þ x  x

ðÞdx:

The truncation error can be rewritten as a function of the integration variable

z that is deﬁned on the interval ½1; 1 as follows:

EzðÞ¼

b  a



nþ2

1

nþ1

b  a

z þ

a þ b



n þ 1ðÞ!

z  z

ðÞz  z

ð Þ z  z

ðÞdz:

The interpolating polynomial function pðxÞ is of degree n and is constructed from

function values obtained at the n + 1 nodes, x

; x

; ...; x

whose locations we seek.

We deﬁne ψ zðÞ¼ z  z

ðÞz  z

ð Þ z z

ðÞ, where ψ zðÞis a polynomial of degree

n + 1. Suppose fðxÞ is a polynomial of degree ≤ 2n + 1, then its (n + 1)th derivative

is a polynomial of degree ≤ n. Let

ω xðÞ¼

nþ1

ξðxÞðÞ

n þ 1ðÞ!

;

where ω xðÞis a polynomial of degree ≤ n. Then, ω zðÞis also a polynomial of degree

≤ n. For any polynomial fxðÞof degree ≤ 2n + 1, the Gaussian quadrature formula

will be exact if the truncation error, EzðÞ¼0, i.e.

1

ω zðÞψ zðÞdz ¼ 0: (B:1)

The above integral is the inner product of two polynomial functions.

The inner product of two continuous functions fðxÞ and gðxÞ is deﬁned as

hf; gi¼

fxðÞgxðÞwðxÞdx; (B:2)

where wðxÞ is a weighting function and wxðÞ

0 for a  x  b. Just as the dot

product of any non-zero vector with itself is a positive number, the inner product

of any function with itself is a positive number, i.e.

hf; fi¼

fxðÞfxðÞwðxÞdx

0: (B:3)