MATLAB-Chap3

1. Elementary Mathematical Functions

You can use the lookfor command to find functions that are relevant to your application. For example, type lookfor imaginary to get a list of the functions that deal with imaginary numbers. You will see listed:

imag Complex imaginary part i Imaginary unit j Imaginary unit

Operations with complex numbers

The rectangular and polar representations of the complex number a + ib.

The magnitude of the product of z of two complex numbers x and y is equal to the product of their magnitudes: |z|=|x||y|

>>x = -3 + 4i; >>y = 6 - 8i;

>>mag_x = abs(x) mag_x=

5.0000

>>mag_y = abs(y) mag_y=

10.0000

>>mag_product = abs(x*y) mag_product=

50.0000

Note that abs(x) denotes absolute value of x.

The angle of the product is equal to the sum of the angles: ∠z=∠x+∠y.

>>angle_x = angle(x) angle_x=

2.2143

>>angle_y = angle(y) angle_y=

−0.9273

>>sum_angles = angle_x + angle_y sum_angles=

1.2870

>>angle_product = angle(x*y) angle_product=

1.2870

Note that angle(x) denotes angle of a complex number x.

Numeric functions

MATLAB will treat a variable as an array automatically. For example, to compute the square roots of 5, 7, and 15, type

>>x = [5, 7, 15]; >>y = sqrt(x) y=

2.23612.63583.8730

Note that sqrt(x) denotes square root of x; x

Trigonometric functions:

We can write sin 2 in text, but MATLAB requires parentheses surrounding the 2 (which is called the function argument or parameter). Thus to evaluate sin 2 in MATLAB, we type sin(2). The MATLAB function name must be followed by a pair of parentheses that surround the argument.

To express in text the sine of the second element of the array x, we would type sin[x(2)]. However, in MATLAB you cannot use square brackets or braces in this way, and you must type sin(x(2)).

To evaluate sin(x2+5) and sin(x+1), you type sin(x.^2 + 5) and sin(sqrt(x)+1), respectively.

Using a function as an argument of another function is called function composition. Be sure to check the order of precedence and the number and placement of parentheses when typing such expressions.

Every left-facing parenthesis requires a right-facing mate. However, this condition does not guarantee that the expression is correct!

Another common mistake involves expressions like sin2x, which means (sinx)2.

In MATLAB we write this expression as (sin(x))^2, not as sin^2(x), sin^2x, sin(x^2), or sin(x)^2!

The MATLAB trigonometric functions operate in radian mode. Thus sin(5) computes the sine of 5 rad, not the sine of 5o.

To convert between degrees and radians, use the relation θradians = (π /180)θdegrees.

2. User-Defined Functions

The first line in a function file must begin with a function definition line that has a list of inputs and outputs. This line distinguishes a function M-file from a script M-file. Its syntax is as follows:

function [output variables] = name(input variables)

Note that the output variables are enclosed in square brackets, while the input variables must be enclosed with parentheses. The function name (here, name) should be the same as the file name in which it is saved (with the .m extension).

Simple function examples

Consider user-defined function fun.

function z = fun(x,y) u = 3*x;

z = u + 6*y.^2;

Note the use of a semicolon at the end of the lines. This prevents the values of u and z from being displayed.

Note also the use of the array exponentiation operator (.^). This enables the function to accept y as an array. Call this function with its output argument:

>>z = fun(3, 7) z=

303

The function uses x=3and y=7to compute z.

Call this function without its output argument and try to access its value. You will see an error message.

>>fun(3, 7) ans=

303>>z

??? Undefined function or variable ‘z’.

Assign the output argument to another variable:

>>q = fun(3,7) q=

303

You can suppress the output by putting a semicolon after the function call. For example, if you type q = fun(3,7); the value of q will be computed but not displayed (because of the semicolon).

The variables x and y are local to the function fun, so unless you pass their values by naming them x and y, their values will not be available in the workspace outside the function. The variable u is also local to the function. For example,

>>x = 3; y = 7;

>>q = fun(x, y); >>x x=

3>>y y=

7>>u

??? Undefined function or variable ‘u’.

Only the order of the arguments is important, not the names of the arguments:

>>x = 7; >>y = 3;

>>z = fun(y, x) z=

303

The second line is equivalent to z = fun(3, 7).

You can use arrays as input arguments:

>>r = fun([2:4], [7:9]) r=

300393498

A function may have more than one output. These are enclosed in square brackets. For example, the function circle computes the area A and circumference C of a circle, given its radius as an input argument.

function [A, C] = circle(r) A = pi*r.^2; C = 2*pi*r;

The function is called as follows, if the radius is 4.

>>[A, C] = circle(4) A=

50.2655

C=

25.1327

A function may have no input arguments and no output list. For example, the function show_date computes and stores the date in the variable today, and displays the value of today.

function show_date today = date

Example of converting between degrees and radians

Write and test a function DR that changes degrees to radians and another function RD that changes radians to degrees. Your functions should be able to accept both scalar and matrix input.

Summary of these processing steps

1. Describe the input and the output

Input: A vector of degree values and a vector of radian values

Output: A table converting degrees to radians and a table converting radians to degrees

2. Develop a hand example

degrees = radians × 180/π radians = degrees × π/180

Degrees to Radians

Degrees Radians 0 0 30 30(π/180) = π/6 = 0.524 60 60(π/180) = π/3 = 1.047 90 90(π/180) = π/2 = 1.571

3. Develop a MATLAB solution

%Example

% clear: remove all variables from memory % clc: clears the MATLAB Command window clear, clc

% Define a vector of degree values degrees = 0:15:180;

% Call the DR function, and use it to find radians radians = DR(degrees);

% Create a table to use in the output degree_radians = [degrees;radians]’ % Define a vector of radian values radians = 0:pi/12:pi;

% Call the RD function, and use it to find degrees degrees = RD(radians);

radians_degrees = [radians;degrees]’

The function called by the program is

function output = DR(x)

% This function changes degrees to radians output = x*pi/180;

and

function output = RD(x)

% This function changes radians to degrees output = x*180/pi;

4. The script M-file must be in the current directory and must be named DR.m and RD.m.

5. The program instantly generates the results in the MATLAB Command window (check yourself).

Variations in the function line

1. One input, one output:

function [area_square] = square(side)

2. Brackets are optional for only one output: function area_square = square(side)

3. Three inputs, one output:

function [volume_box] = box(height, width, length)

4. One input, two outputs:

function [area_circle, circumf] = circle(radius)

5. No output variable: function sqplot(side)

Example 1: Function definition lines

function [dist,vel] = drop(g,v0,t);

% Computes the distance travelled and the % velocity of a dropped object, % as functions of g,

% the initial velocity v0, and % the time t. vel = g*t + v0;

dist = 0.5*g*t.^2 + v0*t;

Example 2: Call the function drop

1) The variable names used in the function definition may, but need not, be used when

the function is called:

>>a = 32.2;

>>initial_speed = 10; >>time = 5;

>>[feet_dropped, speed] = drop(a, initial_speed, time)

2) The input variables need not be assigned values outside the function prior to the function call:

[feet_dropped, speed] = drop(32.2, 10, 5)

3) The inputs and outputs may be arrays:

[feet_dropped, speed]=drop(32.2, 10, [0:1:5])

This function call produces the arrays feet_dropped and speed, each with six values corresponding to the six values of time in the array time.

Local variables

The names of the input variables given in the function definition line are local to that function.

This means that other variable names can be used when you call the function.

All variables inside a function are erased after the function finishes executing, except when the same variable names appear in the output variable list used in the function call.

Global variables

The global command declares certain variables global, and therefore their values are

available to the basic workspace and to other functions that declare these variables global.

The syntax to declare the variables a, x, and q is

global a x q

Any assignment to those variables, in any function or in the base workspace, is available to all the other functions declaring them global.

Finding the zeros of a function

You can use the fzero function to find the zero of a function of a single variable, which is denoted by x. One form of its syntax is

fzero(‘function’, x0)

where function is a string containing the name of the function, and x0 is a user-supplied guess for the zero.

The fzero function returns a value of x that is near x0. It identifies only points where the function crosses the x-axis, not points where the function just touches the axis. For example, fzero(‘cos’, 2) returns the value 1.5708.

Using fzero with user-defined functions

To use the fzero function to find the zeros of more complicated functions, it is more convenient to define the function in a function file.

For example, if y=x+2e−x−3, define the following function file:

Plot of the function y=x+2e−x−3

one near x=3. There is a zero near x=−0.5 and

To find a more precise value of the zero near x=−0.5, type

>>x = fzero(‘f1’, -0.5) x=

−0.5831

Finding the minimum of a function

The fminbnd function finds the minimum of a function of a single variable, which is denoted by x. One form of its syntax is

fminbnd(‘function’, x1, x2)

where function is a string containing the name of the function. The fminbnd function returns a value of x that minimizes the function in the interval x1≤x≤x2.

For example, fminbnd(‘cos’, 0, 4) returns the value 3.1416.

When using fminbnd it is more convenient to define the function in a function file. For example, if y=1−xe−x, define the following function file:

function y = f2(x) y = 1-x.*exp(-x);

To find the value of x that gives a minimum of y for 0≤x≤5, type

>>x = fminbnd(‘f2’, 0, 5)

The answer is x=1. To find the minimum value of y, type y=f2(x). The result is y=0.6321.

A function can have one or more local minima and a global minimum.

If the specified range of the independent variable does not enclose the global minimum, fminbnd will not find the global minimum.

fminbnd will find a minimum that occurs on a boundary.

Example 3: Plot of the function y=0.025x5−0.0625x4−0.333x3+x2.

This function has one local and one global minimum in the interval −1To find the minimum of a function of more than one variable, use the fminsearch function. One form of its syntax is

fminsearch(‘function’, x0)

where function is a string containing the name of the function. The vector x0 is a guess that must be supplied by the user.

22

To minimize the function f=xe−x−y, we first define it in an M-file, using the vector x whose elements are x(1)=x and x(2)=y.

Suppose we guess that the minimum is near x=y=0. The session is

>>fminsearch(‘f4’, [0, 0]) ans=

−0.70710.0000

Thus the minimum occurs at x=−0.7071, y=0.

3. Advanced Function Programming

Function handles

You can create a function handle to any function by using the at sign, @, before the

function name. You can then name the handle if you wish, and use the handle to reference the function. For example, to create a handle to the sine function, you type

>>sine_handle = @sin;

where sine_handle is a user-selected name for the handle.

A common use of a function handle is to pass the function as an argument to another function. For example, we can plot sin x over 0≤x≤6as follows:

>>plot([0:0.01:6], sine_handle, [0:0.01:6])

This is a rather cumbersome way to plot the sine function, but the concept can be extended to create a general purpose plotting function that accepts a function as an input. For example,

function x = gen_plot(fun_handle, interval) plot(interval, fun_handle, interval)

You may call this function to plot the sin x over 0≤x≤6as follows:

>>gen_plot(sine_handle, [0:0.01:6]) or

>>gen_plot(@sin, [0:0.01:6])

For example, this function gen_plot may be used as follows with a function handle array, to create two subplots, one for the sine and one for the cosine.

fh(1) = @sin; fh(2) = @cos; for k = 1:2

subplot(2,1,k)

gen_plot(fh(k),[0:0.01:8]) end

Methods for calling functions

There are four ways to invoke, or “call,” a function into action. These are:

1. As a character string identifying the appropriate function M-file, which is

function y = fun1(x) y = x.^2-4;

The function may be called as follows, to compute the zero over the range 0≤x≤3

>>[x, value] = fzero(‘fun1’, [0, 3])

2. As a function handle to an existing function M-file:

>>[x, value] = fzero(@fun1, [0, 3])

3. As an “inline” function object:

>>fun1 = ‘x.^2-4’;

>>fun_inline = inline(fun1);

>>[x, value] = fzero(fun_inline, [0, 3])

4. As a string expression.

>>fun1 = ‘x.^2-4’;

>>[x, value] = fzero(fun1, [0, 3]) or as

>>[x, value] = fzero(‘x.^2-4’, [0, 3])

Examples of these ways follow for the fzero function used with the user-defined function fun1, which computes y=x2−4.

The function handle method (method 2) is the fastest method, followed by method 1.

In addition to speed improvement, another advantage of using a function handle is that it provides access to subfunctions, which are normally not visible outside of their defining M-file.

Test 3.1-2

>>x = x=

2.0402+1.4705i>>abs(x) ans=

2.5149

>>angle(x) ans=

0.6245

>>real(x) ans=

2.0402

>>imag(x) ans=

1.4705

Test 3.2-1

The function file is f5.m

function y = f5(x)

y = exp(-0.2*x).*sin(x + 2) - 0.1;

You can plot the function to obtain solution estimates to use with fzero, or you can simply try values of x between 0 and 10.

>>fzero(‘f5’, 0) ans=

1.0187

>>fzero(‘f5’, 4) ans=

4.5334

>>fzero(‘f5’, 6) ans=

7.0066

So the solutions are x=1.0187, 4.5334, and 7.0066.

Test 3.2-2

The function file is f6.m

function y = f6(x)

y = 1 + exp(-0.2*x).*sin(x + 2);

You can plot the function to obtain solution estimates to use with fminbnd, or you can simply try values of x between 0 and 10.

>>fminbnd(‘f6’, 0) ans=

2.5150

>>f6(ans) ans=

0.4070

>>fminbnd(‘f6’, 10) ans=

8.7982

>>f6(ans) ans=

0.8312

So the solutions are

(x,y)=(2.5150,0.4070) and (x,y)=(8.7982,0.8312).

Types of user-defined functions

The following types of user-defined functions can be created in MATLAB.

1) Primary function is the first function in an M-file and typically contains the main program. Following the primary function in the same file can be any number of subfunctions, which can serve as subroutines to the primary function.

Usually the primary function is the only function in an M-file that you can call from the MATLAB command line or from another M-file function. You invoke this function using the name of the M-file in which it is defined.

We normally use the same name for the function and its file, but if the function name differs from the file name, you must use the file name to invoke the function.

2) Anonymous functions enable you to create a simple function without needing to create an M-file for it. You can construct an anonymous function either at the MATLAB command line or from within another function or script. Thus, anonymous functions provide a quick way of making a function from any MATLAB expression without the need to create, name, and save a file.

3) Subfunctions are placed in the primary function and are called by the primary function. You can use multiple functions within a single primary function M-file.

4) Nested functions are functions defined within another function. They can help to

improve the readability of your program and also give you more flexible access to variables in the M-file.

The difference between nested functions and subfunctions is that subfunctions normally cannot be accessed outside of their primary function file.

5) Overloaded functions are functions that respond differently to different types of input arguments. They are similar to overloaded functions in any object-oriented language. For example, an overloaded function can be created to treat integer inputs differently than inputs of class double.

6) Private functions enable you to restrict access to a function. They can be called only from an M-file function in the parent directory.

Anonymous functions

Anonymous functions enable you to create a simple function without needing to create an M-file for it. You can construct an anonymous function either at the MATLAB command line or from within another function or script. The syntax for creating an anonymous function from an expression is

fhandle = @(arglist) expr

where arglist is a comma-separated list of input arguments to be passed to the function, and expr is any single, valid MATLAB expression. To create a simple function called sq to calculate the square of a number, type

>>sq = @(x) x.^2;

To improve readability, you may enclose the expression in parentheses, as

sq = @(x) (x.^2);. To execute the function, type the name of the function handle, followed by any input arguments enclosed in parentheses. For example,

>>sq([5, 7]) ans=

2549

You might think that this particular anonymous function will not save you any work because typing sq([5, 7]) requires nine keystrokes, one more than is required to type [5, 7].^2.

Here, however, the anonymous function protects you from forgetting to type the

period (.) required for array exponentiation. Anonymous functions are useful, however, for more complicated functions involving numerous keystrokes.

You can pass the handle of an anonymous function to other functions. For example, to find the minimum of the polynomial 4x2−50x+5over the interval [-10, 10], you type

>>poly1 = @(x) 4*x.^2 - 50*x + 5; >>fminbnd(poly1, -10, 10) ans=

6.2500

If you are not going to use that polynomial again, you can omit the handle definition line and type instead

>>fminbnd(@(x) 4*x.^2 - 50*x + 5, -10, 10)

Multiple input arguments

You can create anonymous functions having more than one input. For example, to define the function x2+y2, type

>>sqrtsum = @(x, y) sqrt(x.^2 + y.^2);

Then type

>>sqrtsum(3, 4) ans=

5

As another example, consider the function defining a plane, z=Ax+By. The scalar variables A and B must be assigned values before you create the function handle. For example,

>>A = 6; B = 4;

>>plane = @(x, y) A*x + B*y; >>z = plane(2, 8) z=

44

Calling one function within another

One anonymous function can call another to implement function composition. Consider the function 5sin(x3). It is composed of the functions g(y)=5sin(y)and f(x)=x3. In the following session the function whose handle is h calls the functions whose handles are f and g.

>>f = @(x) x.^3; >>g = @(x) 5*sin(x); >>h = @(x) g(f(x)); >>h(2) ans=

4.9468

Variables and anonymous functions

Variables can appear in anonymous functions in two ways:

•As variables specified in the argument list, as for example f = @(x) x.^3;, and

•As variables specified in the body of the expression, as for example with the variables A and B in plane = @(x, y) A*x + B*y.

When the function is created MATLAB captures the values of these variables and retains those values for the lifetime of the function handle. If the values of A or B are changed after the handle is created, their values associated with the handle do not change.

This feature has both advantages and disadvantages, so you must keep it in mind.

Subfunctions

A function M-file may contain more than one user-defined function. The first defined function in the file is called the primary function, whose name is the same as the M-file name. All other functions in the file are called subfunctions.

Subfunctions are normally “visible” only to the primary function and other subfunctions in the same file; that is, they normally cannot be called by programs or functions outside the file. However, this limitation can be removed with the use of function handles.

Create the primary function first with a function definition line and its defining code, and name the file with this function name as usual.

Then create each subfunction with its own function definition line and defining code.

The order of the subfunctions does not matter, but function names must be unique within the M-file.

Precedence when calling functions

The order in which MATLAB checks for functions is very important. When a function is called from within an M-file, MATLAB first checks to see if the function is a built-in function such as sin.

If not, it checks to see if it is a subfunction in the file, then checks to see if it is a private function (which is a function M-file residing in the private subdirectory of the calling function).

Then MATLAB checks for a standard M-file on your search path. Thus, because MATLAB checks for a subfunction before checking for private and standard M-file functions, you may use subfunctions with the same name as another existing M-file.

This feature allows you to name subfunctions without being concerned about whether

another function exists with the same name, so you need not choose long function names to avoid conflict.

This feature also protects you from using another function unintentionally.

The following example shows how the MATLAB M-function mean can be superseded by our own definition of the mean, one which gives the root-mean square value.

The function mean is a subfunction.

The function subfun_demo is the primary function.

function y = subfun_demo(a) y = a - mean(a); %

function w = mean(x)

w = sqrt(sum(x.^2))/length(x);

A sample session follows.

>>y = subfun_demo([4, -4]) y=

1.1716−6.8284

If we had used the MATLAB M-function mean, we would have obtained a different answer; that is,

>>a=[4, -4];

>>b = a - mean(a) b=

4−4

Thus the use of subfunctions enables you to reduce the number of files that define your functions. For example, if it were not for the subfunction mean in the previous example, we would have had to define a separate M-file for our mean function and give it a different name so as not to confuse it with the MATLAB function of the same name.

Subfunctions are normally visible only to the primary function and other subfunctions in the same file. However, we can use a function handle to allow access to the subfunction from outside the M-file.

Nested functions

With MATLAB 7 you can now place the definitions of one or more functions within

another function. Functions so defined are said to be nested within the main function. You can also nest functions within other nested functions.

Like any M-file function, a nested function contains the usual components of an M-file function. You must, however, always terminate a nested function with an end statement.

In fact, if an M-file contains at least one nested function, you must terminate all functions, including subfunctions, in the file with an end statement, whether or not they contain nested functions.

The following example constructs a function handle for a nested function and then passes the handle to the MATLAB function fminbnd to find the minimum point on a parabola. The parabola function constructs and returns a function handle f for the nested function p. This handle gets passed to fminbnd.

function f = parabola(a, b, c) f = @p;

function y = p(x)

y = a*x^2 + b*x + c; end end

In the Command window type

>>f = parabola(4, -50, 5); >>fminbnd(f, -10, 10) ans=

6.2500

Note that the function p(x)can see the variables a, b, and c in the calling function’s workspace.

Nested functions might seem to be the same as subfunctions, but they are not. Nested functions have two unique properties:

1. A nested function can access the workspaces of all functions inside of which it is nested. So for example, a variable that has a value assigned to it by the primary function can be read or overwritten by a function nested at any level within the main function.

A variable assigned in a nested function can be read or overwritten by any of the functions containing that function.

2. If you construct a function handle for a nested function, the handle not only stores the information needed to access the nested function; it also stores the values of all variables shared between the nested function and those functions that contain it.

This means that these variables persist in memory between calls made by means of the function handle.

Private functions

Private functions reside in subdirectories with the special name private, and they are visible only to functions in the parent directory.

Assume the directory rsmith is on the MATLAB search path. A subdirectory of rsmith called private may contain functions that only the functions in rsmith can call. Because private functions are invisible outside the parent directory rsmith, they can use the same names as functions in other directories.

Primary functions and subfunctions can be implemented as private functions.

Create a private directory by creating a subdirectory called private using the standard

procedure for creating a directory or a folder on your computer, but do not place the private directory on your path.

Importing spreadsheet files

Some spreadsheet programs store data in the .wk1 format. You can use the command M = wk1read(‘filename’) to import this data into MATLAB and store it in the matrix M.

The command A = xlsread(‘filename’) imports the Microsoft Excel workbook file

filename.xls into the array A. The command [A, B] = xlsread(‘filename’) imports all numeric data into the array A and all text data into the cell array B.

The import wizard

To import ASCII data, you must know how the data in the file is formatted.

For example, many ASCII data files use a fixed (or uniform) format of rows and columns.

For these files, you should know the following.

• How many data items are in each row?

• Are the data items numeric, text strings, or a mixture of both types?

• Does each row or column have a descriptive text header?

• What character is used as the delimiter, that is, the character used to separate the data items in each row? The delimiter is also called the column separator.

You can use the Import Wizard to import many types of ASCII data formats, including data on the clipboard. When you use the Import Wizard to create a variable in the MATLAB workspace, it overwrites any existing variable in the workspace with the same name without issuing a warning.

The Import Wizard presents a series of dialog boxes in which you:

1) Specify the name of the file you want to import, 2) Specify the delimiter used in the file, and 3) Select the variables that you want to import.

Textbook, page 174 and 175, provides a step-by-step procedure for using the Import Wizard.

The first screen in the Import Wizard.

Problem 3.2

The session is:

>>x = -5-8i; >>y = 10-5i; >> ans=

105.4751>> ans=

−2.5930>> ans=

0.8438>> ans=

−1.6657

Problem 3.6

According to guideline of Problem 3-6, the script file is: prob3-6.m

epsilon = 8.854e-12; L = 1; r = 0.001; d = 0.004;

d = input(‘Enter a value for d in meters: ’) C = pi*epsilon*L./log((d-r)./r);

disp(‘The capacitance in farads is: ’) disp(C)

>>prob3-6

Enter a value for d in meters: 0.004 d = 0.0040

The capacitance in farad is: 2.5319e-011

Problem 3.17

The function is: cubicpoly.m

function [a, b, c, d] = cubicpoly(x, y) for k = 1:4

A(:, k) = x.^(4-k); end B = y’; coeff = A\\B a = coeff(1); b = coeff(2); c = coeff(3); d = coeff(4);

The session is

>>x = [-2, 0, 2, 4];

>>y = [-20, 4, 68, 508];

>>[a, b, c, d] = cubicpoly(x, y)

This gives the correct answers. a=

7b=

5c=

−6d=

4