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9.

. Matrix functions

Much of MATLAB's power comes from its matrix functions. The most useful ones are 

eig eigenvalues and eigenvectors chol cholesky factorization svd singular value decomposition inv inverse lu LU factorization qr QR factorization hess hessenberg form schur schur decomposition rref reduced row echelon form expm matrix exponential sqrtm matrix square root poly characteristic polynomial det determinant size size norm 1-norm, 2-norm, F-norm, <infinity>-norm* cond condition number in the 2-norm rank rank * Note: <infinity> stands for mathematical infinity symbol, which cannot be rendered in HTML at this time. -Ed.

MATLAB functions may have single or multiple output arguments. For example,

y = eig(A), or simply eig(A)

produces a column vector containing the eigenvalues of Awhile

[U,D] = eig(A)

produces a matrix U whose columns are the eigenvectors of A and a diagonal matrix D with the eigenvalues of Aon its diagonal. Try it.

10. Command line editing and recall

The command line in MATLAB can be easily edited. The cursor can be positioned with the left/right arrows and the Backspace (or Delete) key used to delete the character to the left of the cursor. Other editing features are also available. On a PC try the Home, End, and Delete keys; on other systems see help cedit or type cedit.

A convenient feature is use of the up/down arrows to scroll through the stack of previous commands. One can, therefore, recall a previous command line, edit it, and execute the revised command line. For small routines, this is much more convenient that using an M-file which requires moving between MATLAB and the editor (see sections 12 and 14). For example, flopcounts (see section 15) for computing the inverse of matrices of various sizes could be compared by repeatedly recalling, editing, and executing

a = rand(8); flops(0), inv(a); flops

If one wanted to compare plots of the functions y = sin mx and y = sin nx on the interval [0,2\pi] for various m and n, one might do the same for the command line:

m=2; n=3; x=0:.01:2*pi; y=sin(m*x); z=cos(n*x); plot(x,y,x,z)11. Submatrices and colon notation

Vectors and submatrices are often used in MATLAB to achieve fairly complex data manipulation effects. "Colon notation" (which is used both to generate vectors and reference submatrices) and subscripting by vectors are keys to efficient manipulation of these objects. Creative use of these features permits one to minimize the use of loops (which slows MATLAB) and to make code simple and readable. Special effort should be made to become familiar with them.

The expression 1:5 (met earlier in for statements) is actually the row vector [1 2 3 4 5]. The numbers need not be integers nor the increment one. For example,

0.2:0.2:1.2

gives [0.2, 0.4, 0.6, 0.8, 1.0, 1.2], and

5:-1:1

gives [5 4 3 2 1]. The following statements will, for example, generate a table of sines. Try it.

x = [0.0:0.1:2.0]';y = sin(x);[x y]

Note that since sin operates entry-wise, it produces a vector y from the vector x.

The colon notation can be used to access submatrices of a matrix. For example,

A(1:4,3)

is the column vector consisting of the first four entries of the third column of A. A colon by itself denotes an entire row or column:

A(:,3)

is the third column of A, and A(1:4,:)is the first four rows. Arbitrary integral vectors can be used as subscripts:

A(:,[2 4])

contains as columns, columns 2 and 4 of A. Such subscripting can be used on both sides of an assignment statement:

A(:,[2 4 5]) = B(:,1:3)

replaces columns 2,4,5 of b with the first three columns of B. Note that the entire altered matrix Ais printed and assigned. Try it.

Columns 2 and 4 of A can be multiplied on the right by the 2-by-2 matrix [1 2;3 4]:

A(:,[2,4]) = A(:,[2,4])*[1 2;3 4]

Once again, the entire altered matrix is printed and assigned.

If x is an n-vector, what is the effect of the statement x = x(n:-1:1)? Try it.

To appreciate the usefulness of these features, compare these MATLAB statements with a Pascal, FORTRAN, or C routine to effect the same.

12. M-files

MATLAB can execute a sequence of statements stored on diskfiles. Such files are called "M-files" because they must have the file type of ".m" as the last part of their filename. Much of your work with MATLAB will be in creating and refining M-files.

There are two types of M-files: script files and function files.

Script files. A script file consists of a sequence of normal MATLAB statements. If the file has the filename, say, rotate.m, then the MATLAB command rotate will cause the statements in the file to be executed. Variables in a script file are global and will change the value of variables of the same name in the environment of the current MATLAB session.

Script files are often used to enter data into a large matrix; in such a file, entry errors can be easily edited out. If, for example, one enters in a diskfile data.m

A = [1 2 3 45 6 7 8];

then the MATLAB statement data will cause the assignment given in data.mto be carried out.

An M-file can reference other M-files, including referencing itself recursively.

Function files. Function files provide extensibility to MATLAB. You can create new functions specific to your problem which will then have the same status as other MATLAB functions. Variables in a function file are by default local. However, version 4.0 permits a variable to be declared global.

We first illustrate with a simple example of a function file.

function a = randint(m,n)%RANDINT Randomly generated integral matrix.% randint(m,n) returns an m-by-n such matrix with entries% between 0 and 9.a = floor(10*rand(m,n));

A more general version of this function is the following:

function a = randint(m,n,a,b)%RANDINT Randomly generated integral matrix.% randint(m,n) returns an m-by-n such matrix with entries% between 0 and 9.% rand(m,n,a,b) return entries between integers a and b.if nargin < 3, a = 0, b = 9; enda = floor((b-a+1)*rand(m,n)) + a;

This should be placed in a diskfile with filename randint.m(corresponding to the function name). The first line declares the function name, input arguments, and output arguments; without this line the file would be a script file. Then a MATLAB statement

z = randint(4,5),

for example, will cause the numbers 4 and 5 to be passed to the variables m and n in the function file with the output result being passed out to the variable z. Since variables in a function file are local, their names are independent of those in the current MATLAB environment.

Note that use of nargin ("number of input arguments") permits one to set a default value of an omitted input variable---such as a and b in the example.

A function may also have multiple output arguments. For example:

function [mean, stdev] = stat(x)% STAT Mean and standard deviation% For a vector x, stat(x) returns the% mean and standard deviation of x.% For a matrix x, stat(x) returns two row vectors containing,% respectively, the mean and standard deviation of each column.[m n] = size(x);if m == 1m = n; % handle case of a row vectorendmean = sum(x)/m;stdev = sqrt(sum(x.^2)/m - mean.^2);

Once this is placed in a diskfile stat.m, a MATLAB command [xm, xd] = stat(x), for example, will assign the mean and standard deviation of the entries in the vector x to xm and xd, respectively. Single assignments can also be made with a function having multiple output arguments. For example, xm = stat(x) (no brackets needed around xm) will assign the mean of x to xm.

The % symbol indicates that the rest of the line is a comment; MATLAB will ignore the rest of the line. However, the first few comment lines, which document the M-file, are available to the on-line help facility and will be displayed if, for example, help stat is entered. Such documentation should always be included in a function file.

This function illustrates some of the MATLAB features that can be used to produce efficient code. Note, for example, that x.^2 is the matrix of squares of the entries of x, that sum is a vector function (section 8), that sqrt is a scalar function (section 7), and that the division in sum(x)/m is a matrix-scalar operation.

The following function, which gives the greatest common divisor of two integers via the Euclidean algorithm, illustrates the use of an error message (see the next section).

function a = gcd(a,b)% GCD Greatest common divisor% gcd(a,b) is the greatest common divisor of% the integers a and b, not both zero.a = round(abs(a)); b = round(abs(b));if a == 0 & b == 0error('The gcd is not defined when both numbers are zero')elsewhile b ~= 0r = rem(a,b)a = b; b = r;endend

Some more advanced features are illustrated by the following function. As noted earlier, some of the input arguments of a function---such as tol in the example, may be made optional through use of nargin ("number of input arguments"). The variable nargout can be similarly used. Note that the fact that a relation is a number (1 when true; 0 when false) is used and that, when while or if evaluates a relation, "nonzero" means "true" and 0 means "false". Finally, the MATLAB function fevalpermits one to have as an input variable a string naming another function.

function [b, steps] = bisect(fun, x, tol)%BISECT Zero of a function of one variable via the bisection method.% bisect(fun,x) returns a zero of the function. fun is a string% containing the name of a real-valued function of a single% real variable; ordinarily functions are defined in M-files.% x is a starting guess. The value returned is near a point% where fun changes sign. For example,% bisect('sin',3) is pi. Note the quotes around sin.%% An optional third input argument sets a tolerence for the% relative accuracy of the result. The default is eps.% An optional second output argument gives a matrix containing a% trace of the steps; the rows are of form [c f(c)]. % Initializationif nargin < 3, tol = eps; endtrace = (nargout == 2);if x ~= 0, dx = x/20; else, dx = 1/20; enda = x - dx; fa = feval(fun,a);b = x + dx; fb = feval(fun,b); % Find change of sign.while (fa > 0) == (fb > 0)dx = 2.0*dx;a = x - dx; fa = feval(fun,a);if (fa > 0) ~= (fb > 0), break, endb = x + dx; fb = feval(fun,b);endif trace, steps = [a fa; b fb] end % Main loopwhile abs(b - a) > 2.0*tol*max(abs(b),1.0)c = a + 0.5*(b - a); fc = feval(fun,c);if trace, steps = [steps; [c fc]];if (fb > 0) == (fc > 0)b = c; fb = fc;elsea = c; fa = fc;endend

Some of MATLAB's functions are built-in while others are distributed as M-files. The actual listing of any M-file---MATLAB's or your own---can be viewed with the MATLAB command type functionname. Try entering type eig, type vander, and type rank.

13. Text strings, error messages, input

Text strings are entered into MATLAB surrounded by single quotes. For example,

s = 'This is a test'

assigns the given text string to the variable s.

Text strings can be displayed with the function disp. For example:

disp('this message is hereby displayed')

Error messages are best displayed with the function error

error('Sorry, the matrix must be symmetric')

since when placed in an M-File, it causes execution to exit the M-file.

In an M-file the user can be prompted to interactively enter input data with the function input. When, for example, the statement

iter = input('Enter the number of iterations: ')

is encountered, the prompt message is displayed and execution pauses while the user keys in the input data. Upon pressing the return key, the data is assigned to the variable iterand execution resumes.

14. Managing M-files

While using MATLAB one frequently wishes to create or edit an M-file and then return to MATLAB. One wishes to keep MATLAB active while editing a file since otherwise all variables would be lost upon exiting.

This can be easily done using the !-feature. If, while in MATLAB, you precede it with an !, any system command---such as those for editing, printing, or copying a file---can be executed without exiting MATLAB. If, for example, the system command ed accesses your editor, the MATLAB command

>> !ed rotate.m

will let you edit the file named rotate.musing your local editor. Upon leaving the editor, you will be returned to MATLAB just where you left it.

As noted in section 1, on systems permitting multiple processes, such as one running Unix, it may be preferable to keep both MATLAB and your local editor active, keeping one process suspended while working in the other. If these processes can be run in multiple windows, as on a workstation, you will want to keep MATLAB active in one window and your editor active in another.

You may consult your instructor or your local computing center for details of the local installation.

Version 4.0 has many debbugging tools. See help dbtype and references given there.

When in MATLAB, the command dir will list the contents of the current directory while the command what will list only the M-files in the directory. The MATLAB commands delete and type can be used to delete a diskfile and print a file to the screen, respectively, and chdir can be used to change the working directory. While these commands may duplicate system commands, they avoid the use of an !.

M-files must be accessible to MATLAB. On most mainframe or workstation network installations, personal M-files which are stored in a subdirectory of one's home directory named matlab will be accessible to MATLAB from any directory in which one is working. See the discussion of MATLABPATH in the User's Guide for further information.

15. Comparing efficiency of algorithms: flops and etime

Two measures of the efficiency of an algorithm are the number of floating point operations (flops) performed and the elapsed time.

The MATLAB function flops keeps a running total of the flops performed. The command flops(0) (not flops = 0!) will reset flops to 0. Hence, entering flops(0) immediately before executing an algorithm and flops immediately after gives the flop count for the algorithm.

The MATLAB function clock gives the current time accurate to a hundreth of a second (see help clock). Given two such times t1 and t2, etime(t2,t1) gives the elapsed time from t1 to t2. One can, for example, measure the time required to solve a given linear system A x = b using Gaussian elimination as follows:

t = clock; x = A\b; time = etime(clock,t)

You may wish to compare this time---and flop count---with that for solving the system using x = inv(A)*b;. Try it. Version 4.0 has the more convenient tic and toc.

It should be noted that, on timesharing machines, etime may not be a reliable measure of the efficiency of an algorithm since the rate of execution depends on how busy the computer is at the time.

16. Output format

While all computations in MATLAB are performed in double precision, the format of the displayed output can be controlled by the following commands. 

format short fixed point with 4 decimal places (the default)  format long fixed point with 14 decimal place format short e scientific notation with 4 decimal places format long e scientific notation with 15 decimal places

Once invoked, the chosen format remains in effect until changed.

The command format compact will suppress most blank lines allowing more information to be placed on the screen or page. It is independent of the other format commands.

17. Hardcopy

Hardcopy is most easily obtained with the diarycommand. The command

diary filename

causes what appears subsequently on the screen (except graphics) to be written to the named diskfile (if the filename is omitted it will be written to a default file named diary) until one gives the command diary off; the command diary onwill cause writing to the file to resume, etc. When finished, you can edit the file as desired and print it out on the local system. The !-feature (see section 14) will permit you to edit and print the file without leaving MATLAB.

18. Graphics

MATLAB can produce both planar plots and 3-D mesh surface plots. To preview some of these capabilities in version 3.5, enter the command plotdemo. Planar plots. The plot command creates linear x-y plots; if x and y are vectors of the same length, the command plot(x,y) opens a graphics window and draws an x-y plot of the elements of x versus the elements of y. You can, for example, draw the graph of the sine function over the interval -4 to 4 with the following commands:

x = -4:.01:4; y = sin(x); plot(x,y)

Try it. The vector x is a partition of the domain with meshsize 0.01 while y is a vector giving the values of sine at the nodes of this partition (recall that sinoperates entrywise).

When in the graphics screen, pressing any key will return you to the command screen while the command shg (show graph) will then return you to the current graphics screen. If your machine supports multiple windows with a separate graphics window, you will want to keep the graphics window exposed---but moved to the side---and the command window active.

As a second example, you can draw the graph of y = e^(-x^2) over the interval -1.5 to 1.5 as follows:

x = -1.5:.01:1.5; y = exp(-x.^2); plot(x,y)

Note that one must precede ^ by a period to ensure that it operates entrywise (see section 3).

Plots of parametrically defined curves can also be made. Try, for example,

t=0:.001:2*pi; x=cos(3*t); y=sin(2*t); plot(x,y)

The command gridwill place grid lines on the current graph.

The graphs can be given titles, axes labeled, and text placed within the graph with the following commands which take a string as an argument.

 

title graph title xlabel x-axis label ylabel y-axis label gtext interactively-positioned text text position text at specified coordinates

For example, the command

title('Best Least Squares Fit')

gives a graph a title. The command gtext('The Spot')allows a mouse or the arrow keys to position a crosshair on the graph, at which the text will be placed when any key is pressed.

By default, the axes are auto-scaled. This can be overridden by the command axis. If c = [xmin,xmax,ymin,ymax] is a 4-element vector, then axis(c) sets the axis scaling to the precribed limits. By itself, axis freezes the current scaling for subsequent graphs; entering axis again returns to auto-scaling. The command axis('square') ensures that the same scale is used on both axes. In version 4.0, axis has been significantly changed; see help axis.

Two ways to make multiple plots on a single graph are illustrated by

x=0:.01:2*pi;y1=sin(x);y2=sin(2*x);y3=sin(4*x);plot(x,y1,x,y2,x,y3)

and by forming a matrix Ycontaining the functional values as columns

x=0:.01:2*pi; Y=[sin(x)', sin(2*x)', sin(4*x)']; plot(x,Y)

Another way is with hold. The command hold freezes the current graphics screen so that subsequent plots are superimposed on it. Entering hold again releases the "hold." The commands hold on and hold offare also available in version 4.0. One can override the default linetypes and pointtypes. For example,

x=0:.01:2*pi; y1=sin(x); y2=sin(2*x); y3=sin(4*x); plot(x,y1,'--',x,y2,':',x,y3,'+')

renders a dashed line and dotted line for the first two graphs while for the third the symbol +is placed at each node. The line- and mark-types are

Linetypes: solid (-), dashed (\tt--). dotted (:), dashdot (-.)Marktypes: point (.), plus (+), star (*), circle (o), x-mark (x)

See help plotfor line and mark colors.

The command subplot can be used to partition the screen so that up to four plots can be viewed simultaneously. See help subplot.

Graphics hardcopy [The features described in this subsection are not available with the Student Edition of Matlab. Graphics hardcopy can be obtained there only with a screen dump: Shift-PrtScr.] A hardcopy of the graphics screen can be most easily obtained with the MATLAB command print. It will send a high-resolution copy of the current graphics screen to the printer, placing the graph on the top half of the page.

In version 4.0 the meta and gpp commands described below have been absorbed into the print command. See help print.

Producing unified hard copy of several plots requires more effort. The Matlab command meta filename stores the current graphics screen in a file named filename.met (a "metafile") in the current directory. Subsequent meta (no filename) commands append a new current graphics screen to the previously named metafile. This metafile---which may now contain several plots---may be processed later with the graphics post-processor (GPP) program to produce high-resolution hardcopy, two plots per page.

The program GPP (graphics post-processor) is a system command, not a MATLAB command. However, in practice it is usually involked from within MATLAB using the "!" feature (see section 14). It acts on a device-independent metafile to produce an output file appropriate for many different hardcopy devices.

The selection of the specific hardcopy device is made with the option key "/d". For example, the system commands

gpp filename /dpsgpp filename /djet

will produce files filename.ps and filename.jet suitable for printing on, respectively, PostScript and HP LaserJet printers. They can be printed using the usual printing command for the local system---for example, lpr filename.ps on a Unix system. Entering the system command gpp with no arguments gives a list of all supported hardcopy devices. On a PC, most of this can be automated by appropriately editing the file gpp.batdistributed with MATLAB.

3-D mesh plots. Three dimensional mesh surface plots are drawn with the function mesh. The command mesh(z) creates a three-dimensional perspective plot of the elements of the matrix z. The mesh surface is defined by the z-coordinates of points above a rectangular grid in the x-y plane. Try mesh(eye(10)).

To draw the graph of a function z = f(x,y) over a rectangle, one first defines vectors xx and yy which give partitions of the sides of the rectangle. With the function meshdom (mesh domain; called meshgrid in version 4.0) one then creates a matrix x, each row of which equals xx and whose column length is the length of yy, and similarly a matrix y, each column of which equals yy, as follows:

[x,y] = meshdom(xx,yy);

One then computes a matrix z, obtained by evaluating f entrywise over the matrices x and y, to which meshcan be applied.

You can, for example, draw the graph of z = e^(-x^2-y^2) over the square [-2,2]\times[-2,2] as follows (try it):

xx = -2:.1:2;yy = xx;[x,y] = meshdom(xx,yy);z = exp(-x.^2 - y.^2);mesh(z)

One could, of course, replace the first three lines of the preceding with

[x,y] = meshdom(-2:.1:2, -2:.1:2);

You are referred to the User's Guide for further details regarding mesh.

In version 4.0, the 3-D graphics capabilities of MATLAB have been considerably expanded. Consult the on-line help for plot3, mesh, and surf

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