• Accessing MATLAB
• Entering matrices
• Matrix operations, array operations
• Statements, expressions, and variables; saving a session
• Matrix building functions
• For, while, if - and relations
• Scalar functions
• Vector functions
• Matrix functions
• Command line editing and recall
• Submatrices and colon notation
• M-files
• Text strings, error messages, input
• Managing M-files
• Comparing efficiency of algorithms: flops and etime
• Output format
• Hardcopy
• Graphics
• Reference (Outside this document)

Accessing MATLAB

On most systems, after logging in one can enter MATLAB with the system command matlab and exit MATLAB with the command exit or quit. On a PC, for example, if properly installed, one may enter MATLAB with the command:

 
C>  matlab

 
>>  quit

On systems permitting multiple processes, such as a Unix system, you will find it convenient, for reasons discussed in section 14, to keep both MATLAB and your local editor active. If you are working on a workstation which runs processes in multiple windows, you will want to keep MATLAB active in one window and your local editor active in another. You should consult your instructor or your local computer center for details of the local installation.

Entering matrices

MATLAB works with essentially only one kind of object-a rectangular numerical matrix with possibly complex entries; all variables represent matrices. In some situations, 1-by-1 matrices are interpreted as scalars and matrices with only one row or one column are interpreted as vectors.

Matrices can be introduced into MATLAB in several different ways:

• Entered by an explicit list of elements,
• Generated by built-in statements and functions,
• Created in M-files (see sections 12 and 14 below),
• Loaded from external data files (see User's Guide).

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

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

When listing a number in exponential form (e.g. 2.34e-9), blank spaces must be avoided. Listing entries of a large matrix is best done in an M-file, where errors can be easily edited away (see sections 12 and 14).

The built-in functions rand, magic, and hilb, for example, provide an easy way to create matrices with which to experiment. The command rand(n) will create an n x n matrix with randomly generated entries distributed uniformly between 0 and 1, while rand(m,n) will create an m x n one. magic(n) will create an integral n x n matrix which is a magic square (rows and columns have common sum); hilb(n) will create the n x n Hilbert matrix, the king of ill-conditioned matrices ( m and n denote, of course, positive integers). Matrices can also be generated with a for-loop (see section 6 below).

Individual matrix and vector entries can be referenced with indices inside parentheses in the usual manner. For example, A(2,3) denotes the entry in the second row, third column of matrix A and x(3) denotes the third coordinate of vector x. Try it. A matrix or a vector will only accept positive integers as indices.

Matrix operations, array operations

The following matrix operations are available in MATLAB:

 +    addition  
 -    subtraction  
 *    multiplication 
 ^    power 
 '    transpose 
 \    left division 
 /    right division

The ``matrix division'' operations deserve special comment. If A is an invertible square matrix and b is a compatible column, resp. row, vector, then

 x=A \ b

    
 x=b/A

    b/A=(A' \ b')'

  *  ,   ^  ,  \  , and /,

  [1,2,3,4].*[1,2,3,4]

  [1,2,3,4].^2

Statements, expressions, and variables; saving a session

MATLAB is an expression language; the expressions you type are interpreted and evaluated. MATLAB statements are usually of the form

variable = expression,

expression

Expressions are usually composed from operators, functions, and variable names. Evaluation of the expression produces a matrix, which is then displayed on the screen and assigned to the variable for future use. If the variable name and = sign are omitted, a variable ans (for answer) is automatically created to which the result is assigned.

A statement is normally terminated with the carriage return. However, a statement can be continued to the next line with three or more periods followed by a carriage return. On the other hand, several statements can be placed on a single line if separated by commas or semicolons.

If the last character of a statement is a semicolon, the printing is suppressed, but the assignment is carried out. This is essential in suppressing unwanted printing of intermediate results.

MATLAB is case-sensitive in the names of commands, functions, and variables. For example, solveUT is not the same as solveut.

The command who will list the variables currently in the workspace. A variable can be cleared from the workspace with the command clear variablename. The command clear alone will clear all nonpermanent variables.

The permanent variable eps (epsilon) gives the machine precision-about on most machines. It is useful in determining tolerences for convergence of iterative processes.

A runaway display or computation can be stopped on most machines without leaving MATLAB with CTRL-C (CTRL-BREAK on a PC).

Saving a session. When one logs out or exits MATLAB all variables are lost. However, invoking the command save before exiting causes all variables to be written to a non-human-readable diskfile named matlab.mat. When one later reenters MATLAB, the command load will restore the workspace to its former state.

Matrix building functions

Convenient matrix building functions are

 eye(10)           identity matrix  
 zeros(10)         matrix of zeros  
 ones(10)          matrix of ones  
 diag(A)           returns diagonal M.E. as vector  
 triu(A)           upper triangular part of a matrix 
 tril(A)           lower triangular part of a matrix  
 rand(10)          randomly generated matrix  
 hilb(5)           Hilbert matrix  
 magic(5)          magic square  
 toeplitz        see   help toeplitz

If x is a vector, diag(x) is the diagonal matrix with x down the diagonal; if A is a square matrix, then diag(A) is a vector consisting of the diagonal of A . What is diag(diag(A))? Try it.

Matrices can be built from blocks. For example, if A is a 3-by-3 matrix, then

 B = [A, zeros(3,2); zeros(2,3), eye(2)]

Matrix building functions

Convenient matrix building functions are

 eye(10)           identity matrix  
 zeros(10)         matrix of zeros  
 ones(10)          matrix of ones  
 diag(A)           returns diagonal M.E. as vector  
 triu(A)           upper triangular part of a matrix 
 tril(A)           lower triangular part of a matrix  
 rand(10)          randomly generated matrix  
 hilb(5)           Hilbert matrix  
 magic(5)          magic square  
 toeplitz        see   help toeplitz

If x is a vector, diag(x) is the diagonal matrix with x down the diagonal; if A is a square matrix, then diag(A) is a vector consisting of the diagonal of A . What is diag(diag(A))? Try it.

Matrices can be built from blocks. For example, if A is a 3-by-3 matrix, then

 B = [A, zeros(3,2); zeros(2,3), eye(2)]

For, while, if - and relations

In their basic forms, these MATLAB flow control statements operate like those in most computer languages.

For. For example, for a given n, the statement

 
x = []; for i = 1:n, x=[x,i^2 ], end

 
x = [];  
        for i = 1:n  
            x = [x,i^2 ]  
        end

 
x = []; for i = n:-1:1, x=[x,i^2 ], end

 
for i = 1:m  
             for j = 1:n  
                 H(i, j) = 1/(i+j-1);  
             end  
        end  
        H

While. The general form of a while loop is

while   relation   
     statements    
end

 
n = 0;    
       while  2^n < a    
            n = n + 1;     
       end    
       n

if  relation     
  statements    
end

 
if n < 0     
              parity = 0;    
         elseif  rem(n,2) == 0    
              parity = 2;    
         else    
              parity = 1;    
         end

Relations. The relational operators in MATLAB are

  <      less than
  >      greater than
  <=     less than or equal 
  >=     greater than or equal  
  ==     equal 
  ~=     not equal.

 
  &      and  
  |      or   
  ~      not.

When applied to scalars, a relation is actually the scalar 1 or 0 depending on whether the relation is true or false. Try 3 < 5, 3 > 5, 3 == 5, and 3 == 3. When applied to matrices of the same size, a relation is a matrix of 0's and 1's giving the value of the relation between corresponding entries. Try a = rand(5), b = triu(a), a == b.

A relation between matrices is interpreted by while and if to be true if each entry of the relation matrix is nonzero. Hence, if you wish to execute statement when matrices A and B are equal you could type

 
if  A == B    
      statement    
end

 
        if  any(any(A ~ B))     
              (   statement   )    
         end

  
if  A == B  else     
                { statement}    
end

 if  A ~= B, (   statement  ), end

The for statement permits any matrix to be used instead of 1:n. See the User's Guide for details of how this feature expands the power of the for statement.

Scalar functions

Certain MATLAB functions operate essentially on scalars, but operate element-wise when applied to a matrix. The most common such functions are

  sin    asin    exp                 abs       round  
  cos    acos    log (natural log)   sqrt      floor 
  tan    atan    rem (remainder)     sign      ceil

Vector functions

Other MATLAB functions operate essentially on a vector (row or column), but act on an m-by-n matrix () in a column-by-column fashion to produce a row vector containing the results of their application to each column. Row-by-row action can be obtained by using the transpose; for example, mean(A')'. A few of these functions are

 
  max        sum       median     any  
  min        prod      mean       all   
  sort                 std

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

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

 
[U,D] = eig(A)

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

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

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

 5:-1:1  gives   [5 4 3 2 1].

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

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 A

with the first three columns of B. Note that the entire altered matrix A is 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]

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.

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 4       
        5 6 7 8    
        ];

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; end    
a = floor((b-a+1)*rand(m,n)) + a;

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 == 1    
       m = n;     % handle case of a row vector     
 end    
 mean = sum(x)/m;    
 stdev = sqrt(sum(x.^ 2)/m - mean.^2);

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

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 == 0    
     error('The gcd is not defined when both numbers are zero')    
else    
     while b  ~= 0    
         r = rem(a,b);    
         a = b;  b = r;    
     end    
end

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 feval permits 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)].    
        
% Initialization    
if nargin < 3, tol = eps; end    
trace = (nargout == 2);    
if x  ~= 0, dx = x/20; else, dx = 1/20; end    
a = 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, end    
     b = x + dx;  fb = feval(fun,b);    
end    
if trace, steps = [a fa; b fb]; end    
        
% Main loop    
while  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]]; end    
     if (fb > 0) == (fc > 0)    
          b = c;  fb = fc;    
       else    
         a = c;  fa = fc;    
     end    
end

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.

Text strings, error messages, input

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

 
s = 'This is a test'

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

 
disp('this message is hereby displayed')

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

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:  ')

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

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.

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 Ax=b using Gaussian elimination as follows:

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

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.

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 places  
 format short e   scientific notation with 4 decimal places  
 format long e    scientific notation with 15 decimal places

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.

Hardcopy

Hardcopy is most easily obtained with the diary command. The command

 
diary {\it filename}

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)

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 over the interval -1.5 to 1.5 as follows:

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

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 grid will 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

 
title('Best Least Squares Fit')

By default, the axes are auto-scaled. This can be overridden by the command axis. If 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)

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

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,'+')

Linetypes: solid (-), dashed (-). dotted (:), dashdot (-.)

Marktypes: point (.), plus (), star (*), circle (o), x-mark (x)

See help plot for 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 Note that most of this section is no longer relevant

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  /dps    
gpp  filename  /djet

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

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

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