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MATLAB Primer Third Edition
Kermit Sigmon Department of Mathematics University of Florida
Department of Mathematics University of Florida Gainesville, FL 32611 sigmon@math.ufl.edu Copyright c 1989, 1992, 1993 by Kermit Sigmon
On the Third Edition
The Third Edition of the MATLAB Primer is based on version 4.0 4.1 of MATLAB. While this edition re ects an extensive general revision of the Second Edition, most signi cant is the new information to help one begin to use the major new features of version 4.0 4.1, the sparse matrix and enhanced graphics capabilities. The plain TEX source and corresponding PostScript le of the latest printing of the MATLAB Primer are always available via anonymous ftp from: Address: math.ufl.edu Directory: pub matlab Files: primer.tex, primer.ps You are advised to download anew each term the latest printing of the Primer since minor improvements and corrections may have been made in the interim. If ftp is unavailable to you, the Primer can be obtained via listserv by sending an email message to listserv@math.ufl.edu containing the single line send matlab primer.tex. Also available at this ftp site are both English primer35.tex, primer35.ps and Spanish primer35sp.tex, primer35sp.ps versions of the Second Edition of the Primer, which was based on version 3.5 of MATLAB. The Spanish translation is by Celestino Montes, University of Seville, Spain. A Spanish translation of the Third Edition is under development. Users of the Primer usually appreciate the convenience and durability of a bound copy with a cover, copy center style. 12-93
Copyright c 1989, 1992, 1993 by Kermit Sigmon The MATLAB Primer may be distributed as desired subject to the following conditions: 1. It may not be altered in any way, except possibly adding an addendum giving information about the local computer installation or MATLAB toolboxes. 2. It, or any part thereof, may not be used as part of a document distributed for a commercial purpose. In particular, it may be distributed via a local copy center or bookstore. Department of Mathematics University of Florida Gainesville, FL 32611 sigmon@math.ufl.edu i
Introduction
MATLAB is an interactive, matrix-based system for scienti c and engineering numeric computation and visualization. You can solve complex numerical problems in a fraction of the time required with a programming language such as Fortran or C. The name MATLAB is derived from MATrix LABoratory. The purpose of this Primer is to help you begin to use MATLAB. It is not intended to be a substitute for the User's Guide and Reference Guide for MATLAB. The Primer can best be used hands-on. You are encouraged to work at the computer as you read the Primer and freely experiment with examples. This Primer, along with the on-line help facility, usually su ce for students in a class requiring use of MATLAB. You should liberally use the on-line help facility for more detailed information. When using MATLAB, the command help functionname will give information about a speci c function. For example, the command help eig will give information about the eigenvalue function eig. By itself, the command help will display a list of topics for which on-line help is available; then help topic will list those speci c functions under this topic for which help is available. The list of functions in the last section of this Primer also gives most of this information. You can preview some of the features of MATLAB by rst entering the command demo and then selecting from the options o ered. The scope and power of MATLAB go far beyond these notes. Eventually you will want to consult the MATLAB User's Guide and Reference Guide. Copies of the complete documentation are often available for review at locations such as consulting desks, terminal rooms, computing labs, and the reserve desk of the library. Consult your instructor or your local computing center to learn where this documentation is located at your institution. MATLAB is available for a number of environments: Sun Apollo VAXstation HP workstations, VAX, MicroVAX, Gould, PC and AT compatibles, 80386 and 80486 computers, Apple Macintosh, and several parallel machines. There is a relatively inexpensive Student Edition available from Prentice Hall publishers. The information in these notes applies generally to all of these environments. MATLAB is licensed by The MathWorks, Inc., 24 Prime Park Way, Natick, MA 01760, 508 653-1415, Fax: 508 653-2997, Email: info@mathworks.com.
Copyright c 1989, 1992, 1993 by Kermit Sigmon
ii
Contents
Page 1. Accessing MATLAB : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Entering matrices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 3. Matrix operations, array operations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 4. Statements, expressions, variables; saving a session : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 5. Matrix building functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 6. For, while, if | and relations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 7. Scalar functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 8. Vector functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 9. Matrix functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 10. Command line editing and recall : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 11. Submatrices and colon notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 12. M- les: script les, function les : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 13. Text strings, error messages, input : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 14. Managing M- les : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 15. Comparing e ciency of algorithms: ops, tic, toc : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 16. Output format : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 17. Hard copy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 18. Graphics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 planar plots 15 , hardcopy 17 , 3-D line plots 18 mesh and surface plots 18 , Handle Graphics 20 19. Sparse matrix computations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 20. Reference : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 22 iii
1. Accessing MATLAB.
matlab
On most systems, after logging in one can enter MATLAB with the system command and exit MATLAB with the MATLAB command quit or exit. However, your local installation may permit MATLAB to be accessed from a menu or by clicking an icon. On systems permitting multiple processes, such as a Unix system or MS Windows, you will nd it convenient, for reasons discussed in section 14, to keep both MATLAB and your local editor active. If you are working on a platform 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.
2. 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 di erent ways: Entered by an explicit list of elements, Generated by built-in statements and functions, Created in a disk le with your local editor, Loaded from external data les or applications see the User's Guide . For example, either of the statements and
A = A 1 4 7 1 2 3; 4 5 6; 7 8 9 = 2 3 5 6 8 9
creates the obvious 3-by-3 matrix and assigns it to a variable A. Try it. The elements within a row of a matrix may be separated by commas as well as a blank. When listing a number in exponential form e.g. 2.34e-9 , blank spaces must be avoided. MATLAB allows complex numbers in all its operations and functions. Two convenient ways to enter complex matrices are: When listing complex numbers e.g. 2+6i in a matrix, blank spaces must be avoided. Either i or j may be used as the imaginary unit. If, however, you use i and j as variables and overwrite their values, you may generate a new imaginary unit with, say, ii = sqrt -1 . 1
A = A = 1 2;3 4 + i* 5 6;7 8 1+5i 2+6i;3+7i 4+8i
Listing entries of a large matrix is best done in an ASCII le with your local editor, where errors can be easily corrected see sections 12 and 14 . The le should consist of a rectangular array of just the numeric matrix entries. If this le is named, say, data.ext where .ext is any extension , the MATLAB command load data.ext will read this le to the variable data in your MATLAB workspace. This may also be done with a script le see section 12 . 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 n matrix with randomly generated entries distributed uniformly between 0 and 1, while rand m,n will create an m n one. magic n will create an integral n n matrix which is a magic square rows, columns, and diagonals have common sum ; hilb n will create the n 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.
3. Matrix operations, array operations.
+
The following matrix operations are available in MATLAB:
,
0
b
n
addition subtraction multiplication power conjugate transpose left division right division
These matrix operations apply, of course, to scalars 1-by-1 matrices as well. If the sizes of the matrices are incompatible for the matrix operation, an error message will result, except in the case of scalar-matrix operations for addition, subtraction, and division as well as for multiplication in which case each entry of the matrix is operated on by the scalar. 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 = Anb is the solution of A x = b and, resp., x = b=A is the solution of x A = b. In left division, if A is square, then it is factored using Gaussian elimination and these factors are used to solve A x = b. If A is not square, it is factored using Householder orthogonalization with column pivoting and the factors are used to solve the under- or over- determined system in the least squares sense. Right division is de ned in terms of left division by b=A = A0 nb0 0 . 2
The matrix operations of addition and subtraction already operate entry-wise but the other matrix operations given above do not|they are matrix operations. It is important to observe that these other operations, , b , n, and , can be made to operate entry-wise by preceding them by a period. For example, either 1,2,3,4 .* 1,2,3,4 or 1,2,3,4 .b 2 will yield 1,4,9,16 . Try it. This is particularly useful when using Matlab graphics.
Array operations.
4. 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, or simply 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 or whos 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 unit roundo |about 10,16 on most machines. It is useful in specifying 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 disk le named matlab.mat. When one later reenters MATLAB, the command load will restore the workspace to its former state.
3
5. Matrix building functions.
eye zeros ones diag triu tril rand hilb magic toeplitz
Convenient matrix building functions are identity matrix matrix of zeros matrix of ones create or extract diagonals upper triangular part of a matrix lower triangular part of a matrix randomly generated matrix Hilbert matrix magic square see help toeplitz
For example, zeros m,n produces an m-by-n matrix of zeros and zeros n produces an n-by-n one. If A is a matrix, then zeros size A produces a matrix of zeros having the same size as A. 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 will build a certain 5-by-5 matrix. Try it.
B = A, zeros 3,2 ; zeros 2,3 , eye 2
In their basic forms, these MATLAB ow control statements operate like those in most computer languages.
6. For, while, if | and relations.
For.
or
For example, for a given n, the statement x = ; for i = 1:n, x= x,ib 2 ,
x = ; for i = 1:n x = x,ib end
end
2
will produce a certain n-vector and the statement x = ; for i = n:-1:1, x= x,ib 2 , end will produce the same vector in reverse order. Try them. Note that a matrix may be empty such as x = .
4
The statements
for i = 1:m for j = 1:n H i, j = 1 i+j-1 ; end end H
will produce and print to the screen the m-by-n hilbert matrix. The semicolon on the inner statement is essential to suppress printing of unwanted intermediate results while the last H displays the nal result. The for statement permits any matrix to be used instead of 1:n. The variable just consecutively assumes the value of each column of the matrix. For example,
s = 0; for c = A s = s + sum c ; end
computes the sum of all entries of the matrix A by adding its column sums Of course, sum sum A does it more e ciently; see section 8 . In fact, since 1:n = 1,2,3,: : : ,n , this column-by-column assigment is what occurs with if i = 1:n,: : : " see section 11 .
While.
The general form of a while loop is while relation statements
end
The statements will be repeatedly executed as long as the relation remains true. For example, for a given number a, the following will compute and display the smallest nonnegative integer n such that 2n a:
n = 0; while 2b n a n = n + 1; end n
If.
The general form of a simple if statement is if relation statements
end
The statements will be executed only if the relation is true. Multiple branching is also possible, as is illustrated by
if n 0 parity = 0;
5
In two-way branching the elseif portion would, of course, be omitted.
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.
Note that =" is used in an assignment statement while ==" is used in a relation. Relations may be connected or quanti ed by the logical operators & and j 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 entering 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 end
statement
but if you wish to execute statement when A and B are not equal, you would type
if any any A end
statement
=
B
or, more simply,
if A == B else end
statement
Note that the seemingly obvious if A = B, statement, end 6
will not give what is intended since statement would execute only if each of the corresponding entries of A and B di er. The functions any and all can be creatively used to reduce matrix relations to vectors or scalars. Two any's are required above since any is a vector operator see section 8 .
7. 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 oor tan atan rem remainder sign ceil Other MATLAB functions operate essentially on a vector row or column , but act on an m-by-n matrix m 2 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 For example, the maximum entry in a matrix A is given by max max A rather than max A . Try it.
8. Vector functions.
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, 1-norm cond condition number in the 2-norm rank rank 7
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 A while produces a matrix U whose columns are the eigenvectors of A and a diagonal matrix D with the eigenvalues of A on its diagonal. Try it.
U,D = eig A
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 a Unix system or a PC the Emacs commands Ctl-a, Ctl-e, Ctl-d, and Ctl-k work; 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- le which requires moving between MATLAB and the editor see sections 12 and 14 . For example, opcounts see section 15 for computing the inverse of matrices of various sizes could be compared by repeatedly recalling, editing, and executing If one wanted to compare plots of the functions y = sin mx and y = sin nx on the interval 0; 2 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 a = rand 8 ; flops 0 , inv a ; flops
Vectors and submatrices are often used in MATLAB to achieve fairly complex data manipulation e ects. Colon notation" which is used both to generate vectors and reference submatrices and subscripting by integral vectors are keys to e cient manipulation of these objects. Creative use of these features to vectorize operations permits one to minimize the use of loops which slows MATLAB and to make code simple and readable. Special e ort 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, gives 5 4 3 2 1 . The following statements will, for example, generate a table of sines. Try it.
5:-1:1 x = 0.0:0.1:2.0 0 ; y = sin x ; x y
11. Submatrices and colon notation.
gives
0.2:0.2:1.2 0.2, 0.4, 0.6, 0.8, 1.0, 1.2
, and
8
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 rst 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 rst 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 rst 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 : Once again, the entire altered matrix is printed and assigned. If x is an n-vector, what is the e ect of the statement x = x n:-1:1 ? Try it. Also try y = fliplr x and y = flipud x' . To appreciate the usefulness of these features, compare these MATLAB statements with a Pascal, FORTRAN, or C routine to e ect the same. MATLAB can execute a sequence of statements stored in disk les. Such les are called M- les" because they must have the le type of .m" as the last part of their lename. Much of your work with MATLAB will be in creating and re ning M- les. M- les are usually created using your local editor. There are two types of M- les: script les and function les. A script le consists of a sequence of normal MATLAB statements. If the le has the lename, say, rotate.m, then the MATLAB command rotate will cause the statements in the le to be executed. Variables in a script le are global and will change the value of variables of the same name in the environment of the current MATLAB session. Script les may be used to enter data into a large matrix; in such a le, entry errors can be easily corrected. If, for example, one enters in a disk le data.m
A = 1 2 3 4 5 6 7 8 ; A :, 2,4 = A :, 2,4 * 1 2;3 4
12. M- les.
Script les.
then the MATLAB statement data will cause the assignment given in data.m to be carried out. However, it is usually easier to use the MATLAB function load see section 2 . An M- le can reference other M- les, including referencing itself recursively. 9
Function les.
Function les provide extensibility to MATLAB. You can create new functions speci c to your problem which will then have the same status as other MATLAB functions. Variables in a function le are by default local. A variable can, however, be declared global see help global . We rst illustrate with a simple example of a function le.
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 and . if nargin 3, a = 0; b = 9; end a = floor b-a+1 *rand m,n + a;
a
b
This should be placed in a disk le with lename randint.m corresponding to the function name . The rst line declares the function name, input arguments, and output arguments; without this line the le would be a script le. 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 le with the output result being passed out to the variable z. Since variables in a function le 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 of x; mean, stdev = stat x both the mean and standard deviation. For a matrix x, stat x acts columnwise. m n = size x ; if m == 1 m = n; handle case of a row vector end mean = sum x m; stdev = sqrt sum x.b m - mean.b ;
2 Once this is placed in a disk le 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 10
2
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. Moreover, the rst few contiguous comment lines, which document the M- le, 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 le. This function illustrates some of the MATLAB features that can be used to produce e cient code. Note, for example, that x.b 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. Thus all operations are vectorized and loops avoided. If you can't vectorize some computations, you can make your for loops go faster by preallocating any vectors or matrices in which output is stored. For example, by including the second statement below, which uses the function zeros, space for storing E in memory is preallocated. Without this MATLAB must resize E one column larger in each iteration, slowing execution.
M = magic 6 ; E = zeros 6,50 ; for j = 1:50 E :,j = eig Mb i ; 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 this 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. Also see eval.
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 MATLAB 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 .
11
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- les. The actual listing of any non-built-in M- le|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.
s = 'This is a test'
Text strings are entered into MATLAB surrounded by single quotes. For example,
assigns the given text string to the variable s. Text strings can be displayed with the function disp. For example: Error messages are best displayed with the function error since when placed in an M-File, it aborts execution of the M- le. 12
error 'Sorry, the matrix must be symmetric' disp 'this message is hereby displayed'
input
In an M- le the user can be prompted to interactively enter input data with the function . 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 iter and execution resumes.
14. Managing M- les.
While using MATLAB one frequently wishes to create or edit an M- le with the local editor and then return to MATLAB. One wishes to keep MATLAB active while editing a le 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 le|can be executed without exiting MATLAB. If, for example, the system command ed accesses your editor, the MATLAB command will let you edit the le named rotate.m using your local editor. Upon leaving the editor, you will be returned to MATLAB just where you left it. However, as noted in section 1, on systems permitting multiple processes, such as one running Unix or MS Windows, 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, you will want to keep MATLAB active in one window and your editor active in another. You should consult your instructor or your local computing center for details of the local installation. Many debugging tools are available. See help dbtype or the list of functions in the last section. When in MATLAB, the command pwd will return the name of the present working directory and cd can be used to change the working directory. Either dir or ls will list the contents of the working directory while the command what lists only the M- les in the directory. The MATLAB commands delete and type can be used to delete a disk le and print an M- le to the screen, respectively. While these commands may duplicate system commands, they avoid the use of an !. You may enjoy entering the command why a few times. M- les must be in a directory accessible to MATLAB. M- les in the present working directory are always accessible. On most mainframe or workstation network installations, personal M- les 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. The current list of directories in MATLAB's search path is obtained by the command path. This command can also be used to add or delete directories from the search path. See help path. 13
!ed rotate.m
15. Comparing e ciency of algorithms: ops, tic and toc.
Two measures of the e ciency of an algorithm are the number of oating point operations ops performed and the elapsed time. The MATLAB function flops keeps a running total of the ops performed. The command flops 0 not flops = 0! will reset ops to 0. Hence, entering flops 0 immediately before executing an algorithm and flops immediately after gives the op count for the algorithm. For example, the number of ops required to solve a given linear system via Gaussian elimination can be obtained with:
flops 0 , x = A b; flops
The elapsed time in seconds can be obtained with the stopwatch timers tic and toc; tic starts the timer and toc returns the elapsed time. Hence, the commands tic, any statement, toc will return the elapsed time for execution of the statement. The elapsed time for solving the linear system above can be obtained, for example, with: You may wish to compare this time|and op count|with that for solving the system using x = inv A *b;. Try it. It should be noted that, on timesharing machines, elapsed time may not be a reliable measure of the e ciency of an algorithm since the rate of execution depends on how busy the computer is at the time. While all computations in MATLAB are performed in double precision, the format of the displayed output can be controlled by the following commands.
format format format format format format format format short long short e long e rat hex bank + tic, x = A b; toc
n
n
16. Output format.
xed point with 4 decimal places the default xed point with 14 decimal places scienti c notation with 4 decimal places scienti c notation with 15 decimal places approximation by ratio of small integers hexadecimal format xed dollars and cents +, -, blank
Once invoked, the chosen format remains in e ect until changed. The command format compact will suppress most blank lines allowing more information to be placed on the screen or page. The command format loose returns to the non-compact format. These commands are independent of the other format commands.
14
Hardcopy is most easily obtained with the diary command. The command diary lename causes what appears subsequently on the screen except graphics to be written to the named disk le if the lename is omitted it will be written to a default le named diary until one gives the command diary off; the command diary on will cause writing to the le to resume, etc. When nished, you can edit the le as desired and print it out on the local system. The !-feature see section 14 will permit you to edit and print the le without leaving MATLAB. MATLAB can produce planar plots of curves, 3-D plots of curves, 3-D mesh surface plots, and 3-D faceted surface plots. The primary commands for these facilities are plot, plot3, mesh, and surf, respectively. An introduction to each of these is given below. To preview some of these capabilities, enter the command demo and select some of the graphics options.
17. Hardcopy.
18. Graphics.
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: 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 sin operates entrywise . You will usually want to keep the current graphics window gure" exposed|but moved to the side|and the command window active. One can have several graphics gures, one of which will at any time be the designated current" gure where graphs from subsequent plotting commands will be placed. If, for example, gure 1 is the current gure, then the command figure 2 or simply figure will open a second gure if necessary and make it the current gure. The command figure 1 will then expose gure 1 and make it again the current gure. The command gcf will return the number of the current gure. As a second example, you can draw the graph of y = e,x2 over the interval -1.5 to 1.5 as follows: Note that one must precede b by a period to ensure that it operates entrywise see section 3 . MATLAB supplies a function fplot to easily and e ciently plot the graph of a function. For example, to plot the graph of the function above, one can rst de ne the function in an M- le called, say, expnormal.m containing 15
x = -1.5:.01:1.5; y = exp -x.b 2 ; plot x,y x = -4:.01:4; y = sin x ; plot x,y
Then the command
function y = expnormal x y = exp -x.b 2 ; fplot 'expnormal', -1.5,1.5
will produce the graph. Try it. Plots of parametrically de ned 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 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 place text on the graph using the mouse text position text at speci ed coordinates For example, the command gives a graph a title. The command gtext 'The Spot' allows one to interactively place the designated text on the current graph by placing the mouse pointer at the desired position and clicking the mouse. To place text in a graph at designated coordinates, one would use the command text see help text . The command grid will place grid lines on the current graph. By default, the axes are auto-scaled. This can be overridden by the command axis. Some features of axis are: axis xmin ,xmax ,ymin ,ymax set axis scaling to prescribed limits axis axis freezes scaling for subsequent graphs axis auto returns to auto-scaling v = axis returns vector v showing current scaling axis square same scale on both axes axis equal same scale and tic marks on both axes axis off turns o axis scaling and tic marks axis on turns on axis scaling and tic marks The axis command should be given after the plot command. Two ways to make multiple plots on a single graph are illustrated by and by forming a matrix Y containing the functional values as columns
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 title 'Best Least Squares Fit'
Another way is with hold. The command hold on freezes the current graphics screen so that subsequent plots are superimposed on it. The axes may, however, become rescaled. Entering hold off releases the hold." 16
One can override the default linetypes, pointtypes and colors. For example, renders a dashed line and dotted line for the rst two graphs while for the third the symbol + is placed at each node. The line- and mark-types are Linetypes: solid - , dashed -- . dotted : , dashdot -. Marktypes: point . , plus + , star * , circle o , x-mark x Colors can be speci ed for the line- and mark-types. Colors: yellow y , magenta m , cyan c , red r green g , blue b , white w , black k For example, plot x,y,'r--' plots a red dashed line. The command subplot can be used to partition the screen so that several small plots can be placed in one gure. See help subplot. Other specialized 2-D plotting functions you may wish to explore via help are:
x=0:.01:2*pi; y1=sin x ; y2=sin 2*x ; y3=sin 4*x ; plot x,y1,'--',x,y2,':',x,y3,'+'
Graphics hardcopy
polar, bar, hist, quiver, compass, feather, rose, stairs, fill
A hardcopy of the current graphics gure can be most easily obtained with the MATLAB command print. Entered by itself, it will send a high-resolution copy of the current graphics gure to the default printer. The printopt M- le is used to specify the default setting used by the print command. If desired, one can change the defaults by editing this le see help printopt . The command print lename saves the current graphics gure to the designated lename in the default le format. If lename has no extension, then an appropriate extension such as .ps, .eps, or .jet is appended. If, for example, PostScript is the default le format, then will create a PostScript le lissajous.ps of the current graphics gure which can subsequently be printed using the system print command. If filename already exists, it will be overwritten unless you use the -append option. The command will append the hopefully di erent current graphics gure to the existing le lissajous.ps. In this way one can save several graphics gures in a single le. The default settings can, of course, be overwritten. For example, will save to an Encapsulated PostScript le saddle.eps the graphics gure 3 | even if it is not the current gure.
print -deps -f3 saddle print -append lissajous print lissajous
17
3-D line plots.
Completely analogous to plot in two dimensions, the command plot3 produces curves in three dimensional space. If x, y, and z are three vectors of the same size, then the command plot3 x,y,z will produce a perspective plot of the piecewise linear curve in 3-space passing through the points whose coordinates are the respective elements of x, y, and z. These vectors are usually de ned parametrically. For example, will produce a helix which is compressed near the x-y plane a slinky" . Try it. Just as for planar plots, a title and axis labels including zlabel can be added. The features of axis command described there also hold for 3-D plots; setting the axis scaling to prescribed limits will, of course, now require a 6-vector. Three dimensional wire mesh surface plots are drawn with the command mesh. The command mesh z creates a three-dimensional perspective plot of the elements of the matrix z. The mesh surface is de ned by the z-coordinates of points above a rectangular grid in the x-y plane. Try mesh eye 10 . Similarly, three dimensional faceted surface plots are drawn with the command surf. Try surf eye 10 . To draw the graph of a function z = f x; y over a rectangle, one rst de nes vectors xx and yy which give partitions of the sides of the rectangle. With the function meshgrid 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: One then computes a matrix z, obtained by evaluating f entrywise over the matrices x and y, to which mesh or surf can be applied. You can, for example, draw the graph of z = e,x2 ,y2 over the square ,2; 2 ,2; 2 as follows try it :
xx = -2:.2:2; yy = xx; x,y = meshgrid xx,yy ; z = exp -x.b 2 - y.b 2 ; mesh z x,y x,y = meshgrid xx,yy ; t=.01:.01:20*pi; x=cos t ; y=sin t ; z=t.b 3; plot3 x,y,z
3-D mesh and surface plots.
One could, of course, replace the rst three lines of the preceding with Try this plot with surf instead of mesh. As noted above, the features of the axis command described in the section on planar plots also hold for 3-D plots as do the commands for titles, axes labelling and the command hold. The color shading of surfaces is set by the shading command. There are three settings for shading: faceted default , interpolated, and flat. These are set by the commands 18
= meshgrid -2:.2:2, -2:.2:2 ;
or shading flat Note that on surfaces produced by surf, the settings interpolated and flat remove the superimposed mesh lines. Experiment with various shadings on the surface produced above. The command shading as well as colormap and view below should be entered after the surf command. The color pro le of a surface is controlled by the colormap command. Available prede ned colormaps include: hsv default , hot, cool, jet, pink, copper, flag, gray, bone The command colormap cool will, for example, set a certain color pro le for the current gure. Experiment with various colormaps on the surface produced above. The command view can be used to specify in spherical or cartesian coordinates the viewpoint from which the 3-D object is to be viewed. See help view. The MATLAB function peaks generates an interesting surface on which to experiment with shading, colormap, and view. Plots of parametrically de ned surfaces can also be made. The MATLAB functions sphere and cylinder will generate such plots of the named surfaces. See type sphere and type cylinder. The following is an example of a similar function which generates a plot of a torus.
shading faceted, shading interp, function x,y,z = torus r,n,a TORUS Generate a torus torus r,n,a generates a plot of a torus with central radius a and lateral radius r. n controls the number of facets on the surface. These input variables are optional with defaults r = 0.5, n = 30, a = 1. x,y,z = torus r,n,a generates three n+1 -by- n+1 matrices so that surf x,y,z will produce the torus. See also SPHERE, CYLINDER if nargin 3, a = 1; end if nargin 2, n = 30; end if nargin 1, r = 0.5; end theta = pi* 0:2:2*n n; phi = 2*pi* 0:2:n ' n; xx = a + r*cos phi *cos theta ; yy = a + r*cos phi *sin theta ; zz = r*sin phi *ones size theta ; if nargout == 0 surf xx,yy,zz ar = a + r sqrt 2 ; axis -ar,ar,-ar,ar,-ar,ar else
19
x = xx; y = yy; z = zz; end
Other 3-D plotting functions you may wish to explore via help are:
Handle Graphics.
meshz, surfc, surfl, contour, pcolor
Beyond those described above, MATLAB's graphics system provides low level functions which permit one to control virtually all aspects of the graphics environment to produce sophisticated plots. Enter the command set 1 and gca,set ans to see some of the properties of gure 1 which one can control. This system is called Handle Graphics, for which one is referred to the MATLAB User's Guide.
19. Sparse Matrix Computations.
In performing matrix computations, MATLAB normally assumes that a matrix is dense; that is, any entry in a matrix may be nonzero. If, however, a matrix contains su ciently many zero entries, computation time could be reduced by avoiding arithmetic operations on zero entries and less memory could be required by storing only the nonzero entries of the matrix. This increase in e ciency in time and storage can make feasible the solution of signi cantly larger problems than would otherwise be possible. MATLAB provides the capability to take advantage of the sparsity of matrices. Matlab has two storage modes, full and sparse, with full the default. The functions full and sparse convert between the two modes. For a matrix A, full or sparse, nnz A returns the number of nonzero elements in A. A sparse matrix is stored as a linear array of its nonzero elements along with their row and column indices. If a full tridiagonal matrix F is created via, say, then the statement S = sparse F will convert F to sparse mode. Try it. Note that the output lists the nonzero entries in column major order along with their row and column indices. The statement F = full S restores S to full storage mode. One can check the storage mode of a matrix A with the command issparse A . A sparse matrix is, of course, usually generated directly rather than by applying the function sparse to a full matrix. A sparse banded matrix can be easily created via the function spdiags by specifying diagonals. For example, a familiar sparse tridiagonal matrix is created by Try it. The integral vector -1,0,1 speci es in which diagonals the columns of e,d,e should be placed use full T to view . Experiment with other values of m and n and, say, -3,0,2 instead of -1,0,1 . See help spdiags for further features of spdiags.
m = 6; n = 6; e = ones n,1 ; d = -2*e; T = spdiags e,d,e , -1,0,1 ,m,n F = floor 10*rand 6 ; F = triu tril F,1 ,-1 ;
20
The sparse analogs of eye, zeros, ones, and randn for full matrices are, respectively, The latter two take a matrix argument and replace only the nonzero entries with ones and normally distributed random numbers, respectively. randn also permits the sparsity structure to be randomized. The command sparse m,n creates a sparse zero matrix. The versatile function sparse permits creation of a sparse matrix via listing its nonzero entries. Try, for example, In general, if the vector s lists the nonzero entries of S and the integral vectors i and j list their corresponding row and column indices, then will create the desired sparse m n matrix S . As another example try
n = 6; e = floor 10*rand n-1,1 ; E = sparse 2:n,1:n-1,e,n,n sparse i,j,s,m,n i = 1 2 3 4 4 4 ; j = 1 2 3 1 2 3 ; s = S = sparse i,j,s,4,3 , full S 5 6 7 8 9 10 ; speye, sparse, spones, sprandn
The arithmetic operations and most MATLAB functions can be applied independent of storage mode. The storage mode of the result? Operations on full matrices always give full results. Selected other results are S=sparse, F=full : Sparse: S+S, S*S, S.*S, S.*F, Sb n, S.b n, SnS Full: S+F, S*F, SnF, FnS Sparse: inv S , chol S , lu S , diag S , max S , sum S For sparse S , eig S is full if S is symmetric but unde ned if S is unsymmetric; svd requires a full argument. A matrix built from blocks, such as A,B;C,D , is sparse if any constituent block is sparse. You may wish to compare, for the two storage modes, the e ciency of solving a tridiagonal system of equations for, say, n = 20; 50; 500; 1000 by entering, recalling and editing the following two command lines:
n=20;e=ones n,1 ;d=-2*e; T=spdiags e,d,e , -1,0,1 ,n,n ; A=full T ; b=ones n,1 ;s=sparse b ;tic,T s;sparsetime=toc, tic,A b;fulltime=toc
n
n
21
There are many MATLAB features which cannot be included in these introductory notes. Listed below are some of the MATLAB functions and operators available, grouped by subject area1. Use the on-line help facility or consult the Reference Guide for more detailed information on the functions. There are many functions beyond these. There exist, in particular, several toolboxes" of functions for speci c areas2. Included among such are signal processing, control systems, robust-control, system identi cation, optimization, splines, chemometrics, -analysis and synthesis, state-space identi cation, neural networks, image processing, symbolic math Maple kernel , and statistics. These can be explored via the command help.
help what type lookfor which demo path cedit version whatsnew info why
Managing Commands and Functions
20. Reference.
help facility list M- les on disk list named M- le keywork search through the help entries locate functions and les run demonstrations control MATLAB's search path set parameters for command line editing and recall display MATLAB version you are running display toolbox README les info about MATLAB and The MathWorks receive ippant answer
who whos save load clear pack size length disp
Managing Variables and the Workspace
list current variables list current variables, long form save workspace variables to disk retrieve variables from disk clear variables and functions from memory consolidate workspace memory size of matrix length of vector display matrix or text
1 Source: MATLAB Reference Guide, version 4.1 2 The toolboxes, which are optional, may not be installed on your system.
22
cd pwd dir, ls delete getenv ! unix diary
Working with Files and the Operating System
change current working directory show current working directory directory listing delete le get environment variable execute operating system command execute operating system command; return result save text of MATLAB session
clc home format echo more
Controlling the Command Window
clear command window send cursor home|to top of screen set output format echo commands inside script commands control paged output in command window
quit startup matlabrc +
b
Starting and Quitting from MATLAB
terminate MATLAB M- le executed when MATLAB is started master startup M- le
Array Operators
Matrix Operators
, n
' .' kron
addition subtraction multiplication power right division left division conjugate transpose transpose Kronecker tensor product
+
, .
.b . .n
addition subtraction multiplication power right division left division
Relational and Logical Operators
= less than or equal = greater than or equal == equal = not equal
greater than
less than
&
and or not xor exclusive or
j
23
= . .. ... , ; : !
assignment statement used to form vectors and matrices; enclose multiple function output variables arithmetic expression precedence; enclose function input variables decimal point parent directory continue statement to next line separate subscripts, function arguments, statements end rows, suppress printing comments subscripting, vector generation execute operating system command
Special Characters
ans eps realmax reammin pi i, j inf NaN ops nargin nargout computer
Special Variables and Constraints
answer when expression not assigned oating point precision largest oating point number smallest positive oating point number
imaginary unit in nity Not-a-Number oating point operation count number of function input arguments number of function output arguments computer type
date clock etime tic, toc cputime
Time and Date
current date wall clock elapsed time function stopwatch timer functions elapsed CPU time
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zeros ones eye diag toeplitz magic compan linspace logspace meshgrid rand randn hilb invhilb vander pascal hadamard hankel rosser wilkinson gallery
matrix of zeros matrix of ones identity diagonal Toeplitz magic square companion linearly spaced vectors logarithmically spaced vectors array for 3-D plots uniformly distributed random numbers normally distributed randon numbers Hilbert inverse Hilbert exact Vandermonde Pascal Hadamard Hankel symmetric eigenvalue test matrix Wilkinson's eigenvalue test matrix two small test matrices
Special Matrices
diag rot90 iplr ipud reshape tril triu .' :
create or extract diagonals rotate matrix 90 degrees ip matrix left-to-right ip matrix up-to-down change size lower triangular part upper triangular part transpose convert matrix to single column; A :
Matrix Manipulation
25
exist any all nd isnan isinf nite isieee isempty issparse isstr strcmp
check if variables or functions exist true if any element of vector is true true if all elements of vector are true nd indices of non-zero elements true for NaNs true for in nite elements true for nite elements true for IEEE oating point arithmetic true for empty matrix true for sparse matrix true for text string compare string variables
Logical Functions
if else elseif end for while break return error
conditionally execute statements used with if used with if terminate if, for, while repeat statements for a speci c number of times repeat statments while condition is true terminate execution of for or while loops return to invoking function display message and abort function
Control Flow
input keyboard menu pause function eval feval global nargchk
prompt for user input invoke keyboard as if it were a script le generate menu of choices for user input wait for user response de ne function execute string with MATLAB expression evaluate function speci ed by string de ne global variables validate number of input arguments
Programming
26
string abs blanks eval num2str int2str str2num isstr strcmp upper lower hex2num hex2dec dec2hex
about character strings in MATLAB convert string to numeric values a string of blanks evaluate string with MATLAB expression convert number to string convert integer to string convert string to number true for string variables compare string variables convert string to uppercase convert string to lowercase convert hex string to oating point number convert hex string to decimal integer convert decimal integer to hex string
Text and Strings
dbstop dbclear dbcont dbdown dbstack dbstatus dbstep dbtype dbup dbdown dbquit
set breakpoint remove breakpoint remove execution change local workspace context list who called whom list all breakpoints execute one or more lines list M- le with line numbers change local workspace context opposite of dbup quit debug mode
Debugging
saxis sound auread auwrite lin2mu mu2lin
Sound Processing Functions
sound axis scaling convert vector to sound Read Sun audio le Write Sun audio le linear to mu-law conversion mu-law to linear conversion
27
abs angle sqrt real imag conj gcd lcm round x oor ceil sign rem exp log log10
Elementary Math Functions
absolute value or complex magnitude phase angle square root real part imaginary part complex conjugate greatest common divisor least common multiple round to nearest integer round toward zero round toward ,1 round toward 1 signum function remainder exponential base e natural logarithm log base 10
sin, asin, sinh, asinh cos, acos, cosh, acosh tan, atan, tanh, atanh cot, acot, coth, acoth sec, asec, sech, asech csc, acsc, csch, acsch
sine, arcsine, hyperbolic sine, hyperbolic arcsine cosine, arccosine, hyperbolic cosine, hyperbolic arccosine tangent, arctangent, hyperbolic tangent, hyperbolic arctangent cotangent, arccotangent, hyperbolic cotan., hyperbolic arccotan. secant, arcsecant, hyperbolic secant, hyperbolic arcsecant cosecant, arccosecant, hyperbolic cosecant, hyperbolic arccosecant
Special Functions
Trigonometric Functions
bessel beta gamma rat rats erf erfinv ellipke ellipj expint log2 pow2
bessel function beta function gamma function rational approximation rational output error function inverse error function complete elliptic integral Jacobian elliptic integral exponential integral dissect oating point numbers scale oating point numbers
28
inv lu rref chol qr nnls lscov null orth eig hess schur cdf2rdf rsf2csf balance qz polyeig svd pinv
Matrix Decompositions and Factorizations
inverse factors from Gaussian elimination reduced row echelon form Cholesky factorization orthogonal-triangular decomposition nonnegative least squares least squares in presence of know covariance null space orthogonalization eigenvalues and eigenvectors Hessenberg form Schur decomposition complex diagonal form to real block diagonal form real block diagonal form to complex diagonal form diagonal scaling for eigenvalue accuracy generalized eigenvalues polynomial eigenvalue solver singular value decomposition pseudoinverse
cond rcond condest norm normest rank
condition number in 2-norm LINPACK reciprocal condition number estimator Hager Higham condition number estimator 1-norm,2-norm,F-norm,1-norm 2-norm estimator rank
Matrix Conditioning
expm expm1 expm2 expm3 logm sqrtm funm poly det trace
matrix exponential M- le implementation of expm matrix exponential via Taylor series matrix exponential via eigenvalues and eigenvectors matrix logarithm matrix square root evaluate general matrix function characteristic polynomial determinant trace
Elementary Matrix Functions
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poly roots roots1 polyval polyvalm conv deconv residue poly t polyder
construct polynomial with speci ed roots polynomial roots|companion matrix method polynomial roots|Laguerre's method evaluate polynomial evaluate polynomial with matrix argument multiply polynomials divide polynomials partial-fraction expansion residues t polynomial to data di erentiate polynomial
Column-wise Data Analysis
Polynomials
max min mean median std sort sum prod cumsum cumprod hist abs angle conv deconv corrcoef cov lter lter2 cplxpair unwrap nextpow2 t t2 i t i t2 tshift
largest component smallest component average or mean value median value standard deviation sort in ascending order sum of elements product of elements cumulative sum of elements cumulative product of elements histogram
complex magnitude phase angle convolution and polynomial multiplication deconvolution and polynomial division correlation coe cients covariance matrix one-dimensional digital lter two-dimensional digital lter sort numbers into complex pairs remove phase angle jumps across 360 boundaries next higher power of 2 radix-2 fast Fourier transform two-dimensional FFT inverse fast Fourier transform inverse 2-D FFT zero-th lag to center of spectrum
Signal Processing
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di gradient del2 subspace spline interp1 interp2 interpft griddata
Finite Differences and Data Interpolation
approximate derivatives approximate gradient ve point discrete Laplacian angle between two subspaces cubic spline interpolation 1-D data interpolation 2-D data interpolation 1-D data interpolation via FFT method data gridding
quad quad8 trapz
Numerical Integration
adaptive 2-panel Simpson's Rule adaptive 8-panel Newton-Cotes Rule trapezoidal method
ode23 2nd 3rd order Runge-Kutta method ode23p solve via ode23, displaying plot ode45 4th 5th order Runge-Kutta-Fehlberg method
Nonlinear Equations and Optimization
Differential Equation Solution
fmin fmins fsolve fzero fplot
minimize function of one variable minimize function of several variables solution to a system of nonlinear equations nd zeros of a function of several variables nd zero of function of one variable plot graph of a function
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plot loglog semilogx semilogy ll polar bar stairs errorbar hist rose compass feather fplot
Two Dimensional Graphs
linear plot log-log scale plot semilog scale plot semilog scale plot draw lled 2-D polygons polar coordinate plot bar graph stairstep plot error bar plot histogram plot angle histogram plot compass plot feather plot plot function
title xlabel ylabel zlabel grid text gtext ginput
Graph Annotation
graph title x-axis label y-axis label z-axis label for 3-D plots grid lines text annotation mouse placement of text graphical input from mouse
Figure Window Axis Creation and Control
gure gcf clf close hold ishold subplot axes gca axis caxis
create gure graph window get handle to current gure clear current gure close gure hold current graph return hold status create axes in tiled positions create axes in arbitrary positions get handle to to current axes control axis scaling and appearance control pseudocolor axis scaling
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print print graph or save graph to le printopt con gure local printer defaults orient set paper orientation
Three Dimensional Graphs
Graph Hardcopy and Storage
mesh meshc meshz surf surfc surfl plot3 ll3 contour contour3 clabel contourc pcolor quiver image waterfall slice
3-D mesh surface combination mesh contour plot 3-D mesh with zero plane 3-D shaded surface combination surface contour plot 3-D shaded surface with lighting plot lines and points in 3-D space draw lled 3-D polygons in 3-D space contour plot 3-D contour plot contour plot elevation labels contour plot computation used by contour pseudocolor checkerboard plot quiver plot display image waterfall plot volumetric visualization plot
3-D Graph Appearance
view viewmtx hidden shading axis caxis specular di use surfnorm colormap brighten spinmap rgbplot hsv2rgb rgb2hsv
3-D graph viewpoint speci cation view transformation matrices mesh hidden line removal mode color shading mode axis scaling and apearance pseudocolor axis scaling specular re ectance di use re ectance surface normals color lookup table see below brighten or darken color map spin color map plot colormap hsv to rgb color map conversion rgb to hsv color map conversion
33
hsv jet gray hot cool bone copper pink ag
hue-saturation-value default variant of hsv linear gray-scale black-red-yellow-white shades of cyan and magenta gray-scale with tinge of blue linear copper tone pastel shades of pink alternating red, white, blue, and black
3-D Objects
Color Maps
sphere generate sphere cylinder generate cylinder peaks generate demo surface
Movies and Animation
moviein getframe movie
initialize movie frame memory get movie frame play recorded movie frames
gure axes line text patch surface image uicontrol uimenu
Handle Graphics Objects
create gure window create axes create line create text create patch create surface create image create user interface control create user interface menu
set get reset delete drawnow
Handle Graphics Operations
set object properties get object properties reset object properties delete object ush pending graphics events
34
spdiags speye sprandn spones sprandsym spfun sparse full nd spconvert issparse nnz nonzeros nzmax spalloc spy gplot colmmd colperm dmperm randperm symmmd symrcm condest normest sprank spaugment spparms symbfact sparsefun
sparse matrix formed from diagonals sparse identity matrix sparse random matrix replace nonzero entries with ones sparse symmetric random matrix apply function to nonzero entries create sparse matrix; convert full matrix to sparse convert sparse matrix to full matrix nd indices of nonzero entries convert from sparse matrix external format true if matrix is sparse number of nonzero entries nonzero entries amount of storage allocated for nonzero entries allocate memory for nonzero entries visualize sparsity structure plot graph, as in graph theory" column minimum degree order columns based on nonzero count Dulmage-Mendelsohn decomposition random permutation vector symmetric minimum degree reverse Cuthill-McKee ordering estimate 1-norm condition estimate 2-norm structural rank form least squares augmented system set parameters for sparse matrix routines symbolic factorization analysis sparse auxillary functions and parameters
Sparse Matrix Functions
35
MATLAB Primer (ang)
MATLAB Primer Third Edition
Kermit Sigmon Department of Mathematics University of Florida
Department of Mathematics University of Florida Gainesville, FL 32611 sigmon@math.ufl.edu Copyright c 1989, 1992, 1993 by Kermit Sigmon
On the Third Edition
The Third Edition of the MATLAB Primer is based on version 4.0 4.1 of MATLAB. While this edition re ects an extensive general revision of the Second Edition, most signi cant is the new information to help one begin to use the major new features of version 4.0 4.1, the sparse matrix and enhanced graphics capabilities. The plain TEX source and corresponding PostScript le of the latest printing of the MATLAB Primer are always available via anonymous ftp from: Address: math.ufl.edu Directory: pub matlab Files: primer.tex, primer.ps You are advised to download anew each term the latest printing of the Primer since minor improvements and corrections may have been made in the interim. If ftp is unavailable to you, the Primer can be obtained via listserv by sending an email message to listserv@math.ufl.edu containing the single line send matlab primer.tex. Also available at this ftp site are both English primer35.tex, primer35.ps and Spanish primer35sp.tex, primer35sp.ps versions of the Second Edition of the Primer, which was based on version 3.5 of MATLAB. The Spanish translation is by Celestino Montes, University of Seville, Spain. A Spanish translation of the Third Edition is under development. Users of the Primer usually appreciate the convenience and durability of a bound copy with a cover, copy center style. 12-93
Copyright c 1989, 1992, 1993 by Kermit Sigmon The MATLAB Primer may be distributed as desired subject to the following conditions: 1. It may not be altered in any way, except possibly adding an addendum giving information about the local computer installation or MATLAB toolboxes. 2. It, or any part thereof, may not be used as part of a document distributed for a commercial purpose. In particular, it may be distributed via a local copy center or bookstore. Department of Mathematics University of Florida Gainesville, FL 32611 sigmon@math.ufl.edu i
Introduction
MATLAB is an interactive, matrix-based system for scienti c and engineering numeric computation and visualization. You can solve complex numerical problems in a fraction of the time required with a programming language such as Fortran or C. The name MATLAB is derived from MATrix LABoratory. The purpose of this Primer is to help you begin to use MATLAB. It is not intended to be a substitute for the User's Guide and Reference Guide for MATLAB. The Primer can best be used hands-on. You are encouraged to work at the computer as you read the Primer and freely experiment with examples. This Primer, along with the on-line help facility, usually su ce for students in a class requiring use of MATLAB. You should liberally use the on-line help facility for more detailed information. When using MATLAB, the command help functionname will give information about a speci c function. For example, the command help eig will give information about the eigenvalue function eig. By itself, the command help will display a list of topics for which on-line help is available; then help topic will list those speci c functions under this topic for which help is available. The list of functions in the last section of this Primer also gives most of this information. You can preview some of the features of MATLAB by rst entering the command demo and then selecting from the options o ered. The scope and power of MATLAB go far beyond these notes. Eventually you will want to consult the MATLAB User's Guide and Reference Guide. Copies of the complete documentation are often available for review at locations such as consulting desks, terminal rooms, computing labs, and the reserve desk of the library. Consult your instructor or your local computing center to learn where this documentation is located at your institution. MATLAB is available for a number of environments: Sun Apollo VAXstation HP workstations, VAX, MicroVAX, Gould, PC and AT compatibles, 80386 and 80486 computers, Apple Macintosh, and several parallel machines. There is a relatively inexpensive Student Edition available from Prentice Hall publishers. The information in these notes applies generally to all of these environments. MATLAB is licensed by The MathWorks, Inc., 24 Prime Park Way, Natick, MA 01760, 508 653-1415, Fax: 508 653-2997, Email: info@mathworks.com.
Copyright c 1989, 1992, 1993 by Kermit Sigmon
ii
Contents
Page 1. Accessing MATLAB : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Entering matrices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 3. Matrix operations, array operations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 4. Statements, expressions, variables; saving a session : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 5. Matrix building functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 6. For, while, if | and relations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 7. Scalar functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 8. Vector functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 9. Matrix functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 10. Command line editing and recall : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 11. Submatrices and colon notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 12. M- les: script les, function les : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 13. Text strings, error messages, input : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 14. Managing M- les : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 15. Comparing e ciency of algorithms: ops, tic, toc : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 16. Output format : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 17. Hard copy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 18. Graphics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 planar plots 15 , hardcopy 17 , 3-D line plots 18 mesh and surface plots 18 , Handle Graphics 20 19. Sparse matrix computations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 20. Reference : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 22 iii
1. Accessing MATLAB.
matlab
On most systems, after logging in one can enter MATLAB with the system command and exit MATLAB with the MATLAB command quit or exit. However, your local installation may permit MATLAB to be accessed from a menu or by clicking an icon. On systems permitting multiple processes, such as a Unix system or MS Windows, you will nd it convenient, for reasons discussed in section 14, to keep both MATLAB and your local editor active. If you are working on a platform 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.
2. 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 di erent ways: Entered by an explicit list of elements, Generated by built-in statements and functions, Created in a disk le with your local editor, Loaded from external data les or applications see the User's Guide . For example, either of the statements and
A = A 1 4 7 1 2 3; 4 5 6; 7 8 9 = 2 3 5 6 8 9
creates the obvious 3-by-3 matrix and assigns it to a variable A. Try it. The elements within a row of a matrix may be separated by commas as well as a blank. When listing a number in exponential form e.g. 2.34e-9 , blank spaces must be avoided. MATLAB allows complex numbers in all its operations and functions. Two convenient ways to enter complex matrices are: When listing complex numbers e.g. 2+6i in a matrix, blank spaces must be avoided. Either i or j may be used as the imaginary unit. If, however, you use i and j as variables and overwrite their values, you may generate a new imaginary unit with, say, ii = sqrt -1 . 1
A = A = 1 2;3 4 + i* 5 6;7 8 1+5i 2+6i;3+7i 4+8i
Listing entries of a large matrix is best done in an ASCII le with your local editor, where errors can be easily corrected see sections 12 and 14 . The le should consist of a rectangular array of just the numeric matrix entries. If this le is named, say, data.ext where .ext is any extension , the MATLAB command load data.ext will read this le to the variable data in your MATLAB workspace. This may also be done with a script le see section 12 . 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 n matrix with randomly generated entries distributed uniformly between 0 and 1, while rand m,n will create an m n one. magic n will create an integral n n matrix which is a magic square rows, columns, and diagonals have common sum ; hilb n will create the n 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.
3. Matrix operations, array operations.
+
The following matrix operations are available in MATLAB:
,
0
b
n
addition subtraction multiplication power conjugate transpose left division right division
These matrix operations apply, of course, to scalars 1-by-1 matrices as well. If the sizes of the matrices are incompatible for the matrix operation, an error message will result, except in the case of scalar-matrix operations for addition, subtraction, and division as well as for multiplication in which case each entry of the matrix is operated on by the scalar. 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 = Anb is the solution of A x = b and, resp., x = b=A is the solution of x A = b. In left division, if A is square, then it is factored using Gaussian elimination and these factors are used to solve A x = b. If A is not square, it is factored using Householder orthogonalization with column pivoting and the factors are used to solve the under- or over- determined system in the least squares sense. Right division is de ned in terms of left division by b=A = A0 nb0 0 . 2
The matrix operations of addition and subtraction already operate entry-wise but the other matrix operations given above do not|they are matrix operations. It is important to observe that these other operations, , b , n, and , can be made to operate entry-wise by preceding them by a period. For example, either 1,2,3,4 .* 1,2,3,4 or 1,2,3,4 .b 2 will yield 1,4,9,16 . Try it. This is particularly useful when using Matlab graphics.
Array operations.
4. 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, or simply 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 or whos 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 unit roundo |about 10,16 on most machines. It is useful in specifying 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 disk le named matlab.mat. When one later reenters MATLAB, the command load will restore the workspace to its former state.
3
5. Matrix building functions.
eye zeros ones diag triu tril rand hilb magic toeplitz
Convenient matrix building functions are identity matrix matrix of zeros matrix of ones create or extract diagonals upper triangular part of a matrix lower triangular part of a matrix randomly generated matrix Hilbert matrix magic square see help toeplitz
For example, zeros m,n produces an m-by-n matrix of zeros and zeros n produces an n-by-n one. If A is a matrix, then zeros size A produces a matrix of zeros having the same size as A. 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 will build a certain 5-by-5 matrix. Try it.
B = A, zeros 3,2 ; zeros 2,3 , eye 2
In their basic forms, these MATLAB ow control statements operate like those in most computer languages.
6. For, while, if | and relations.
For.
or
For example, for a given n, the statement x = ; for i = 1:n, x= x,ib 2 ,
x = ; for i = 1:n x = x,ib end
end
2
will produce a certain n-vector and the statement x = ; for i = n:-1:1, x= x,ib 2 , end will produce the same vector in reverse order. Try them. Note that a matrix may be empty such as x = .
4
The statements
for i = 1:m for j = 1:n H i, j = 1 i+j-1 ; end end H
will produce and print to the screen the m-by-n hilbert matrix. The semicolon on the inner statement is essential to suppress printing of unwanted intermediate results while the last H displays the nal result. The for statement permits any matrix to be used instead of 1:n. The variable just consecutively assumes the value of each column of the matrix. For example,
s = 0; for c = A s = s + sum c ; end
computes the sum of all entries of the matrix A by adding its column sums Of course, sum sum A does it more e ciently; see section 8 . In fact, since 1:n = 1,2,3,: : : ,n , this column-by-column assigment is what occurs with if i = 1:n,: : : " see section 11 .
While.
The general form of a while loop is while relation statements
end
The statements will be repeatedly executed as long as the relation remains true. For example, for a given number a, the following will compute and display the smallest nonnegative integer n such that 2n a:
n = 0; while 2b n a n = n + 1; end n
If.
The general form of a simple if statement is if relation statements
end
The statements will be executed only if the relation is true. Multiple branching is also possible, as is illustrated by
if n 0 parity = 0;
5
In two-way branching the elseif portion would, of course, be omitted.
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.
Note that =" is used in an assignment statement while ==" is used in a relation. Relations may be connected or quanti ed by the logical operators & and j 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 entering 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 end
statement
but if you wish to execute statement when A and B are not equal, you would type
if any any A end
statement
=
B
or, more simply,
if A == B else end
statement
Note that the seemingly obvious if A = B, statement, end 6
will not give what is intended since statement would execute only if each of the corresponding entries of A and B di er. The functions any and all can be creatively used to reduce matrix relations to vectors or scalars. Two any's are required above since any is a vector operator see section 8 .
7. 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 oor tan atan rem remainder sign ceil Other MATLAB functions operate essentially on a vector row or column , but act on an m-by-n matrix m 2 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 For example, the maximum entry in a matrix A is given by max max A rather than max A . Try it.
8. Vector functions.
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, 1-norm cond condition number in the 2-norm rank rank 7
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 A while produces a matrix U whose columns are the eigenvectors of A and a diagonal matrix D with the eigenvalues of A on its diagonal. Try it.
U,D = eig A
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 a Unix system or a PC the Emacs commands Ctl-a, Ctl-e, Ctl-d, and Ctl-k work; 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- le which requires moving between MATLAB and the editor see sections 12 and 14 . For example, opcounts see section 15 for computing the inverse of matrices of various sizes could be compared by repeatedly recalling, editing, and executing If one wanted to compare plots of the functions y = sin mx and y = sin nx on the interval 0; 2 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 a = rand 8 ; flops 0 , inv a ; flops
Vectors and submatrices are often used in MATLAB to achieve fairly complex data manipulation e ects. Colon notation" which is used both to generate vectors and reference submatrices and subscripting by integral vectors are keys to e cient manipulation of these objects. Creative use of these features to vectorize operations permits one to minimize the use of loops which slows MATLAB and to make code simple and readable. Special e ort 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, gives 5 4 3 2 1 . The following statements will, for example, generate a table of sines. Try it.
5:-1:1 x = 0.0:0.1:2.0 0 ; y = sin x ; x y
11. Submatrices and colon notation.
gives
0.2:0.2:1.2 0.2, 0.4, 0.6, 0.8, 1.0, 1.2
, and
8
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 rst 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 rst 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 rst 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 : Once again, the entire altered matrix is printed and assigned. If x is an n-vector, what is the e ect of the statement x = x n:-1:1 ? Try it. Also try y = fliplr x and y = flipud x' . To appreciate the usefulness of these features, compare these MATLAB statements with a Pascal, FORTRAN, or C routine to e ect the same. MATLAB can execute a sequence of statements stored in disk les. Such les are called M- les" because they must have the le type of .m" as the last part of their lename. Much of your work with MATLAB will be in creating and re ning M- les. M- les are usually created using your local editor. There are two types of M- les: script les and function les. A script le consists of a sequence of normal MATLAB statements. If the le has the lename, say, rotate.m, then the MATLAB command rotate will cause the statements in the le to be executed. Variables in a script le are global and will change the value of variables of the same name in the environment of the current MATLAB session. Script les may be used to enter data into a large matrix; in such a le, entry errors can be easily corrected. If, for example, one enters in a disk le data.m
A = 1 2 3 4 5 6 7 8 ; A :, 2,4 = A :, 2,4 * 1 2;3 4
12. M- les.
Script les.
then the MATLAB statement data will cause the assignment given in data.m to be carried out. However, it is usually easier to use the MATLAB function load see section 2 . An M- le can reference other M- les, including referencing itself recursively. 9
Function les.
Function les provide extensibility to MATLAB. You can create new functions speci c to your problem which will then have the same status as other MATLAB functions. Variables in a function le are by default local. A variable can, however, be declared global see help global . We rst illustrate with a simple example of a function le.
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 and . if nargin 3, a = 0; b = 9; end a = floor b-a+1 *rand m,n + a;
a
b
This should be placed in a disk le with lename randint.m corresponding to the function name . The rst line declares the function name, input arguments, and output arguments; without this line the le would be a script le. 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 le with the output result being passed out to the variable z. Since variables in a function le 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 of x; mean, stdev = stat x both the mean and standard deviation. For a matrix x, stat x acts columnwise. m n = size x ; if m == 1 m = n; handle case of a row vector end mean = sum x m; stdev = sqrt sum x.b m - mean.b ;
2 Once this is placed in a disk le 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 10
2
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. Moreover, the rst few contiguous comment lines, which document the M- le, 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 le. This function illustrates some of the MATLAB features that can be used to produce e cient code. Note, for example, that x.b 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. Thus all operations are vectorized and loops avoided. If you can't vectorize some computations, you can make your for loops go faster by preallocating any vectors or matrices in which output is stored. For example, by including the second statement below, which uses the function zeros, space for storing E in memory is preallocated. Without this MATLAB must resize E one column larger in each iteration, slowing execution.
M = magic 6 ; E = zeros 6,50 ; for j = 1:50 E :,j = eig Mb i ; 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 this 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. Also see eval.
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 MATLAB 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 .
11
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- les. The actual listing of any non-built-in M- le|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.
s = 'This is a test'
Text strings are entered into MATLAB surrounded by single quotes. For example,
assigns the given text string to the variable s. Text strings can be displayed with the function disp. For example: Error messages are best displayed with the function error since when placed in an M-File, it aborts execution of the M- le. 12
error 'Sorry, the matrix must be symmetric' disp 'this message is hereby displayed'
input
In an M- le the user can be prompted to interactively enter input data with the function . 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 iter and execution resumes.
14. Managing M- les.
While using MATLAB one frequently wishes to create or edit an M- le with the local editor and then return to MATLAB. One wishes to keep MATLAB active while editing a le 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 le|can be executed without exiting MATLAB. If, for example, the system command ed accesses your editor, the MATLAB command will let you edit the le named rotate.m using your local editor. Upon leaving the editor, you will be returned to MATLAB just where you left it. However, as noted in section 1, on systems permitting multiple processes, such as one running Unix or MS Windows, 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, you will want to keep MATLAB active in one window and your editor active in another. You should consult your instructor or your local computing center for details of the local installation. Many debugging tools are available. See help dbtype or the list of functions in the last section. When in MATLAB, the command pwd will return the name of the present working directory and cd can be used to change the working directory. Either dir or ls will list the contents of the working directory while the command what lists only the M- les in the directory. The MATLAB commands delete and type can be used to delete a disk le and print an M- le to the screen, respectively. While these commands may duplicate system commands, they avoid the use of an !. You may enjoy entering the command why a few times. M- les must be in a directory accessible to MATLAB. M- les in the present working directory are always accessible. On most mainframe or workstation network installations, personal M- les 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. The current list of directories in MATLAB's search path is obtained by the command path. This command can also be used to add or delete directories from the search path. See help path. 13
!ed rotate.m
15. Comparing e ciency of algorithms: ops, tic and toc.
Two measures of the e ciency of an algorithm are the number of oating point operations ops performed and the elapsed time. The MATLAB function flops keeps a running total of the ops performed. The command flops 0 not flops = 0! will reset ops to 0. Hence, entering flops 0 immediately before executing an algorithm and flops immediately after gives the op count for the algorithm. For example, the number of ops required to solve a given linear system via Gaussian elimination can be obtained with:
flops 0 , x = A b; flops
The elapsed time in seconds can be obtained with the stopwatch timers tic and toc; tic starts the timer and toc returns the elapsed time. Hence, the commands tic, any statement, toc will return the elapsed time for execution of the statement. The elapsed time for solving the linear system above can be obtained, for example, with: You may wish to compare this time|and op count|with that for solving the system using x = inv A *b;. Try it. It should be noted that, on timesharing machines, elapsed time may not be a reliable measure of the e ciency of an algorithm since the rate of execution depends on how busy the computer is at the time. While all computations in MATLAB are performed in double precision, the format of the displayed output can be controlled by the following commands.
format format format format format format format format short long short e long e rat hex bank + tic, x = A b; toc
n
n
16. Output format.
xed point with 4 decimal places the default xed point with 14 decimal places scienti c notation with 4 decimal places scienti c notation with 15 decimal places approximation by ratio of small integers hexadecimal format xed dollars and cents +, -, blank
Once invoked, the chosen format remains in e ect until changed. The command format compact will suppress most blank lines allowing more information to be placed on the screen or page. The command format loose returns to the non-compact format. These commands are independent of the other format commands.
14
Hardcopy is most easily obtained with the diary command. The command diary lename causes what appears subsequently on the screen except graphics to be written to the named disk le if the lename is omitted it will be written to a default le named diary until one gives the command diary off; the command diary on will cause writing to the le to resume, etc. When nished, you can edit the le as desired and print it out on the local system. The !-feature see section 14 will permit you to edit and print the le without leaving MATLAB. MATLAB can produce planar plots of curves, 3-D plots of curves, 3-D mesh surface plots, and 3-D faceted surface plots. The primary commands for these facilities are plot, plot3, mesh, and surf, respectively. An introduction to each of these is given below. To preview some of these capabilities, enter the command demo and select some of the graphics options.
17. Hardcopy.
18. Graphics.
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: 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 sin operates entrywise . You will usually want to keep the current graphics window gure" exposed|but moved to the side|and the command window active. One can have several graphics gures, one of which will at any time be the designated current" gure where graphs from subsequent plotting commands will be placed. If, for example, gure 1 is the current gure, then the command figure 2 or simply figure will open a second gure if necessary and make it the current gure. The command figure 1 will then expose gure 1 and make it again the current gure. The command gcf will return the number of the current gure. As a second example, you can draw the graph of y = e,x2 over the interval -1.5 to 1.5 as follows: Note that one must precede b by a period to ensure that it operates entrywise see section 3 . MATLAB supplies a function fplot to easily and e ciently plot the graph of a function. For example, to plot the graph of the function above, one can rst de ne the function in an M- le called, say, expnormal.m containing 15
x = -1.5:.01:1.5; y = exp -x.b 2 ; plot x,y x = -4:.01:4; y = sin x ; plot x,y
Then the command
function y = expnormal x y = exp -x.b 2 ; fplot 'expnormal', -1.5,1.5
will produce the graph. Try it. Plots of parametrically de ned 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 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 place text on the graph using the mouse text position text at speci ed coordinates For example, the command gives a graph a title. The command gtext 'The Spot' allows one to interactively place the designated text on the current graph by placing the mouse pointer at the desired position and clicking the mouse. To place text in a graph at designated coordinates, one would use the command text see help text . The command grid will place grid lines on the current graph. By default, the axes are auto-scaled. This can be overridden by the command axis. Some features of axis are: axis xmin ,xmax ,ymin ,ymax set axis scaling to prescribed limits axis axis freezes scaling for subsequent graphs axis auto returns to auto-scaling v = axis returns vector v showing current scaling axis square same scale on both axes axis equal same scale and tic marks on both axes axis off turns o axis scaling and tic marks axis on turns on axis scaling and tic marks The axis command should be given after the plot command. Two ways to make multiple plots on a single graph are illustrated by and by forming a matrix Y containing the functional values as columns
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 title 'Best Least Squares Fit'
Another way is with hold. The command hold on freezes the current graphics screen so that subsequent plots are superimposed on it. The axes may, however, become rescaled. Entering hold off releases the hold." 16
One can override the default linetypes, pointtypes and colors. For example, renders a dashed line and dotted line for the rst two graphs while for the third the symbol + is placed at each node. The line- and mark-types are Linetypes: solid - , dashed -- . dotted : , dashdot -. Marktypes: point . , plus + , star * , circle o , x-mark x Colors can be speci ed for the line- and mark-types. Colors: yellow y , magenta m , cyan c , red r green g , blue b , white w , black k For example, plot x,y,'r--' plots a red dashed line. The command subplot can be used to partition the screen so that several small plots can be placed in one gure. See help subplot. Other specialized 2-D plotting functions you may wish to explore via help are:
x=0:.01:2*pi; y1=sin x ; y2=sin 2*x ; y3=sin 4*x ; plot x,y1,'--',x,y2,':',x,y3,'+'
Graphics hardcopy
polar, bar, hist, quiver, compass, feather, rose, stairs, fill
A hardcopy of the current graphics gure can be most easily obtained with the MATLAB command print. Entered by itself, it will send a high-resolution copy of the current graphics gure to the default printer. The printopt M- le is used to specify the default setting used by the print command. If desired, one can change the defaults by editing this le see help printopt . The command print lename saves the current graphics gure to the designated lename in the default le format. If lename has no extension, then an appropriate extension such as .ps, .eps, or .jet is appended. If, for example, PostScript is the default le format, then will create a PostScript le lissajous.ps of the current graphics gure which can subsequently be printed using the system print command. If filename already exists, it will be overwritten unless you use the -append option. The command will append the hopefully di erent current graphics gure to the existing le lissajous.ps. In this way one can save several graphics gures in a single le. The default settings can, of course, be overwritten. For example, will save to an Encapsulated PostScript le saddle.eps the graphics gure 3 | even if it is not the current gure.
print -deps -f3 saddle print -append lissajous print lissajous
17
3-D line plots.
Completely analogous to plot in two dimensions, the command plot3 produces curves in three dimensional space. If x, y, and z are three vectors of the same size, then the command plot3 x,y,z will produce a perspective plot of the piecewise linear curve in 3-space passing through the points whose coordinates are the respective elements of x, y, and z. These vectors are usually de ned parametrically. For example, will produce a helix which is compressed near the x-y plane a slinky" . Try it. Just as for planar plots, a title and axis labels including zlabel can be added. The features of axis command described there also hold for 3-D plots; setting the axis scaling to prescribed limits will, of course, now require a 6-vector. Three dimensional wire mesh surface plots are drawn with the command mesh. The command mesh z creates a three-dimensional perspective plot of the elements of the matrix z. The mesh surface is de ned by the z-coordinates of points above a rectangular grid in the x-y plane. Try mesh eye 10 . Similarly, three dimensional faceted surface plots are drawn with the command surf. Try surf eye 10 . To draw the graph of a function z = f x; y over a rectangle, one rst de nes vectors xx and yy which give partitions of the sides of the rectangle. With the function meshgrid 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: One then computes a matrix z, obtained by evaluating f entrywise over the matrices x and y, to which mesh or surf can be applied. You can, for example, draw the graph of z = e,x2 ,y2 over the square ,2; 2 ,2; 2 as follows try it :
xx = -2:.2:2; yy = xx; x,y = meshgrid xx,yy ; z = exp -x.b 2 - y.b 2 ; mesh z x,y x,y = meshgrid xx,yy ; t=.01:.01:20*pi; x=cos t ; y=sin t ; z=t.b 3; plot3 x,y,z
3-D mesh and surface plots.
One could, of course, replace the rst three lines of the preceding with Try this plot with surf instead of mesh. As noted above, the features of the axis command described in the section on planar plots also hold for 3-D plots as do the commands for titles, axes labelling and the command hold. The color shading of surfaces is set by the shading command. There are three settings for shading: faceted default , interpolated, and flat. These are set by the commands 18
= meshgrid -2:.2:2, -2:.2:2 ;
or shading flat Note that on surfaces produced by surf, the settings interpolated and flat remove the superimposed mesh lines. Experiment with various shadings on the surface produced above. The command shading as well as colormap and view below should be entered after the surf command. The color pro le of a surface is controlled by the colormap command. Available prede ned colormaps include: hsv default , hot, cool, jet, pink, copper, flag, gray, bone The command colormap cool will, for example, set a certain color pro le for the current gure. Experiment with various colormaps on the surface produced above. The command view can be used to specify in spherical or cartesian coordinates the viewpoint from which the 3-D object is to be viewed. See help view. The MATLAB function peaks generates an interesting surface on which to experiment with shading, colormap, and view. Plots of parametrically de ned surfaces can also be made. The MATLAB functions sphere and cylinder will generate such plots of the named surfaces. See type sphere and type cylinder. The following is an example of a similar function which generates a plot of a torus.
shading faceted, shading interp, function x,y,z = torus r,n,a TORUS Generate a torus torus r,n,a generates a plot of a torus with central radius a and lateral radius r. n controls the number of facets on the surface. These input variables are optional with defaults r = 0.5, n = 30, a = 1. x,y,z = torus r,n,a generates three n+1 -by- n+1 matrices so that surf x,y,z will produce the torus. See also SPHERE, CYLINDER if nargin 3, a = 1; end if nargin 2, n = 30; end if nargin 1, r = 0.5; end theta = pi* 0:2:2*n n; phi = 2*pi* 0:2:n ' n; xx = a + r*cos phi *cos theta ; yy = a + r*cos phi *sin theta ; zz = r*sin phi *ones size theta ; if nargout == 0 surf xx,yy,zz ar = a + r sqrt 2 ; axis -ar,ar,-ar,ar,-ar,ar else
19
x = xx; y = yy; z = zz; end
Other 3-D plotting functions you may wish to explore via help are:
Handle Graphics.
meshz, surfc, surfl, contour, pcolor
Beyond those described above, MATLAB's graphics system provides low level functions which permit one to control virtually all aspects of the graphics environment to produce sophisticated plots. Enter the command set 1 and gca,set ans to see some of the properties of gure 1 which one can control. This system is called Handle Graphics, for which one is referred to the MATLAB User's Guide.
19. Sparse Matrix Computations.
In performing matrix computations, MATLAB normally assumes that a matrix is dense; that is, any entry in a matrix may be nonzero. If, however, a matrix contains su ciently many zero entries, computation time could be reduced by avoiding arithmetic operations on zero entries and less memory could be required by storing only the nonzero entries of the matrix. This increase in e ciency in time and storage can make feasible the solution of signi cantly larger problems than would otherwise be possible. MATLAB provides the capability to take advantage of the sparsity of matrices. Matlab has two storage modes, full and sparse, with full the default. The functions full and sparse convert between the two modes. For a matrix A, full or sparse, nnz A returns the number of nonzero elements in A. A sparse matrix is stored as a linear array of its nonzero elements along with their row and column indices. If a full tridiagonal matrix F is created via, say, then the statement S = sparse F will convert F to sparse mode. Try it. Note that the output lists the nonzero entries in column major order along with their row and column indices. The statement F = full S restores S to full storage mode. One can check the storage mode of a matrix A with the command issparse A . A sparse matrix is, of course, usually generated directly rather than by applying the function sparse to a full matrix. A sparse banded matrix can be easily created via the function spdiags by specifying diagonals. For example, a familiar sparse tridiagonal matrix is created by Try it. The integral vector -1,0,1 speci es in which diagonals the columns of e,d,e should be placed use full T to view . Experiment with other values of m and n and, say, -3,0,2 instead of -1,0,1 . See help spdiags for further features of spdiags.
m = 6; n = 6; e = ones n,1 ; d = -2*e; T = spdiags e,d,e , -1,0,1 ,m,n F = floor 10*rand 6 ; F = triu tril F,1 ,-1 ;
20
The sparse analogs of eye, zeros, ones, and randn for full matrices are, respectively, The latter two take a matrix argument and replace only the nonzero entries with ones and normally distributed random numbers, respectively. randn also permits the sparsity structure to be randomized. The command sparse m,n creates a sparse zero matrix. The versatile function sparse permits creation of a sparse matrix via listing its nonzero entries. Try, for example, In general, if the vector s lists the nonzero entries of S and the integral vectors i and j list their corresponding row and column indices, then will create the desired sparse m n matrix S . As another example try
n = 6; e = floor 10*rand n-1,1 ; E = sparse 2:n,1:n-1,e,n,n sparse i,j,s,m,n i = 1 2 3 4 4 4 ; j = 1 2 3 1 2 3 ; s = S = sparse i,j,s,4,3 , full S 5 6 7 8 9 10 ; speye, sparse, spones, sprandn
The arithmetic operations and most MATLAB functions can be applied independent of storage mode. The storage mode of the result? Operations on full matrices always give full results. Selected other results are S=sparse, F=full : Sparse: S+S, S*S, S.*S, S.*F, Sb n, S.b n, SnS Full: S+F, S*F, SnF, FnS Sparse: inv S , chol S , lu S , diag S , max S , sum S For sparse S , eig S is full if S is symmetric but unde ned if S is unsymmetric; svd requires a full argument. A matrix built from blocks, such as A,B;C,D , is sparse if any constituent block is sparse. You may wish to compare, for the two storage modes, the e ciency of solving a tridiagonal system of equations for, say, n = 20; 50; 500; 1000 by entering, recalling and editing the following two command lines:
n=20;e=ones n,1 ;d=-2*e; T=spdiags e,d,e , -1,0,1 ,n,n ; A=full T ; b=ones n,1 ;s=sparse b ;tic,T s;sparsetime=toc, tic,A b;fulltime=toc
n
n
21
There are many MATLAB features which cannot be included in these introductory notes. Listed below are some of the MATLAB functions and operators available, grouped by subject area1. Use the on-line help facility or consult the Reference Guide for more detailed information on the functions. There are many functions beyond these. There exist, in particular, several toolboxes" of functions for speci c areas2. Included among such are signal processing, control systems, robust-control, system identi cation, optimization, splines, chemometrics, -analysis and synthesis, state-space identi cation, neural networks, image processing, symbolic math Maple kernel , and statistics. These can be explored via the command help.
help what type lookfor which demo path cedit version whatsnew info why
Managing Commands and Functions
20. Reference.
help facility list M- les on disk list named M- le keywork search through the help entries locate functions and les run demonstrations control MATLAB's search path set parameters for command line editing and recall display MATLAB version you are running display toolbox README les info about MATLAB and The MathWorks receive ippant answer
who whos save load clear pack size length disp
Managing Variables and the Workspace
list current variables list current variables, long form save workspace variables to disk retrieve variables from disk clear variables and functions from memory consolidate workspace memory size of matrix length of vector display matrix or text
1 Source: MATLAB Reference Guide, version 4.1 2 The toolboxes, which are optional, may not be installed on your system.
22
cd pwd dir, ls delete getenv ! unix diary
Working with Files and the Operating System
change current working directory show current working directory directory listing delete le get environment variable execute operating system command execute operating system command; return result save text of MATLAB session
clc home format echo more
Controlling the Command Window
clear command window send cursor home|to top of screen set output format echo commands inside script commands control paged output in command window
quit startup matlabrc +
b
Starting and Quitting from MATLAB
terminate MATLAB M- le executed when MATLAB is started master startup M- le
Array Operators
Matrix Operators
, n
' .' kron
addition subtraction multiplication power right division left division conjugate transpose transpose Kronecker tensor product
+
, .
.b . .n
addition subtraction multiplication power right division left division
Relational and Logical Operators
= less than or equal = greater than or equal == equal = not equal
greater than
less than
&
and or not xor exclusive or
j
23
= . .. ... , ; : !
assignment statement used to form vectors and matrices; enclose multiple function output variables arithmetic expression precedence; enclose function input variables decimal point parent directory continue statement to next line separate subscripts, function arguments, statements end rows, suppress printing comments subscripting, vector generation execute operating system command
Special Characters
ans eps realmax reammin pi i, j inf NaN ops nargin nargout computer
Special Variables and Constraints
answer when expression not assigned oating point precision largest oating point number smallest positive oating point number
imaginary unit in nity Not-a-Number oating point operation count number of function input arguments number of function output arguments computer type
date clock etime tic, toc cputime
Time and Date
current date wall clock elapsed time function stopwatch timer functions elapsed CPU time
24
zeros ones eye diag toeplitz magic compan linspace logspace meshgrid rand randn hilb invhilb vander pascal hadamard hankel rosser wilkinson gallery
matrix of zeros matrix of ones identity diagonal Toeplitz magic square companion linearly spaced vectors logarithmically spaced vectors array for 3-D plots uniformly distributed random numbers normally distributed randon numbers Hilbert inverse Hilbert exact Vandermonde Pascal Hadamard Hankel symmetric eigenvalue test matrix Wilkinson's eigenvalue test matrix two small test matrices
Special Matrices
diag rot90 iplr ipud reshape tril triu .' :
create or extract diagonals rotate matrix 90 degrees ip matrix left-to-right ip matrix up-to-down change size lower triangular part upper triangular part transpose convert matrix to single column; A :
Matrix Manipulation
25
exist any all nd isnan isinf nite isieee isempty issparse isstr strcmp
check if variables or functions exist true if any element of vector is true true if all elements of vector are true nd indices of non-zero elements true for NaNs true for in nite elements true for nite elements true for IEEE oating point arithmetic true for empty matrix true for sparse matrix true for text string compare string variables
Logical Functions
if else elseif end for while break return error
conditionally execute statements used with if used with if terminate if, for, while repeat statements for a speci c number of times repeat statments while condition is true terminate execution of for or while loops return to invoking function display message and abort function
Control Flow
input keyboard menu pause function eval feval global nargchk
prompt for user input invoke keyboard as if it were a script le generate menu of choices for user input wait for user response de ne function execute string with MATLAB expression evaluate function speci ed by string de ne global variables validate number of input arguments
Programming
26
string abs blanks eval num2str int2str str2num isstr strcmp upper lower hex2num hex2dec dec2hex
about character strings in MATLAB convert string to numeric values a string of blanks evaluate string with MATLAB expression convert number to string convert integer to string convert string to number true for string variables compare string variables convert string to uppercase convert string to lowercase convert hex string to oating point number convert hex string to decimal integer convert decimal integer to hex string
Text and Strings
dbstop dbclear dbcont dbdown dbstack dbstatus dbstep dbtype dbup dbdown dbquit
set breakpoint remove breakpoint remove execution change local workspace context list who called whom list all breakpoints execute one or more lines list M- le with line numbers change local workspace context opposite of dbup quit debug mode
Debugging
saxis sound auread auwrite lin2mu mu2lin
Sound Processing Functions
sound axis scaling convert vector to sound Read Sun audio le Write Sun audio le linear to mu-law conversion mu-law to linear conversion
27
abs angle sqrt real imag conj gcd lcm round x oor ceil sign rem exp log log10
Elementary Math Functions
absolute value or complex magnitude phase angle square root real part imaginary part complex conjugate greatest common divisor least common multiple round to nearest integer round toward zero round toward ,1 round toward 1 signum function remainder exponential base e natural logarithm log base 10
sin, asin, sinh, asinh cos, acos, cosh, acosh tan, atan, tanh, atanh cot, acot, coth, acoth sec, asec, sech, asech csc, acsc, csch, acsch
sine, arcsine, hyperbolic sine, hyperbolic arcsine cosine, arccosine, hyperbolic cosine, hyperbolic arccosine tangent, arctangent, hyperbolic tangent, hyperbolic arctangent cotangent, arccotangent, hyperbolic cotan., hyperbolic arccotan. secant, arcsecant, hyperbolic secant, hyperbolic arcsecant cosecant, arccosecant, hyperbolic cosecant, hyperbolic arccosecant
Special Functions
Trigonometric Functions
bessel beta gamma rat rats erf erfinv ellipke ellipj expint log2 pow2
bessel function beta function gamma function rational approximation rational output error function inverse error function complete elliptic integral Jacobian elliptic integral exponential integral dissect oating point numbers scale oating point numbers
28
inv lu rref chol qr nnls lscov null orth eig hess schur cdf2rdf rsf2csf balance qz polyeig svd pinv
Matrix Decompositions and Factorizations
inverse factors from Gaussian elimination reduced row echelon form Cholesky factorization orthogonal-triangular decomposition nonnegative least squares least squares in presence of know covariance null space orthogonalization eigenvalues and eigenvectors Hessenberg form Schur decomposition complex diagonal form to real block diagonal form real block diagonal form to complex diagonal form diagonal scaling for eigenvalue accuracy generalized eigenvalues polynomial eigenvalue solver singular value decomposition pseudoinverse
cond rcond condest norm normest rank
condition number in 2-norm LINPACK reciprocal condition number estimator Hager Higham condition number estimator 1-norm,2-norm,F-norm,1-norm 2-norm estimator rank
Matrix Conditioning
expm expm1 expm2 expm3 logm sqrtm funm poly det trace
matrix exponential M- le implementation of expm matrix exponential via Taylor series matrix exponential via eigenvalues and eigenvectors matrix logarithm matrix square root evaluate general matrix function characteristic polynomial determinant trace
Elementary Matrix Functions
29
poly roots roots1 polyval polyvalm conv deconv residue poly t polyder
construct polynomial with speci ed roots polynomial roots|companion matrix method polynomial roots|Laguerre's method evaluate polynomial evaluate polynomial with matrix argument multiply polynomials divide polynomials partial-fraction expansion residues t polynomial to data di erentiate polynomial
Column-wise Data Analysis
Polynomials
max min mean median std sort sum prod cumsum cumprod hist abs angle conv deconv corrcoef cov lter lter2 cplxpair unwrap nextpow2 t t2 i t i t2 tshift
largest component smallest component average or mean value median value standard deviation sort in ascending order sum of elements product of elements cumulative sum of elements cumulative product of elements histogram
complex magnitude phase angle convolution and polynomial multiplication deconvolution and polynomial division correlation coe cients covariance matrix one-dimensional digital lter two-dimensional digital lter sort numbers into complex pairs remove phase angle jumps across 360 boundaries next higher power of 2 radix-2 fast Fourier transform two-dimensional FFT inverse fast Fourier transform inverse 2-D FFT zero-th lag to center of spectrum
Signal Processing
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di gradient del2 subspace spline interp1 interp2 interpft griddata
Finite Differences and Data Interpolation
approximate derivatives approximate gradient ve point discrete Laplacian angle between two subspaces cubic spline interpolation 1-D data interpolation 2-D data interpolation 1-D data interpolation via FFT method data gridding
quad quad8 trapz
Numerical Integration
adaptive 2-panel Simpson's Rule adaptive 8-panel Newton-Cotes Rule trapezoidal method
ode23 2nd 3rd order Runge-Kutta method ode23p solve via ode23, displaying plot ode45 4th 5th order Runge-Kutta-Fehlberg method
Nonlinear Equations and Optimization
Differential Equation Solution
fmin fmins fsolve fzero fplot
minimize function of one variable minimize function of several variables solution to a system of nonlinear equations nd zeros of a function of several variables nd zero of function of one variable plot graph of a function
31
plot loglog semilogx semilogy ll polar bar stairs errorbar hist rose compass feather fplot
Two Dimensional Graphs
linear plot log-log scale plot semilog scale plot semilog scale plot draw lled 2-D polygons polar coordinate plot bar graph stairstep plot error bar plot histogram plot angle histogram plot compass plot feather plot plot function
title xlabel ylabel zlabel grid text gtext ginput
Graph Annotation
graph title x-axis label y-axis label z-axis label for 3-D plots grid lines text annotation mouse placement of text graphical input from mouse
Figure Window Axis Creation and Control
gure gcf clf close hold ishold subplot axes gca axis caxis
create gure graph window get handle to current gure clear current gure close gure hold current graph return hold status create axes in tiled positions create axes in arbitrary positions get handle to to current axes control axis scaling and appearance control pseudocolor axis scaling
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print print graph or save graph to le printopt con gure local printer defaults orient set paper orientation
Three Dimensional Graphs
Graph Hardcopy and Storage
mesh meshc meshz surf surfc surfl plot3 ll3 contour contour3 clabel contourc pcolor quiver image waterfall slice
3-D mesh surface combination mesh contour plot 3-D mesh with zero plane 3-D shaded surface combination surface contour plot 3-D shaded surface with lighting plot lines and points in 3-D space draw lled 3-D polygons in 3-D space contour plot 3-D contour plot contour plot elevation labels contour plot computation used by contour pseudocolor checkerboard plot quiver plot display image waterfall plot volumetric visualization plot
3-D Graph Appearance
view viewmtx hidden shading axis caxis specular di use surfnorm colormap brighten spinmap rgbplot hsv2rgb rgb2hsv
3-D graph viewpoint speci cation view transformation matrices mesh hidden line removal mode color shading mode axis scaling and apearance pseudocolor axis scaling specular re ectance di use re ectance surface normals color lookup table see below brighten or darken color map spin color map plot colormap hsv to rgb color map conversion rgb to hsv color map conversion
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hsv jet gray hot cool bone copper pink ag
hue-saturation-value default variant of hsv linear gray-scale black-red-yellow-white shades of cyan and magenta gray-scale with tinge of blue linear copper tone pastel shades of pink alternating red, white, blue, and black
3-D Objects
Color Maps
sphere generate sphere cylinder generate cylinder peaks generate demo surface
Movies and Animation
moviein getframe movie
initialize movie frame memory get movie frame play recorded movie frames
gure axes line text patch surface image uicontrol uimenu
Handle Graphics Objects
create gure window create axes create line create text create patch create surface create image create user interface control create user interface menu
set get reset delete drawnow
Handle Graphics Operations
set object properties get object properties reset object properties delete object ush pending graphics events
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spdiags speye sprandn spones sprandsym spfun sparse full nd spconvert issparse nnz nonzeros nzmax spalloc spy gplot colmmd colperm dmperm randperm symmmd symrcm condest normest sprank spaugment spparms symbfact sparsefun
sparse matrix formed from diagonals sparse identity matrix sparse random matrix replace nonzero entries with ones sparse symmetric random matrix apply function to nonzero entries create sparse matrix; convert full matrix to sparse convert sparse matrix to full matrix nd indices of nonzero entries convert from sparse matrix external format true if matrix is sparse number of nonzero entries nonzero entries amount of storage allocated for nonzero entries allocate memory for nonzero entries visualize sparsity structure plot graph, as in graph theory" column minimum degree order columns based on nonzero count Dulmage-Mendelsohn decomposition random permutation vector symmetric minimum degree reverse Cuthill-McKee ordering estimate 1-norm condition estimate 2-norm structural rank form least squares augmented system set parameters for sparse matrix routines symbolic factorization analysis sparse auxillary functions and parameters
Sparse Matrix Functions
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