EECE 340 Tutorial, Spring 2011

MATLAB Primer Second Edition Kermit Sigmon Department of Mathematics University of Florida Department of Mathematics e University of Florida e Gainesville, FL 32611 sigmon@math.ufl.edu e sigmon@ufpine.bitnet Copyright ©1989, 1992 by Kermit Sigmon Copyright ©1989, 1992 by Kermit Sigmon The MATLAB Primer may be distributed as desired subject to the following con- ditions: 1. It may not be altered in any way, except possibly adding an addendum giving information about the local computer installation. 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. Users usually appreciate the convenience and durability of a bound copy with a cover, copy center style. Department of Mathematics e University of Florida e Gainesville, FL 32611 sigmon@math.ufl.edu e sigmon@ufpine.bitnet INTRODUCTION MATLAB is an interactive, matrix-based system for scientific and engineering calcu- lations. You can solve complex numerical problems without actually writing a program. The name MATLAB is an abbreviation for MATrix LABoratory. The purpose of this Primer is to help you begin to use MATLAB. They can best be used hands-on. You are encouraged to work at the computer as you read the Primer and freely experiment with examples. You should liberally use the on-line help facility for more detailed information. After entering MATLAB as described in section 1, the command help will display a list of functions for which on-line help is available; the command help functionname will give information about a specific function. The command help eig, for example, will give information about the eigenvalue function eig. You can preview some of the features of MATLAB by entering the command demo. 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. The documentation which accompanies the Student Edition of MATLAB is an excellent source. Copies of the complete User’s Guide 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 the User’s Guides are 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 com- puters, 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. While the second edition of the Primer is based on version 3.5 of MATLAB, it is compatible with version 4.0 with the fundamental differences noted. An edition of the Primer based on version 4.0 is under development. The plain T&X source (and a PostScript file primer35.ps) of this second edition of the MATLAB Primer is available via anonymous ftp from math.ufl.edu as the file primer35.tex in the directory pub/matlab. If ftp is unavailable to you, it can be obtained via listserv by sending an email message to listserv@math.uf1l.edu which contains the single line send matlab/primer35.tex. It can also be obtained by sending a request to the author at sigmon@math.ufl.edu. The latest edition of the Primer will always be available as indicated above as the file primer.tex. A Spanish version is also available there. MATLAB is licensed by The MathWorks, Inc., Cochituate Place, 24 Prime Park Way, Natick, MA 01760, (508)653-1415, Fax: (508)653-2997, Email: info@mathworks.com. Copyright ©1989, 1992 by Kermit Sigmon (6-93) ii CONTENTS Page 1. Accessing MATLAB ......... 0. ccc cece cece cece nent ence etn eben ene ene nen eneeeee 1 2. Entering matrices ....... 0. eee e eee eee beeen ene e nent eens ees 1 3. Matrix operations, array operations ........... 00. c eect t een ene ene ees 2 4. Statements, expressions, variables; saving a session ........... 0. cece eee eee eens 3 5. Matrix building functions .......... 0... nett e een eee eens 3 6. For, while, if — and relations .......... 0... c ccc cece cence cence tence nen tenes 4 7. Scalar functions 2.0.0.0... cent eee eee n ete e teens 6 8. Vector functions 1.2... 6. ccc nee n nen en tet e bene n ene en ences 6 9. Matrix functions 2.0.0... 0c n tenn eee beeen eae 7 10. Command line editing and recall ... 0.0.0.0 nce cece eee 7 11. Submatrices and colon notation .......... 0... c cece cece eee cence cnet een en ences 8 12. M-files 20.0... ence nnn een een tne e eee n eee en eens 8 13. Text strings, error messages, input .......... 0... ccc eect nen eee eens 12 14. Managing M-files 2.0.0... cece nee nen n ene e nent bene neeaes 12 15. Comparing efficiency of algorithms: flops and etime ..................0 020s ee eee 13 16. Output format 2.0.0... ccc nnn nee nee n teen e nnn enn en eee 13 17. Hard copy ...... ccc een eee nee nee n eee een ene e eee 14 18. Graphics 2.2.0.0... ccc cece nen nee e nent e tenn ent en ene eeeeneeaee 14 19. Reference 2.0... 0... cece nee n eee eben tebe enna nes 17 ili 1. Accessing MATLAB. On most systems, after logging in one can enter MATLAB with the system command matlab and exit MATLAB with the command exit or quit. On a PC, for example, if properly installed, one may enter MATLAB with the command: Cc> matlab and exit it with the command: >> quit On systems permitting multiple processes, such as a Unix system, you will find it convenient, for reasons discussed in section 14, to keep both MATLAB and your local editor active. If you are working on a workstation which runs processes in multiple windows, you will want to keep MATLAB active in one window and your local editor active in another. You should consult your instructor or your local computer center for details of the local installation. 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 different ways: e Entered by an explicit list of elements, e Generated by built-in statements and functions, e Created in M-files (see sections 12 and 14 below), e Loaded from external data files (see User’s Guide). For example, either of the statements A= (123; 456; 789] and A=[ 123 456 789] 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. Listing entries of a large matrix is best done in an M-file, where errors can be easily edited away (see sections 12 and 14). The built-in functions rand, magic, and hilb, for example, provide an easy way to create matrices with which to experiment. The command rand (n) will create an n x n matrix with randomly generated entries distributed uniformly between 0 and 1, while 1 rand(m,n) will create an m x n one. magic(n) will create an integral n x n matrix which is a magic square (rows and columns have common sum); hilb(n) will create the n x n Hilbert matrix, the king of ill-conditioned matrices (m and n denote, of course, positive integers). Matrices can also be generated with a for-loop (see section 6 below). Individual matrix and vector entries can be referenced with indices inside parentheses in the usual manner. For example, A(2,3) denotes the entry in the second row, third column of matrix A and (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: + addition - subtraction * multiplication i power / 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 6 is a compatible column, resp. row, vector, then x = A\b is the solution of A * x = b and, resp., xz = 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 «a = 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 defined in terms of left division by b/A = (A‘\b’)’. Array operations. The matrix operations of addition and subtraction already op- erate entry-wise but the other matrix operations given above do not—they are matrix operations. It is important to observe that these other operations, x, ~, \, 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].72 will yield [1,4,9,16]. Try it. This is par- ticularly useful when using Matlab graphics. 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. Eval- uation of the expression produces a matrix, which is then displayed on the screen and assigned to the variable for future use. If the variable name and = sign are omitted, a variable ans (for answer) is automatically created to which the result is assigned. A statement is normally terminated with the carriage return. However, a statement can be continued to the next line with three or more periods followed by a carriage return. On the other hand, several statements can be placed on a single line if separated by commas or semicolons. If the last character of a statement is a semicolon, the printing is suppressed, but the assignment is carried out. This is essential in suppressing unwanted printing of intermediate results. MATLAB is case-sensitive in the names of commands, functions, and variables. For example, solveUT is not the same as solveut. The command who will list the variables currently in the workspace. A variable can be cleared from the workspace with the command clear variablename. The command clear alone will clear all nonpermanent variables. The permanent variable eps (epsilon) gives the machine precision—about 10~'° on most machines. It is useful in determining tolerences for convergence of iterative processes. A runaway display or computation can be stopped on most machines without leaving MATLAB with CTRL-C (CTRL-BREAK on a PC). Saving a session. When one logs out or exits MATLAB all variables are lost. How- ever, invoking the command save before exiting causes all variables to be written to a non-human-readable diskfile named matlab.mat. When one later reenters MATLAB, the command load will restore the workspace to its former state. 5. Matrix building functions. Convenient matrix building functions are eye identity matrix ZeTOS matrix of zeros ones matrix of ones diag see below triu upper triangular part of a matrix tril lower triangular part of a matrix rand randomly generated matrix hilb Hilbert matrix magic magic square toeplitz see help toeplitz 3 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(A) produces a matrix of zeros of 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 B = (A, zeros(3,2); zeros(2,3), eye(2)] will build a certain 5-by-5 matrix. Try it. 6. For, while, if — and relations. In their basic forms, these MATLAB flow control statements operate like those in most computer languages. For. For example, for a given n, the statement x = [1]; fori iin, x=[x,i7~ 2], end or x= (1; for i = 1i:n x = [x,i72] end will produce a certain n-vector and the statement x = []; for i = n:-1:1, x=[x,i72], end will produce the same vector in reverse order. Try them. Note that a matrix may be empty (such as x = []). The statements for i = 1i:m for j = i:n H(i, j) = 1/(itj-1); end end H will produce and print to the screen the m-by-n hilbert matrix. The semicolon on the inner statement suppresses printing of unwanted intermediate results while the last H displays the final result. 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 exam- ple, for a given number a, the following will compute and display the smallest nonnegative integer n such that 2” > a: n= 0; while 2n<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 ifn<0 parity = 0; elseif rem(n,2) == parity = 2; else parity = 1; end In two-way branching the elseif portion would, of course, be omitted. Relations. The relational operators in MATLAB are < less than > greater than <= less than or equal = greater than or equal == equal v= not equal. Note that “=” is used in an assignment statement while “==” is used in a relation. Relations may be connected or quantified by the logical operators & and | or ~ not. When applied to scalars, a relation is actually the scalar 1 or 0 depending on whether the relation is true or false. Try 3 < 5, 3 > 5, 3 == 5, and 3 == 3. When applied to matrices of the same size, a relation is a matrix of 0’s and L’s giving the value of the relation between corresponding entries. Try a = rand(5), b = triu(a), a == 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 == statement end but if you wish to execute statement when A and B are not equal, you would type if any(any(A ~= B)) statement end or, more simply, if A == B else statement end Note that the seemingly obvious if A ~= 8B, statement, end will not give what is intended since statement would execute only if each of the correspond- ing entries of A and B differ. 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). The for statement permits any matrix to be used instead of 1:n. See the User’s Guide for details of how this feature expands the power of the for statement. 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 floor tan atan rem (remainder) sign ceil 8. Vector functions. 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.