Introduction to MATLAB for Engineering Students, Exercises of Engineering

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INTRODUCTION TO MATLAB FOR ENGINEERING STUDENTS David Houcque Northwestern University Contents 1 Tutorial lessons 1 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Basic features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 A minimum MATLAB session . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.1 Starting MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.2 Using MATLAB as a calculator . . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Quitting MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 Creating MATLAB variables . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.2 Overwriting variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.3 Error messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.4 Making corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.5 Controlling the hierarchy of operations or precedence . . . . . . . . . 6 1.4.6 Controlling the appearance of floating point number . . . . . . . . . . 8 1.4.7 Managing the workspace . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.8 Keeping track of your work session . . . . . . . . . . . . . . . . . . . 9 1.4.9 Entering multiple statements per line . . . . . . . . . . . . . . . . . . 10 1.4.10 Miscellaneous commands . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.11 Getting help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Tutorial lessons 2 12 2.1 Mathematical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 i 7.2.1 Preparing for debugging . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.2.2 Setting breakpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.2.3 Running with breakpoints . . . . . . . . . . . . . . . . . . . . . . . . 52 7.2.4 Examining values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2.5 Correcting and ending debugging . . . . . . . . . . . . . . . . . . . . 53 7.2.6 Ending debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2.7 Correcting an M-file . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 A Summary of commands 55 B Release notes for Release 14 with Service Pack 2 60 B.1 Summary of changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 B.2 Other changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 B.3 Further details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 C Main characteristics of MATLAB 64 C.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 C.2 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 C.3 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 C.4 Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 iv List of Tables 1.1 Basic arithmetic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Hierarchy of arithmetic operations . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Elementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Predefined constant values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Attributes for plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1 Elementary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Special matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Array operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Summary of matrix and array operations . . . . . . . . . . . . . . . . . . . . 32 4.3 Matrix functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1 Anatomy of a M-File function . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 Main differences between scripts and functions . . . . . . . . . . . . . . . . . 40 5.3 Example of input and output arguments . . . . . . . . . . . . . . . . . . . . 41 5.4 disp and fprintf commands . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.5 Conversion codes for output command fprintf . . . . . . . . . . . . . . . . 43 6.1 Relational and logical operators . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.2 Operator precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A.1 Arithmetic operators and special characters . . . . . . . . . . . . . . . 55 A.2 Array operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 A.3 Relational and logical operators . . . . . . . . . . . . . . . . . . . . . . 56 A.4 Managing workspace and file commands . . . . . . . . . . . . . . . . . 57 v A.5 Predefined variables and math constants . . . . . . . . . . . . . . . . . 57 A.6 Elementary matrices and arrays . . . . . . . . . . . . . . . . . . . . . . 58 A.7 Arrays and Matrices: Basic information . . . . . . . . . . . . . . . . . 58 A.8 Arrays and Matrices: operations and manipulation . . . . . . . . . . 58 A.9 Arrays and Matrices: matrix analysis and linear equations . . . . . 59 vi Acknowledgements I would like to thank Dean Stephen Carr for his constant support. I am grateful to a number of people who offered helpful advice and comments. I want to thank the EA1 instructors (Fall Quarter 2004), in particular Randy Freeman, Jorge Nocedal, and Allen Taflove for their helpful reviews on some specific parts of the document. I also want to thank Malcomb MacIver, EA3 Honors instructor (Spring 2005) for helping me to better understand the animation of system dynamics using MATLAB. I am particularly indebted to the many students (340 or so) who have used these materials, and have communicated their comments and suggestions. Finally, I want to thank IT personnel for helping setting up the classes and other computer related work: Rebecca Swierz, Jesse Becker, Rick Mazec, Alan Wolff, Ken Kalan, Mike Vilches, and Daniel Lee. About the author David Houcque has more than 25 years’ experience in the modeling and simulation of struc- tures and solid continua including 14 years in industry. In industry, he has been working as R&D engineer in the fields of nuclear engineering, oil rig platform offshore design, oil reser- voir engineering, and steel industry. All of these include working in different international environments: Germany, France, Norway, and United Arab Emirates. Among other things, he has a combined background experience: scientific computing and engineering expertise. He earned his academic degrees from Europe and the United States. Here at Northwestern University, he is working under the supervision of Professor Brian Moran, a world-renowned expert in fracture mechanics, to investigate the integrity assess- ment of the aging highway bridges under severe operating conditions and corrosion. ix Chapter 1 Tutorial lessons 1 1.1 Introduction The tutorials are independent of the rest of the document. The primarily objective is to help you learn quickly the first steps. The emphasis here is “learning by doing”. Therefore, the best way to learn is by trying it yourself. Working through the examples will give you a feel for the way that MATLAB operates. In this introduction we will describe how MATLAB handles simple numerical expressions and mathematical formulas. The name MATLAB stands for MATrix LABoratory. MATLAB was written originally to provide easy access to matrix software developed by the LINPACK (linear system package) and EISPACK (Eigen system package) projects. MATLAB [1] is a high-performance language for technical computing. It integrates computation, visualization, and programming environment. Furthermore, MATLAB is a modern programming language environment: it has sophisticated data structures, contains built-in editing and debugging tools, and supports object-oriented programming. These factors make MATLAB an excellent tool for teaching and research. MATLAB has many advantages compared to conventional computer languages (e.g., C, FORTRAN) for solving technical problems. MATLAB is an interactive system whose basic data element is an array that does not require dimensioning. The software package has been commercially available since 1984 and is now considered as a standard tool at most universities and industries worldwide. It has powerful built-in routines that enable a very wide variety of computations. It also has easy to use graphics commands that make the visualization of results immediately available. Specific applications are collected in packages referred to as toolbox. There are toolboxes for signal processing, symbolic computation, control theory, simulation, optimiza- tion, and several other fields of applied science and engineering. In addition to the MATLAB documentation which is mostly available on-line, we would 1 recommend the following books: [2], [3], [4], [5], [6], [7], [8], and [9]. They are excellent in their specific applications. 1.2 Basic features As we mentioned earlier, the following tutorial lessons are designed to get you started quickly in MATLAB. The lessons are intended to make you familiar with the basics of MATLAB. We urge you to complete the exercises given at the end of each lesson. 1.3 A minimum MATLAB session The goal of this minimum session (also called starting and exiting sessions) is to learn the first steps: • How to log on • Invoke MATLAB • Do a few simple calculations • How to quit MATLAB 1.3.1 Starting MATLAB After logging into your account, you can enter MATLAB by double-clicking on the MATLAB shortcut icon (MATLAB 7.0.4) on your Windows desktop. When you start MATLAB, a special window called the MATLAB desktop appears. The desktop is a window that contains other windows. The major tools within or accessible from the desktop are: • The Command Window • The Command History • The Workspace • The Current Directory • The Help Browser • The Start button 2 Table 1.1: Basic arithmetic operators Symbol Operation Example + Addition 2 + 3 − Subtraction 2− 3 ∗ Multiplication 2 ∗ 3 / Division 2/3 1.3.3 Quitting MATLAB To end your MATLAB session, type quit in the Command Window, or select File −→ Exit MATLAB in the desktop main menu. 1.4 Getting started After learning the minimum MATLAB session, we will now learn to use some additional operations. 1.4.1 Creating MATLAB variables MATLAB variables are created with an assignment statement. The syntax of variable as- signment is variable name = a value (or an expression) For example, >> x = expression where expression is a combination of numerical values, mathematical operators, variables, and function calls. On other words, expression can involve: • manual entry • built-in functions • user-defined functions 5 1.4.2 Overwriting variable Once a variable has been created, it can be reassigned. In addition, if you do not wish to see the intermediate results, you can suppress the numerical output by putting a semicolon (;) at the end of the line. Then the sequence of commands looks like this: >> t = 5; >> t = t+1 t = 6 1.4.3 Error messages If we enter an expression incorrectly, MATLAB will return an error message. For example, in the following, we left out the multiplication sign, *, in the following expression >> x = 10; >> 5x ??? 5x | Error: Unexpected MATLAB expression. 1.4.4 Making corrections To make corrections, we can, of course retype the expressions. But if the expression is lengthy, we make more mistakes by typing a second time. A previously typed command can be recalled with the up-arrow key ↑. When the command is displayed at the command prompt, it can be modified if needed and executed. 1.4.5 Controlling the hierarchy of operations or precedence Let’s consider the previous arithmetic operation, but now we will include parentheses. For example, 1 + 2× 3 will become (1 + 2)× 3 >> (1+2)*3 ans = 9 and, from previous example 6 >> 1+2*3 ans = 7 By adding parentheses, these two expressions give different results: 9 and 7. The order in which MATLAB performs arithmetic operations is exactly that taught in high school algebra courses. Exponentiations are done first, followed by multiplications and divisions, and finally by additions and subtractions. However, the standard order of precedence of arithmetic operations can be changed by inserting parentheses. For example, the result of 1+2×3 is quite different than the similar expression with parentheses (1+2)×3. The results are 7 and 9 respectively. Parentheses can always be used to overrule priority, and their use is recommended in some complex expressions to avoid ambiguity. Therefore, to make the evaluation of expressions unambiguous, MATLAB has estab- lished a series of rules. The order in which the arithmetic operations are evaluated is given in Table 1.2. Table 1.2: Hierarchy of arithmetic operations Precedence Mathematical operations First The contents of all parentheses are evaluated first, starting from the innermost parentheses and working outward. Second All exponentials are evaluated, working from left to right Third All multiplications and divisions are evaluated, working from left to right Fourth All additions and subtractions are evaluated, starting from left to right MATLAB arithmetic operators obey the same precedence rules as those in most com- puter programs. For operators of equal precedence, evaluation is from left to right. Now, consider another example: 1 2 + 32 + 4 5 × 6 7 In MATLAB, it becomes >> 1/(2+3^2)+4/5*6/7 ans = 0.7766 or, if parentheses are missing, 7 1.4.9 Entering multiple statements per line It is possible to enter multiple statements per line. Use commas (,) or semicolons (;) to enter more than one statement at once. Commas (,) allow multiple statements per line without suppressing output. >> a=7; b=cos(a), c=cosh(a) b = 0.6570 c = 548.3170 1.4.10 Miscellaneous commands Here are few additional useful commands: • To clear the Command Window, type clc • To abort a MATLAB computation, type ctrl-c • To continue a line, type . . . 1.4.11 Getting help To view the online documentation, select MATLAB Help from Help menu or MATLAB Help directly in the Command Window. The preferred method is to use the Help Browser. The Help Browser can be started by selecting the ? icon from the desktop toolbar. On the other hand, information about any command is available by typing >> help Command Another way to get help is to use the lookfor command. The lookfor command differs from the help command. The help command searches for an exact function name match, while the lookfor command searches the quick summary information in each function for a match. For example, suppose that we were looking for a function to take the inverse of a matrix. Since MATLAB does not have a function named inverse, the command help inverse will produce nothing. On the other hand, the command lookfor inverse will produce detailed information, which includes the function of interest, inv. >> lookfor inverse 10 Note - At this particular time of our study, it is important to emphasize one main point. Because MATLAB is a huge program; it is impossible to cover all the details of each function one by one. However, we will give you information how to get help. Here are some examples: • Use on-line help to request info on a specific function >> help sqrt • In the current version (MATLAB version 7), the doc function opens the on-line version of the help manual. This is very helpful for more complex commands. >> doc plot • Use lookfor to find functions by keywords. The general form is >> lookfor FunctionName 1.5 Exercises Note: Due to the teaching class during this Fall 2005, the problems are temporarily removed from this section. 11 Chapter 2 Tutorial lessons 2 2.1 Mathematical functions MATLAB offers many predefined mathematical functions for technical computing which contains a large set of mathematical functions. Typing help elfun and help specfun calls up full lists of elementary and special functions respectively. There is a long list of mathematical functions that are built into MATLAB. These functions are called built-ins. Many standard mathematical functions, such as sin(x), cos(x), tan(x), ex, ln(x), are evaluated by the functions sin, cos, tan, exp, and log respectively in MATLAB. Table 2.1 lists some commonly used functions, where variables x and y can be numbers, vectors, or matrices. Table 2.1: Elementary functions cos(x) Cosine abs(x) Absolute value sin(x) Sine sign(x) Signum function tan(x) Tangent max(x) Maximum value acos(x) Arc cosine min(x) Minimum value asin(x) Arc sine ceil(x) Round towards +∞ atan(x) Arc tangent floor(x) Round towards −∞ exp(x) Exponential round(x) Round to nearest integer sqrt(x) Square root rem(x) Remainder after division log(x) Natural logarithm angle(x) Phase angle log10(x) Common logarithm conj(x) Complex conjugate In addition to the elementary functions, MATLAB includes a number of predefined 12 1 2 3 4 5 6 −1 0 1 2 3 4 5 Figure 2.1: Plot for the vectors x and y >> x = 0:pi/100:2*pi; >> y = sin(x); >> plot(x,y) Notes: • 0:pi/100:2*pi yields a vector that – starts at 0, – takes steps (or increments) of π/100, – stops when 2π is reached. • If you omit the increment, MATLAB automatically increments by 1. 2.2.3 Adding titles, axis labels, and annotations MATLAB enables you to add axis labels and titles. For example, using the graph from the previous example, add an x- and y-axis labels. Now label the axes and add a title. The character \pi creates the symbol π. An example of 2D plot is shown in Figure 2.2. 15 0 1 2 3 4 5 6 7 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x = 0:2π S in e of x Plot of the Sine function Figure 2.2: Plot of the Sine function >> xlabel(’x = 0:2\pi’) >> ylabel(’Sine of x’) >> title(’Plot of the Sine function’) The color of a single curve is, by default, blue, but other colors are possible. The desired color is indicated by a third argument. For example, red is selected by plot(x,y,’r’). Note the single quotes, ’ ’, around r. 2.2.4 Multiple data sets in one plot Multiple (x, y) pairs arguments create multiple graphs with a single call to plot. For example, these statements plot three related functions of x: y1 = 2 cos(x), y2 = cos(x), and y3 = 0.5 ∗ cos(x), in the interval 0 ≤ x ≤ 2π. >> x = 0:pi/100:2*pi; >> y1 = 2*cos(x); >> y2 = cos(x); >> y3 = 0.5*cos(x); >> plot(x,y1,’--’,x,y2,’-’,x,y3,’:’) >> xlabel(’0 \leq x \leq 2\pi’) >> ylabel(’Cosine functions’) >> legend(’2*cos(x)’,’cos(x)’,’0.5*cos(x)’) 16 >> title(’Typical example of multiple plots’) >> axis([0 2*pi -3 3]) The result of multiple data sets in one graph plot is shown in Figure 2.3. 0 1 2 3 4 5 6 −3 −2 −1 0 1 2 3 0 ≤ x ≤ 2π C os in e fu nc tio ns Typical example of multiple plots 2*cos(x) cos(x) 0.5*cos(x) Figure 2.3: Typical example of multiple plots By default, MATLAB uses line style and color to distinguish the data sets plotted in the graph. However, you can change the appearance of these graphic components or add annotations to the graph to help explain your data for presentation. 2.2.5 Specifying line styles and colors It is possible to specify line styles, colors, and markers (e.g., circles, plus signs, . . . ) using the plot command: plot(x,y,’style_color_marker’) where style_color_marker is a triplet of values from Table 2.3. To find additional information, type help plot or doc plot. 17 Similarly, column vectors are created in a similar way, however, semicolon (;) must separate the components of a column vector, >> w = [1;4;7;10;13] w = 1 4 7 10 13 On the other hand, a row vector is converted to a column vector using the transpose operator. The transpose operation is denoted by an apostrophe or a single quote (’). >> w = v’ w = 1 4 7 10 13 Thus, v(1) is the first element of vector v, v(2) its second element, and so forth. Furthermore, to access blocks of elements, we use MATLAB’s colon notation. For example, to access the first three elements of v, we write, >> v(1:3) ans = 1 4 7 Or, all elements from the third through the last elements, >> v(3,end) ans = 7 10 13 where end signifies the last element in the vector. If v is a vector, writing >> v(:) produces a column vector, whereas writing >> v(1:end) produces a row vector. 20 3.2.2 Entering a matrix A matrix is an array of numbers. To type a matrix into MATLAB you must • begin with a square bracket, [ • separate elements in a row with spaces or commas (,) • use a semicolon (;) to separate rows • end the matrix with another square bracket, ]. Here is a typical example. To enter a matrix A, such as, A =   1 2 3 4 5 6 7 8 9   (3.1) we will enter each element of the matrix A as follow, >> A = [1 2 3; 4 5 6; 7 8 9] MATLAB then displays the 3× 3 matrix, A = 1 2 3 4 5 6 7 8 9 Note that the use of semicolons (;) here is different from their use mentioned earlier to suppress output or to write multiple commands in a single line. Once we have entered the matrix, it is automatically stored and remembered in the workspace. We can refer to it simply as matrix A. We can then view a particular element in a matrix by specifying its location. We write >> A(2,1) ans = 4 A(2,1) is an element located in the second row and first column. Its value is 4. 21 3.2.3 Matrix indexing We select elements in a matrix just as we did for vectors, but now we need two indices. The element of row i and column j of the matrix A is denoted by A(i,j). Thus, A(i,j) in MATLAB refers to the element Aij of matrix A. The first index is the row number and the second index is the column number. For example, A(1,3) is an element of first row and third column. Here, A(1,3)=3. Correcting any entry is easy through indexing. Here we substitute A(3,3)=9 by A(3,3)=0. The result is >> A(3,3) = 0 A = 1 2 3 4 5 6 7 8 0 Single elements of a matrix are accessed as A(i,j), where i ≥ 1 and j ≥ 1 (zero or negative subscripts are not supported in MATLAB). 3.2.4 Colon operator The colon operator will prove very useful and understanding how it works is the key to efficient and convenient usage of MATLAB. It occurs in several different forms. Often we must deal with matrices or vectors that are too large to enter one ele- ment at a time. For example, suppose we want to enter a vector x consisting of points (0, 0.1, 0.2, 0.3, · · · , 5). We can use the command >> x = 0:0.1:5; The row vector has 51 elements. The colon operator can also be used to pick out a certain row or column. For example, the statement A(m:n,k:l specifies rows m to n and column k to l. Subscript expressions refer to portions of a matrix. For example, >> A(2,:) ans = 4 5 6 is the second row elements of A. The colon operator can also be used to extract a sub-matrix from A. >> A(:,2:3) 22 >> A(2:3,2:3) ans = 5 6 8 9 >> A(end:-1:1,end) ans = 9 6 3 >> A([1 3],[2 3]) ans = 2 3 8 9 3.2.7 Deleting row or column To delete a row or column of a matrix, use the empty vector operator, [ ]. For the (full) matrix A from the previous example >> A(3,:) = [] A = 1 2 3 4 5 6 Third row of matrix A is now deleted. To restore the third row, we use a technique for creating a matrix >> A = [A(1,:);A(2,:);[7 8 0]] A = 1 2 3 4 5 6 7 8 0 Matrix A is now restored to its original form. 3.2.8 Dimension To determine the dimensions of a matrix or vector, use the command size. For example, 25 >> size(A) ans = 3 3 means 3 rows and 3 columns. Or more explicitly with, >> [m,n]=size(A) 3.2.9 Continuation If it is not possible to type the entire input on the same line, use consecutive periods, called an ellipsis . . ., to signal continuation, then continue the input on the next line. B = [4/5 7.23*tan(x) sqrt(6); ... 1/x^2 0 3/(x*log(x)); ... x-7 sqrt(3) x*sin(x)]; Note that blank spaces around +, −, = signs are optional, but they improve readability. 3.2.10 Transposing a matrix The transpose operation is denoted by an apostrophe or a single quote (’). It flips a matrix about its main diagonal and it turns a row vector into a column vector. Thus, >> A’ ans = 1 4 7 2 5 8 3 6 0 By using linear algebra notation, the transpose of m× n real matrix A is the n×m matrix that results from interchanging the rows and columns of A. The transpose matrix is denoted AT . 3.2.11 Concatenating matrices Matrices can be made up of sub-matrices. Here is an example. First, let’s recall our previous matrix A. 26 A = 1 2 3 4 5 6 7 8 9 The new matrix B will be, >> B = [A 10*A; -A [1 0 0; 0 1 0; 0 0 1]] B = 1 2 3 10 20 30 4 5 6 40 50 60 7 8 9 70 80 90 -1 -2 -3 1 0 0 -4 -5 -6 0 1 0 -7 -8 -9 0 0 1 3.2.12 Matrix generators MATLAB provides functions that generates elementary matrices. The matrix of zeros, the matrix of ones, and the identity matrix are returned by the functions zeros, ones, and eye, respectively. Table 3.1: Elementary matrices Matrix Description eye(m,n) Returns an m-by-n matrix with 1 on the main diagonal eye(n) Returns an n-by-n square identity matrix zeros(m,n) Returns an m-by-n matrix of zeros ones(m,n) Returns an m-by-n matrix of ones diag(A) Extracts the diagonal of matrix A rand(m,n) Returns an m-by-n matrix of random numbers For a complete list of elementary matrices and matrix manipulations, type help elmat or doc elmat. Here are some examples: 1. >> b=ones(3,1) b = 1 1 1 Equivalently, we can define b as >> b=[1;1;1] 27 Chapter 4 Array operations and Linear equations MATLAB has two different types of arithmetic operations: matrix arithmetic operations and array arithmetic operations. 4.1 Array operations MATLAB has two different types of arithmetic operations: matrix arithmetic operations and array arithmetic operations. 4.1.1 Matrix arithmetic operations As we mentioned earlier, MATLAB allows arithmetic operations: +, −, ∗, and ˆ to be carried out on matrices. Thus, A+B or B+A is valid if A and B are of the same size A*B is valid if A’s number of column equals B’s number of rows Aˆ2 is valid if A is square and equals A*A α*A or A*α multiplies each element of A by α 4.1.2 Array arithmetic operations On the other hand, array arithmetic operations or array operations for short are done element-by-element. The period character (.) distinguishes the array operations from the matrix operations. However, since the matrix and array operations are the same for addition 30 (+) and subtraction (-), the character pairs (.+) and (.-) are not used. The list of array operators is shown below in Table 4.2. Operators Description .* Element-by-element multiplication ./ Element-by-element division .^ Element-by-element exponentiation Table 4.1: Array operators If A and B are two matrices of the same size with elements A = [aij] and B = [bij], then the command >> C = A.*B produces another matrix C of the same size with elements cij = aijbij. For example, using the same 3× 3 matrices, A =   1 2 3 4 5 6 7 8 9   , B =   10 20 30 40 50 60 70 80 90   we have >> C = A.*B C = 10 40 90 160 250 360 490 640 810 To raise a scalar to a power, we use for example the command 10^2. If we want the operation to be applied to each element of a matrix, we use .^2. For example, if we want to produce a new matrix whose elements are the square of the elements of the matrix A, we enter >> A.^2 ans = 1 4 9 16 25 36 49 64 81 The relations below summarize the above operations. To simplify, let’s consider two vectors U and V with elements U = [ui] and V = [vj]. 31 U. ∗V produces [u1v1 u2v2 . . . unvn] U./V produces [u1/v1 u2/v2 . . . un/vn] U.ˆV produces [uv1 1 uv2 2 . . . uvn n ] 4.1.3 Summary In order to avoid any confusion on this topic of “Matrix and Array Operations”, the above description can be summarized in Table 4.2. Operation Matrix Array Addition + + Subtraction − − Multiplication ∗ .∗ Division / ./ Left division \ .\ Exponentiation ˆ .̂ Table 4.2: Summary of matrix and array operations 4.2 Solving linear equations One of the problems encountered most frequently in scientific computation is the solution of systems of simultaneous linear equations. With matrix notation, a system of simultaneous linear equations is written Ax = b (4.1) when there are as many equations as unknown, A is a given square matrix of order n, b is a given column vector of n components, and x is an unknown column vector of n components. Students of linear algebra learn that the solution to Ax = b can be written as x = A−1b, where A−1 is the inverse of A−1 is the inverse of A. However, in the vast majority of practical computational problems, it is unnecessary and inadvisable to actually compute A−1. The inverse requires more arithmetic and produces a less accurate answer. For example, consider the following system of linear equations    x + 2y + 3z = 1 4x + 5y + 6z = 1 7x + 8y = 1 32 det Determinant diag Diagonal matrices and diagonals of a matrix eig Eigenvalues and eigenvectors inv Matrix inverse norm Matrix and vector norms rank Number of linearly independent rows or columns Table 4.3: Matrix functions 4.3 Exercises Note: Due to the teaching class during this Fall Quarter 2005, the problems are temporarily removed from this section. 35 Chapter 5 Introduction to M-File programming 5.1 Introduction So far in this manual, all the commands were executed in the Command Window. The problem is that the commands entered in the Command Window cannot be saved and executed again for several times. Therefore, a different way of executing commands with MATLAB is • first, to create a file with a list of commands, • then, save it, and • finally, run the file. If needed, corrections or changes can be made to the commands in the file. The files that are used for this purpose are called script files or scripts for short. This section covers the following topics: • M-File Scripts • M-File Functions 5.2 M-File Scripts A script file is an external file that contains a sequence of MATLAB statements. Script files have a filename extension .m and are often called M-files. M-files can be scripts that simply execute a series of MATLAB statements, or they can be functions that can accept arguments and can produce one or more outputs. 36 5.2.1 Examples Here are two simple scripts. Example 1 Consider the system of equations:    x + 2y + 3z = 1 3x + 3y + 4z = 1 2x + 3y + 3z = 2 Find the solution x to the system of equations. Solution: • Use the MATLAB editor to create a file: File → New → M-file. • Enter the following statements in the file: A = [1 2 3; 3 3 4; 2 3 3]; b = [1; 1; 2]; x = A\b • Save the file, for example, example1.m. • Run the file, in the command line, by typing: >> example1 x = -0.5000 1.5000 -0.5000 When execution completes, the variables (A, b, and x) remain in the workspace. To see a listing of them, enter whos at the command prompt. Note: The MATLAB editor is both a text editor specialized for creating M-files and a graphical MATLAB debugger. The MATLAB editor has numerous menus for tasks such as saving, viewing, and debugging. Because it performs some simple checks and also uses color to differentiate between various elements of codes, this text editor is recommended as the tool of choice for writing and editing M-files. There is another way to open the editor: 37 Both functions and scripts can have all of these parts, except for the function definition line which applies to function only. In addition, it is important to note that function name must begin with a letter, and must be no longer than than the maximum of 63 characters. Furthermore, the name of the text file that you save will consist of the function name with the extension .m. Thus, the above example file would be factorial.m. Table 5.2 summarizes the differences between scripts and functions. Table 5.2: Main differences between scripts and functions Scripts Functions - Do not accept input - Can accept input arguments and arguments or return output return output arguments. arguments. - Store variables in a - Store variables in a workspace workspace that is shared internal to the function. with other scripts - Are useful for automating - Are useful for extending the MATLAB a series of commands language for your application 40 5.3.2 Input and output arguments As mentioned above, the input arguments are listed inside parentheses following the function name. The output arguments are listed inside the brackets on the left side. They are used to transfer the output from the function file. The general form looks like this function [outputs] = function_name(inputs) Function file can have none, one, or several output arguments. Table 5.3 illustrates some possible combinations of input and output arguments. Table 5.3: Example of input and output arguments function C=FtoC(F) One input argument and one output argument function area=TrapArea(a,b,h) Three inputs and one output function [h,d]=motion(v,angle) Two inputs and two outputs 5.4 Input to a script file When a script file is executed, the variables that are used in the calculations within the file must have assigned values. The assignment of a value to a variable can be done in three ways. 1. The variable is defined in the script file. 2. The variable is defined in the command prompt. 3. The variable is entered when the script is executed. We have already seen the two first cases. Here, we will focus our attention on the third one. In this case, the variable is defined in the script file. When the file is executed, the user is prompted to assign a value to the variable in the command prompt. This is done by using the input command. Here is an example. % This script file calculates the average of points % scored in three games. % The point from each game are assigned to a variable % by using the ‘input’ command. game1 = input(’Enter the points scored in the first game ’); 41 game2 = input(’Enter the points scored in the second game ’); game3 = input(’Enter the points scored in the third game ’); average = (game1+game2+game3)/3 The following shows the command prompt when this script file (saved as example3) is executed. >> example3 >> Enter the points scored in the first game 15 >> Enter the points scored in the second game 23 >> Enter the points scored in the third game 10 average = 16 The input command can also be used to assign string to a variable. For more infor- mation, see MATLAB documentation. A typical example of M-file function programming can be found in a recent paper which related to the solution of the ordinary differential equation (ODE) [12]. 5.5 Output commands As discussed before, MATLAB automatically generates a display when commands are exe- cuted. In addition to this automatic display, MATLAB has several commands that can be used to generate displays or outputs. Two commands that are frequently used to generate output are: disp and fprintf. The main differences between these two commands can be summarized as follows (Table 5.4). Table 5.4: disp and fprintf commands disp . Simple to use. . Provide limited control over the appearance of output fprintf . Slightly more complicated than disp. . Provide total control over the appearance of output 42 Chapter 6 Control flow and operators 6.1 Introduction MATLAB is also a programming language. Like other computer programming languages, MATLAB has some decision making structures for control of command execution. These decision making or control flow structures include for loops, while loops, and if-else-end constructions. Control flow structures are often used in script M-files and function M-files. By creating a file with the extension .m, we can easily write and run programs. We do not need to compile the program since MATLAB is an interpretative (not compiled) language. MATLAB has thousand of functions, and you can add your own using m-files. MATLAB provides several tools that can be used to control the flow of a program (script or function). In a simple program as shown in the previous Chapter, the commands are executed one after the other. Here we introduce the flow control structure that make possible to skip commands or to execute specific group of commands. 6.2 Control flow MATLAB has four control flow structures: the if statement, the for loop, the while loop, and the switch statement. 6.2.1 The ‘‘if...end’’ structure MATLAB supports the variants of “if” construct. • if ... end • if ... else ... end 45 • if ... elseif ... else ... end The simplest form of the if statement is if expression statements end Here are some examples based on the familiar quadratic formula. 1. discr = b*b - 4*a*c; if discr < 0 disp(’Warning: discriminant is negative, roots are imaginary’); end 2. discr = b*b - 4*a*c; if discr < 0 disp(’Warning: discriminant is negative, roots are imaginary’); else disp(’Roots are real, but may be repeated’) end 3. discr = b*b - 4*a*c; if discr < 0 disp(’Warning: discriminant is negative, roots are imaginary’); elseif discr == 0 disp(’Discriminant is zero, roots are repeated’) else disp(’Roots are real’) end It should be noted that: • elseif has no space between else and if (one word) • no semicolon (;) is needed at the end of lines containing if, else, end • indentation of if block is not required, but facilitate the reading. • the end statement is required 46 6.2.2 Relational and logical operators A relational operator compares two numbers by determining whether a comparison is true or false. Relational operators are shown in Table 6.1. Table 6.1: Relational and logical operators Operator Description > Greater than < Less than >= Greater than or equal to <= Less than or equal to == Equal to ∼= Not equal to & AND operator | OR operator ∼ NOT operator Note that the “equal to” relational operator consists of two equal signs (==) (with no space between them), since = is reserved for the assignment operator. 6.2.3 The ‘‘for...end’’ loop In the for ... end loop, the execution of a command is repeated at a fixed and predeter- mined number of times. The syntax is for variable = expression statements end Usually, expression is a vector of the form i:s:j. A simple example of for loop is for ii=1:5 x=ii*ii end It is a good idea to indent the loops for readability, especially when they are nested. Note that MATLAB editor does it automatically. Multiple for loops can be nested, in which case indentation helps to improve the readability. The following statements form the 5-by-5 symmetric matrix A with (i, j) element i/j for j ≥ i: 47 % write some variable length strings to a file op = fopen(’weekdays.txt’,’wt’); fprintf(op,’Sunday\nMonday\nTuesday\nWednesday\n’); fprintf(op,’Thursday\nFriday\nSaturday\n’); fclose(op); This file (weekdays.txt) can be opened with any program that can read .txt file. 6.4 Exercises Note: Due to the teaching class during this Fall Quarter 2005, the problems are temporarily removed from this section. 50 Chapter 7 Debugging M-files 7.1 Introduction This section introduces general techniques for finding errors in M-files. Debugging is the process by which you isolate and fix errors in your program or code. Debugging helps to correct two kind of errors: • Syntax errors - For example omitting a parenthesis or misspelling a function name. • Run-time errors - Run-time errors are usually apparent and difficult to track down. They produce unexpected results. 7.2 Debugging process We can debug the M-files using the Editor/Debugger as well as using debugging functions from the Command Window. The debugging process consists of • Preparing for debugging • Setting breakpoints • Running an M-file with breakpoints • Stepping through an M-file • Examining values • Correcting problems • Ending debugging 51 7.2.1 Preparing for debugging Here we use the Editor/Debugger for debugging. Do the following to prepare for debugging: • Open the file • Save changes • Be sure the file you run and any files it calls are in the directories that are on the search path. 7.2.2 Setting breakpoints Set breakpoints to pause execution of the function, so we can examine where the problem might be. There are three basic types of breakpoints: • A standard breakpoint, which stops at a specified line. • A conditional breakpoint, which stops at a specified line and under specified conditions. • An error breakpoint that stops when it produces the specified type of warning, error, NaN, or infinite value. You cannot set breakpoints while MATLAB is busy, for example, running an M-file. 7.2.3 Running with breakpoints After setting breakpoints, run the M-file from the Editor/Debugger or from the Command Window. Running the M-file results in the following: • The prompt in the Command Window changes to K>> indicating that MATLAB is in debug mode. • The program pauses at the first breakpoint. This means that line will be executed when you continue. The pause is indicated by the green arrow. • In breakpoint, we can examine variable, step through programs, and run other calling functions. 52 Appendix A Summary of commands Table A.1: Arithmetic operators and special characters Character Description + Addition − Subtraction ∗ Multiplication (scalar and array) / Division (right) ˆ Power or exponentiation : Colon; creates vectors with equally spaced elements ; Semi-colon; suppresses display; ends row in array , Comma; separates array subscripts . . . Continuation of lines % Percent; denotes a comment; specifies output format ′ Single quote; creates string; specifies matrix transpose = Assignment operator ( ) Parentheses; encloses elements of arrays and input arguments [ ] Brackets; encloses matrix elements and output arguments 55 Table A.2: Array operators Character Description .∗ Array multiplication ./ Array (right) division .ˆ Array power .\ Array (left) division .′ Array (nonconjugated) transpose Table A.3: Relational and logical operators Character Description < Less than ≤ Less than or equal to > Greater than ≥ Greater than or equal to == Equal to ∼= Not equal to & Logical or element-wise AND | Logical or element-wise OR && Short-circuit AND | | Short-circuit OR 56 Table A.4: Managing workspace and file commands Command Description cd Change current directory clc Clear the Command Window clear (all) Removes all variables from the workspace clear x Remove x from the workspace copyfile Copy file or directory delete Delete files dir Display directory listing exist Check if variables or functions are defined help Display help for MATLAB functions lookfor Search for specified word in all help entries mkdir Make new directory movefile Move file or directory pwd Identify current directory rmdir Remove directory type Display contents of file what List MATLAB files in current directory which Locate functions and files who Display variables currently in the workspace whos Display information on variables in the workspace Table A.5: Predefined variables and math constants Variable Description ans Value of last variable (answer) eps Floating-point relative accuracy i Imaginary unit of a complex number Inf Infinity (∞) eps Floating-point relative accuracy j Imaginary unit of a complex number NaN Not a number pi The number π (3.14159 . . .) 57 Appendix B Release notes for Release 14 with Service Pack 2 B.1 Summary of changes MATLAB 7 Release 14 with Service Pack 2 (R14SP2) includes several new features. The major focus of R14SP2 is on improving the quality of the product. This document doesn’t attempt to provide a complete specification of every single feature, but instead provides a brief introduction to each of them. For full details, you should refer to the MATLAB documentation (Release Notes). The following key points may be relevant: 1. Spaces before numbers - For example: A* .5, you will typically get a mystifying message saying that A was previously used as a variable. There are two workarounds: (a) Remove all the spaces: A*.5 (b) Or, put a zero in front of the dot: A * 0.5 2. RHS empty matrix - The right-hand side must literally be the empty matrix [ ]. It cannot be a variable that has the value [ ], as shown here: rhs = []; A(:,2) = rhs ??? Subscripted assignment dimension mismatch 60 3. New format option - We can display MATLAB output using two new formats: short eng and long eng. • short eng – Displays output in engineering format that has at least 5 digits and a power that is a multiple of three. >> format short eng >> pi ans = 3.1416e+000 • long eng – Displays output in engineering format that has 16 significant digits and a power that is a multiple of three. >> format long eng >> pi ans = 3.14159265358979e+000 4. Help - To get help for a subfunction, use >> help function_name>subfunction_name In previous versions, the syntax was >> help function_name/subfunction_name This change was introduced in R14 (MATLAB 7.0) but was not documented. Use the MathWorks Web site search features to look for the latest information. 5. Publishing - Publishing to LATEXnow respects the image file type you specify in pref- erences rather than always using EPSC2-files. • The Publish image options in Editor/Debugger preferences for Publishing Images have changed slightly. The changes prevent you from choosing invalid formats. • The files created when publishing using cells now have more natural extensions. For example, JPEG-files now have a .jpg instead of a .jpeg extension, and EPSC2- files now have an .eps instead of an .epsc2 extension. • Notebook will no longer support Microsoft Word 97 starting in the next release of MATLAB. 6. Debugging - Go directly to a subfunction or using the enhanced Go To dialog box. Click the Name column header to arrange the list of function alphabetically, or click the Line column header to arrange the list by the position of the functions in the file. 61 B.2 Other changes 1. There is a new command mlint, which will scan an M-file and show inefficiencies in the code. For example, it will tell you if you’ve defined a variable you’ve never used, if you’ve failed to pre-allocate an array, etc. These are common mistakes in EA1 which produce runnable but inefficient code. 2. You can comment-out a block of code without putting % at the beginning of each line. The format is %{ Stuff you want MATLAB to ignore... %} The delimiters %{ and %} must appear on lines by themselves, and it may not work with the comments used in functions to interact with the help system (like the H1 line). 3. There is a new function linsolve which will solve Ax = b but with the user’s choice of algorithm. This is in addition to left division x = A\b which uses a default algorithm. 4. The eps constant now takes an optional argument. eps(x) is the same as the old eps*abs(x). 5. You can break an M-file up into named cells (blocks of code), each of which you can run separately. This may be useful for testing/debugging code. 6. Functions now optionally end with the end keyword. This keyword is mandatory when working with nested functions. B.3 Further details 1. You can dock and un-dock windows from the main window by clicking on an icon. Thus you can choose to have all Figures, M-files being edited, help browser, command window, etc. All appear as panes in a single window. 2. Error messages in the command window resulting from running an M-file now include a clickable link to the offending line in the editor window containing the M-file. 3. You can customize figure interactively (labels, line styles, etc.) and then automatically generate the code which reproduces the customized figure. 62 • MATLAB is interpreted (not compiled), errors are easy to fix • MATLAB is optimized to be relatively fast when performing matrix operations • MATLAB does have some object-oriented elements C.3 Weaknesses • MATLAB is not a general purpose programming language such as C, C++, or FOR- TRAN • MATLAB is designed for scientific computing, and is not well suitable for other appli- cations • MATLAB is an interpreted language, slower than a compiled language such as C++ • MATLAB commands are specific for MATLAB usage. Most of them do not have a direct equivalent with other programming language commands C.4 Competition • One of MATLAB’s competitors is Mathematica, the symbolic computation program. • MATLAB is more convenient for numerical analysis and linear algebra. It is frequently used in engineering community. • Mathematica has superior symbolic manipulation, making it popular among physicists. • There are other competitors: – Scilab – GNU Octave – Rlab 65 Bibliography [1] The MathWorks Inc. MATLAB 7.0 (R14SP2). The MathWorks Inc., 2005. [2] S. J. Chapman. MATLAB Programming for Engineers. Thomson, 2004. [3] C. B. Moler. Numerical Computing with MATLAB. Siam, 2004. [4] C. F. Van Loan. Introduction to Scientific Computing. Prentice Hall, 1997. [5] D. J. Higham and N. J. Higham. MATLAB Guide. Siam, second edition edition, 2005. [6] K. R. Coombes, B. R. Hunt, R. L. Lipsman, J. E. Osborn, and G. J. Stuck. Differential Equations with MATLAB. John Wiley and Sons, 2000. [7] A. Gilat. MATLAB: An introduction with Applications. John Wiley and Sons, 2004. [8] J. Cooper. A MATLAB Companion for Multivariable Calculus. Academic Press, 2001. [9] J. C. Polking and D. Arnold. ODE using MATLAB. Prentice Hall, 2004. [10] D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice-Hall, 1989. [11] J. W. Demmel. Applied Numerical Linear Algebra. Siam, 1997. [12] D. Houcque. Applications of MATLAB: Ordinary Differential Equations. Internal communication, Northwestern University, pages 1–12, 2005. 66