1.1 ABOUT MATLAB and MATLAB GUI (Graphical User Interface)

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1 Chapter 1 Introduction The Taylor Series is one of the most important tools in numerical analysis. It constitutes the foundation of numerical methods and will be used in most of the chapters of this text. From the Taylor Series, we can derive the formulas and error estimates of the many numerical techniques used. This chapter contains a review of the Taylor Series, and a brief introduction to MATLAB R. 1.1 ABOUT MATLAB and MATLAB GUI (Graphical User Interface) MATLAB (MATrix LABoratory) is a powerful interactive system for matrixbased computation designed for scientific and engineering use. It is good for many forms of numeric computation and visualization. MATLAB language is a highlevel matrix/array language with control flow statements, functions, data structures, input/output, and object-oriented programming features. To fully use the power and computing capabilities of this software program in classrooms and laboratories by teachers and students in science and engineering, part of this text is inted to introduce the computational power of MATLAB to modern numerical methods. MATLAB has several advantages. There are three major elements that have contributed to its immense popularity. First, it is extremely easy to use since data can be easily entered, especially for algorithms that are adaptable to a table format. This is an important feature because it allows students to experiment with many numerical problems in a short period of time. Second, it includes highlevel commands for two-dimensional and three-dimensional data visualization, and presentation graphics. Plots are easily obtained from within a script or in command mode. Third, the most evident power of a MATLAB is its speed of calculation. The program gives instantaneous feedback. Because of their popularity, MATLAB and other software such as MAPLE and Mathematica are now available in most university microcomputer laboratories. One of the primary objectives of this text is to give students a clear step-by-step explanation of the algorithm corresponding to each numerical method used. To 1

2 2 INTRODUCTION accomplish this objective, we have developed MATLAB M-functions contained in a supplementary CD-ROM at the back of the book. These M-files can be used for the application or illustration of all numerical methods discussed in this text. Each M-function will enable students to solve numerical problems simply by entering data and executing the M-functions. It is well known that the best way to learn computer programming is to write computer programs. Therefore, we believe that by understanding the basic theory underlying each numerical method and the algorithm behind it, students will have the necessary tools to write their own programs in a high-level computer language such as C or FORTRAN. Another future of MATLAB is that it provides the Graphical User Interface (GUI). It is a pictorial interface to a program inted to provide students with a familiar environment in which to work. This environment contains push buttons, toggle buttons, lists, menus, text boxes, and so forth, all of which are already familiar to the user, so that they can concentrate on using the application rather than on the mechanics involved in doing things. They do not have to type commands at the command line to run the MATLAB functions. For this reason, we introduced GUI in this edition to make it easier for the students to run the M-functions of the book. A readme file named SUN Package readme contained in the directory NMETH of the CD attached at the back cover of the book, explains in details the steps to follow to run the MATLAB functions using GUI. In addition, Appix D shows the use of GUI for solving several examples. 1.2 ANINTRODUCTIONTOMATLAB In this section we give to the reader a brief tutorial introduction to MATLAB. For additional information we urge the reader to use the reference and user s guides of MATLAB Matrices and matrix computation MATLAB treats all variables as matrices. They are assigned to expressions by using an equal sign and their names are case-sensitive. For example, >> A = [4-2 5; 6 1 7; ] A= New rows may be indicated by a new line or by a semicolon. A column vector may be given as >> x=[2;8;9]orx=[2 8 9] x=

3 AN INTRODUCTION TO MATLAB Elements of the matrix can be a number or an expression, like >> x = [ /4 2^3] x= One can define an array with a particular structure by using the command As an example x=a:step:b >> y = [0: 0.2 : 1] y= >> y=[0: pi/3:pi] y= >> y = [20: -5 : 2] y= MATLAB has a number of special matrices that can be generated by built-in functions >> ones(2) a matrix of all 1 s. >> zeros(2,4) a2 4 matrix of zeros. >> rand(2,4)

4 4 INTRODUCTION a2 4 random matrix with uniformly distributed random elements. >> eyes(3) the 3 3identitymatrix. The diag function either creates a matrix with specified values on a diagonal or extracts the diagonal entries. For example, >> v=[325]; >> D=diag(v) D= a3 3 diagonal matrix with v on the main diagonal. To extract the diagonal entries of an existing matrix, the diag function is used: >> C=[2-47;318;-156]; >> u=diag(c) u= The linspace function generates row vectors with equally spaced elements. The form of the function is linspace(firstvalue, lastvalue, numvalues) where firstvalue and lastvalue are the starting and ing values in the sequence of elements and NumValues is the number of elements to be created. For example, >> evens = linspace(0,10,6) evens = The most useful matrix functions are

5 AN INTRODUCTION TO MATLAB 5 eig inv lu qr rank eigenvalues and eigenvectors inverse LU decomposition QR factorization rank A component-by-component addition of two vectors of the same dimensions is indicated as >> x = [1 5 8]; >> y = [2 4 3]; >> x+y Multiplying a vector or matrix by a scalar will scale each element of the vector or matrix by the value of the scalar. >> C = 2*[1 3; 4 2] C= >> v = 3*[ ] v= The component-by-component multiplication of the vectors x and y is indicated as >> x. * y The inner, or dot, product of two vectors is a scaler and can be obtained by multiplying a row vector and a column vector. For example, >> u=[ ]; v=[2; 3; 10; 5]; >> u*v 31

6 6 INTRODUCTION The transpose of a matrix or a vector is denoted by a prime symbol. For example >> x The matrix operations of multiplication, power, and division are indicated as >> B = [1 3; 4 2]; >> A = [2 5; 0 6]; >> A*B >> A^ >> A/B >> A.*B Note that the three operations of multiplication, division, and power can operate elementwise by preceding them with a period Polynomials MATLAB provides a number of useful built-in functions that operates on polynomials. The coefficients of a polynomial are specified by the entries of a vector. For example, the polynomial p(x) =2x 3 5x is specified by [ ]. To evaluate the polynomial at a given x we use the polyval function. >> c=[2-508]; >> polyval(c,2)

7 AN INTRODUCTION TO MATLAB 7 4 evaluates the polynomials p(x) =2x 3 5x 2 +8atx =2. The function roots finds the n roots of a polynomial of degree n. For example, to find the roots of the polynomial p(x) =x 3 5x 2 17x + 21, enter >> a = [ ]; >> roots(a) Output format While all computations in MATLAB are performed in double precision, the format of the displayed output can be controlled by the following commands: format short format long format short e format long e format rat fixed point with 4 decimal places (the default) fixed point with 14 decimal places scientific notation with 4 decimal places scientific notation with 15 decimal places approximation by ratio of small integers Once invoked, the chosen format remains in effect until changed. It is possible to make the printed output from a MATLAB function look good by using the disp and fprintf functions. disp(x) displays the array X, without printing the array name. If X is a string, the text is displayed. For example, disp( The input matrix A is ) will display the text placed in single quotes. The function disp also allows us to print numerical values placed in text strings. For example, disp([ Newton s method converges after,num2str(iter), iterations ]) will write on the screen Newton s method converges after 9 iterations. Here the function num2str(iter) converts a number to a string. The other function fprintf is more flexible and has the form

8 8 INTRODUCTION fprintf( filename, format,list) where filename is optional; if it is omitted, output goes to the screen. format is a format control string containing conversion specifications or any optional text. The conversion specifications control the output of array elements. Common conversion specifications include: %P.Qe %P.Qf %P.Qg for exponential fixed point to automatically select the shorter of %P.Qe or %P.Qf where P and Q are integers that set the field width and the number of decimal places, respectively. list is a list of variable names separated by commas, and the special format \n produces a new line. For example, the statements >> x=pi;y=4.679;z=9; >> fprintf( \n x = %8.7f\n y=%4.3f\n z = %1.0f\n,x,y,z) prints on the screen x = y = z= Planar plots Plotting functions is very easy in MATLAB. Graphs of two-dimensional functions can be generated by using the MATLAB command plot. You can, for example, plot the function y = x over the interval [ 1, 2] with the following commands: >> x = [-1: 0.02: 2]; >> y=x.ˆ2 +1; >> plot(x,y) The plot is shown in Figure 1.1. The command plot(x,y, r+: ) will result in r=red with + for points and dotted line. Use help plot for many options. To plot a function f1 defined by an M-file (see Subsection 1.2.7) in an interval [a, b], we use the command >> fplot( f1,[a,b]).

9 AN INTRODUCTION TO MATLAB FIGURE 1.1 Plot of the function f(x) =x 2 +1 in [ 1, 2]. Other features can be added to a given graph using commands such as grid, xlabel, ylabel, title, etc... For example, the command grid on will create grid lines on the graph. Plots of parametrically defined curves can also be made. Try, for example, t=[0: 0.01: 4*pi]; x=cos(2*t); y=sin(3*t); plot(x,y). Multiple plots on a single graph are illustrated by >> x=[0:0.1:2*pi]; >> y1=cos(x); y2=sin(x); y3=cos(2*x); >> plot(x,y1,x,y2,x,y3) D mesh plots Three-dimensional or 3-D mesh surface plots are drawn with the function mesh. The command mesh(z) creates a three-dimensional perspective plot of the elements of the matrix z. The following example will generate the plot shown in Figure 1.2 of the function z = e x2 y 2 over the square [ 4, 4] [ 4, 4]. >> [x y]=meshgrid(-4.0:0.2:4.0,-4.0:0.2:4.0); >> z=exp(-x.^2-y.^2); >> mesh(x,y,z) The first command creates a matrix whose entries are the points of a grid in

10 10 INTRODUCTION FIGURE 1.2 Three-dimensional surface. the square 4 x 4, 4 y 4 with grid 0.2 units wide and 0.2 units tall. The second command creates a matrix whose entries are the values of the function z(x, y) at the grid points. The third command uses this information to construct the graph. Other commands related to 3-D graphics are plot3 and surf Function files MATLAB contains a vast collection of built-in functions from elementary functions like sine and cosine, to more sophisticated functions like matrix inversions, matrix eigenvalues, and fast Fourier transforms. Table 1.1 contains a small selection of elementary mathematical MATLAB functions and Appix B contains additional lists of functions grouped by subject area. Note that all these function names must be in lower cases. MATLAB functions may have single or multiple arguments. For example, [U,D]=eig(A) produces a matrix U whose columns are the eigenvectors of A and a diagonal matrix D with the eigenvalues of A on its diagonal Defining functions MATLAB allows users to define their own functions by constructing an M-file in the M-file Editor/Debugger. The first line of a function has the form

11 AN INTRODUCTION TO MATLAB 11 cos(x) sin(x) tan(x) sqrt(x) abs(x) exp(x) log(x) log10(x) cosh(x) tanh(x) asin(x) expm(a) cosine of x sine of x tangent of x square root of x absolute value of x exponential of x log to the base e of x log to the base 10 of x hyperbolic cosine of x hyperbolic tangent of x inverse sine function of x exponential of matrix A Table 1.1 Elementary mathematical MATLAB functions. function y = function name(input arguments) As an example, let us define the function f1(x) =2x 3 3x + 1 as an m.file. Using the Editor/Debugger or any text editor like emacs, or vi, we enter function y=f1(x) y=2*x.ˆ3-3*x+1; Once this function is saved as an M-file named f1.m, we can use the MATLAB command window to compute, for example, >> f1(2) 11 User defined functions can themselves have functions as input parameters. They can be evaluated internally by using the function feval. For example, >> x1=fzero( f1,0) x1= gives the zero of f1 closest to 0. We can pass one function f1.m to another function f2.m in MATLAB by using symbol: >> f2(@f1) For example, the built-in function quad(f,a,b) in MATLAB approximately integrates the function y = f(x) froma to b. To use this built-in function to approximately integrate y =cos(πx) +2 from x = 1 tox = 1, we begin by writing a

12 12 INTRODUCTION MATLAB function called f1.m: function y=f1(x) y=cos(pi*x)+2; If we are in the correct directory, we type in MATLAB: >> Relations and loops Like most computer languages, MATLAB offers a variety of flow control statements like for, while, andif. The statements, that we use to control the flow are called relations. The Relational operators in MATLAB are == equal <= less than or equal >= greater than or equal = notequal < less than > greater than Note that = is used in an assignment statement while == is used in a relation. Logical operators & and or not if, for, while statements The general form of an if statement is if <if expression> <statements> elseif <if expression> <statements>... else

13 AN INTRODUCTION TO MATLAB 13 <statements> Here is a simple example. If i, j =1,..., 10, then if i==j A(i,j)=1; else A(i,j)=0; will result in a diagonal matrix A with 1 s in its diagonal. The general form of a for loop is for <loop variable= loop expression> <statements> For example, S=0; for i=1:100 S=S+1; S results in the output S= 100 You can also nest for statements. Hilbert matrix, H. The following statements create an m n for i=1:m for j=1:n H(i,j)=1/(i+j-1); The switch command is ideal for decision making. The general form of the switch command is switch switch expression case case-expression commands

14 14 INTRODUCTION... otherwise commands Here is a code that finds the number of days in a month. For simplicity we will ignore the effect of leap years on the number of days in February and assume that there are always 28 days in February. The switch command is ideal for this kind of code. We implement it as follows: switch month case {1,3,5,7,8,10,12} days = 31; case {4,6,9,11} days = 30; case 2 days = 28; The variable month carries the number of the month, and deping on its value the first case, second case, or third case will be executed deping on the outcome of comparing the switch expression (month) to the members of the various case expressions. The same result could also have been accomplished with an if-structure, but in this case the switch-command resulted in a more compact and elegant code. Finally, MATLAB also has its own version of the while loop, which repeatedly executes a group of statements while a condition remains true. Its form is while < while expression > < condition statements> Given a positive number n the following statements compute and display the even powers of 3 less than n. k=0; while 3^k < n if rem(k,2)==0 3^k k=k+1; else k=k+1;