Outline. 1. Introduction to MATLAB 2. Symbolic Toolbox 3. Plots 4. Calculus 5. matrices and vectors 6. M-files 7. Programming 8.

Transcription

1 MATLAB

2 Outline 1. Introduction to MATLAB 2. Symbolic Toolbox 3. Plots 4. Calculus 5. matrices and vectors 6. M-files 7. Programming 8. simulink

3 1. INTRODUCTION

4 Introduction MATLAB stands for MATrix LABoratory Initially developed by a lecturer in 1970 s to help students learn linear algebra. Everything is a matrix - easy to do linear algebra It was later marketed and further developed under MathWorks Inc. (founded in 1984) Matlab is a software package which can be used to perform analysis and solve mathematical and engineering problems High level language for technical computing It has excellent programming features and graphics capability easy to learn and flexible. Available in many operating systems Windows, Macintosh, Unix, DOS It has several toolboxes to solve specific problems.

5 What is Matlab? Matlab is basically a high level language which has many specialized toolboxes for making things easier for us How high? Matlab High Level Languages such as C, Pascal, Fortran etc. Assembly language

6 Why Matlab? Common Uses for Matlab in Research Mathematical computations Data Acquisition Multi-platform, multi-format data importing Analysis Tools (existing,custom) Statistics Graphing Modeling Used a lot for problem solving Cybernetics Signal processing Image processing Pattern recognition

7 Why Matlab? Multi-platform, Multi Format data importing Data can be loaded into Matlab from almost any format and platform Binary data files (eg. REX, PLEXON etc.) Ascii Text (eg. Eyelink I, II) Analog/Digital Data files PC UNIX Subject Subject Subject 3 87

8 Analysis Tools Why Matlab? A considerable library of analysis tools exist for data analysis Provides a framework for the design, creation, and implementation of any custom analysis tool imaginable

9 Why Matlab? Graphing, Visualization A Comprehensive array of plotting options available from 2 to 4 dimensions Full control of formatting, axes, and other visual representational elements

10 Why Matlab? Modeling Models of complex dynamic system interactions can be designed to test experimental data

11 Strengths of MATLAB MATLAB is relatively easy to learn MATLAB code is optimized to be relatively quick when performing matrix operations MATLAB may behave like a calculator or as a programming language MATLAB is interpreted, errors are easier to fix Although primarily procedural, MATLAB does have some object-oriented elements

12 No need for types. i.e., int a; double b; float c; Variables All variables are created with double precision unless specified and they are matrices. Example: >>x=5; >>x1=2; After these statements, the variables are 1x1 matrices with double precision

13 Weaknesses of MATLAB MATLAB is NOT a general purpose programming language MATLAB is an interpreted language (making it for the most part slower than a compiled language such as C++) MATLAB is designed for scientific computation and is not suitable for some things (such as parsing text)

14 Where to get MATLAB Mathworks: Student version is affordable and complete.

15 The MATLAB System Development Environment Mathematical Function Library MATLAB language Application Programming Language (not discussed today)

16 Menu and toolbar MATLAB Desktop Workspace History Command

17 Matlab Screen Command Window type commands Current Directory View folders and m-files Workspace View program variables Double click on a variable to see it in the Array Editor Command History view past commands save a whole session using diary

18 User interface Command window Workspace Editor Help

19 Workspace User interface

20 Editor User interface

21 Help Contents Index Search Demo User interface Type help help function, e.g. help plot Running demos type demos type help demos Using the Help Browser (.html,.pdf) View getstart.pdf, graphg.pdf, using_ml.pdf

22 Useful Commands The two commands used most by Matlab users are >>help functionname >>lookfor keyword Keep a diary of your work >>Diary >> >> >>Diary off

23 File I/O Matlab has a native file format to save and load workspaces. Use keywords load and save. In addition MATLAB knows a large number of popular formats. Type help fileformats for a listing. In addition MATLAB supports C style low level file I/O. Type help fprintf for more information.

24 Note % is the neglect sign for Matlab (equivalent of // in C). Anything after it on the same line is neglected by Matlab compiler. Insert comment lines, starts with %, so that you understand your program Sometimes slowing down the execution is done deliberately for observation purposes. You can use the command pause for this purpose pause %wait until any key pause(3) %wait 3 seconds

25 Workspace Matlab remembers old commands and variables as well each function maintains its own scope the keyword clear removes all variables from workspace the keyword who lists the variables

26 Some Key Commands Executing Commands Basic Calculation Operators: + Addition - Subtraction * Multiplication / Division ^ Exponentiation

27 Order of computations to the power of multiplication, division subtraction, addition If you want to change the priorities use brackets

28 Number format Short: 4 digits Long: 14 digits round-off errors may become significant If you carry out more complex computations

29 2. Symbolic Math Toolbox

30 What is Symbolic Math? Symbolic mathematics deals with equations before you plug in the numbers. Calculus integration, differentiation, Taylor series expansion, Linear Algebra inverses, determinants, eigenvalues, Simplification algebraic and trigonometric expressions Equation Solutions algebraic and differential equations Transforms Fourier, Laplace, Z transforms and inverse transforms, You must use syms to declare the variables you plan to use to be symbolic variables

31 Matlab's Symbolic Toolbox The Symbolic Toolbox allows one to use Matlab for symbolic math calculations. The Symbolic Toolbox is a separately licensed option for Matlab. (The license server separately counts usage of the Symbolic Toolbox.) Recent versions use a symbolic computation engine called MuPAD. Older versions used Maple. Matlab translates the commands you use to work with the appropriate engine. One could also use Maple or Mathematica for symbolic math calculations. Strategy: Use the symbolic toolbox only to develop the equations you will need. Then use those equations with nonsymbolic Matlab to implement your program.

32 Symbolic Objects Use sym to create a symbolic number, and double to convert to a normal number. >> sqrt(2) ans = >> var = sqrt(sym(2)) var = 2^(1/2) >> double(var) ans = >> sym(2)/sym(5) + sym(1)/sym(3) ans = 11/15

33 Symbolic variables Use syms to define symbolic variables. (Or use sym to create an abbreviated symbol name.) >> syms m n b c x >> th = sym('theta') >> sin(th) ans = sin(theta) >> sin(th)^2 + cos(th)^2 ans = cos(theta)^2 + sin(theta)^2 >> y = m*x + b y = b + m*x

34 Substituting into symbolic expressions The subs function substitutes values or expressions for variables in a symbolic expression. >> clear >> syms m x b >> y = m*x + b y = b + m*x >> subs(y,x,3) ans = b + 3*m >> subs(y, [m b], [2 3]) ans = 2*x + 3 >> subs(y, [b m x], [3 2 4]) ans = 11 The symbolic expression itself is unchanged. >> y y = b + m*x

35 Substitutions, continued Variables can hold symbolic expressions. >> syms th z >> f = cos(th) f = cos(th) >> subs(f,pi) ans = -1 Expressions can be substituted into variables. >> subs(f, z*pi) ans = cos(pi*z)

36 Differentiation Use diff to do symbolic differentiation. >> clear >> syms m x b th n y >> y = m*x + b; >> diff(y, x) ans = m >> diff(y, b) ans = 1 >> p = sin(th)^n p = sin(th)^n >> diff(p, th) ans = n*cos(th)*sin(th)^(n - 1)

37 Integration >> clear >> syms m b x >> y = m*x + b; Indefinite integrals >> int(y, x) ans = (m*x^2)/2 + b*x >> int(y, b) ans = (b + m*x)^2/2 >> int(1/(1+x^2)) ans = atan(x) Definite integrals >> int(y,x,2,5) ans = 3*b + (21*m)/2 >> int(1/(1+x^2),x,0,1) ans = pi/4

38 Solving algebraic equations >> clear >> syms a b c d x >> solve('a*x^2 + b*x + c = 0') ans = % Quadratic equation! -(b + (b^2-4*a*c)^(1/2))/(2*a) -(b - (b^2-4*a*c)^(1/2))/(2*a) >> solve('a*x^3 + b*x^2 + c*x + d = 0') Nasty-looking expression >> pretty(ans) Debatable better-looking expression Another example: >> solve('m*x + b - (n*x + c)', 'x') ans = -(b - c)/(m - n) >> solve('m*x + b - (n*x + c)', 'b') ans = c - m*x + n*x >> collect(ans, 'x') ans = c - x*(m - n)

39 Solving systems of equations Systems of equations can be solved. >> [x, y] = solve('x^2 + x*y + y = 3',... 'x^2-4*x + 3 = 0') Two solutions: x = [ 1 ; 3 ] y = [ 1 ; -3/2 ] >> [x, y] = solve('m*x + b = y', 'y = n*x + c') Unique solution: x = -(b - c)/(m - n) y = -(b*n - c*m)/(m - n) If there is no analytic solution, a numeric solution is attempted. >> [x,y] = solve('sin(x+y) - exp(x)*y = 0',... 'x^2 - y = 2') x = y =

40 Solving differential equations We want to solve: Use D to represent differentiation against the independent variable. >> y = dsolve('dy = -a*y') y = C5/exp(a*t) Initial values can be added: >> y = dsolve('dy = -a*y', 'y(0) = 1') y = 1/exp(a*t)

41 Simplifying expressions >> clear; syms th >> cos(th)^2 + sin(th)^2 ans = cos(th)^2 + sin(th)^2 >> simplify(ans) ans = 1 >> simple(cos(th)^2 + sin(th)^2) >> [result,how] = simple(cos(th)^ sin(th)^2) result = 1 how = simplify >> [result,how] = simple(cos(th)+i*sin(th)) result = exp(i*th) how = rewrite(exp)

42 3. Plotting Graphs

43 Many Options Matlab has a powerful plotting engine (built in functions) that can generate a wide variety of plots. 2-D plots, 3-D plots, contour plots, line plots, polar plots

44 Use the help command and you find the built in functions >>> help graph2d >>> help graph3d e.g. plot polar loglog mesh semilog plotyy surf

45 Graphics: 2D-plotting Example: y=f(x) with f(x) = x 2 Plot f(x) in the interval x [-2,2].

46 Another Example: Plot the function e -x/3 sin(x) between 0 x 4π Create an x-array of 100 samples between 0 and 4π. >>x=linspace(0,4*pi,100); Calculate sin(.) of the x-array >>y=sin(x); Calculate e -x/3 of the x-array >>y1=exp(-x/3); Multiply the arrays y and y1 >>y2=y*y1; THIS IS WRONG!!!

47 Plot the function e -x/3 sin(x) between 0 x 4π Multiply the arrays y and y1 correctly by using the. command 0.7 >>y2=y.*y1; Plot the y2-array >>plot(y2)

48 Data visualisation plotting graphs Example on plot 2 dimensional plot >>> x=linspace(0,(2*pi),100); >>> y1=sin(x); >>> y2=cos(x); >>> plot(x,y1,'r-') >>> hold Current plot held >>> plot(x,y2,'g--') >>> Add title, labels and legend title xlabel ylabel legend You can add to the plot by using copy and paste (if you copy the plot into e.g. MSWord. You can also insert titles, labels, legends by using commands in the plot environment

49 y1 and y2 Data visualisation plotting graphs Example on plot sin(x) cos(x) angular frequency (rad/s)

50 Graphics - Annotation

51 Use title, xlabel, Graphics - Annotation ylabel and legend for annotation >> title('demo plot'); >> xlabel('x Axis'); >> ylabel('y Axis'); >> legend([pl1, pl2], 'x^2', 'x^3');

52 Graphics: Multiple Graphs in One Plot Window >>t = 0:pi/100:2*pi; >>y1=sin(t); >>y2=sin(t+pi/2); >>plot(t,y1,t,y2) >>grid on

53 Graphics: Multiple Plots The command subplot(m,n,o) devides the plotting window into several panes. The command defines a matrix of n plots in rows 1 m, whereby o defines the position of the subplots >>t = 0:pi/100:2*pi; >>y1=sin(t); >>y2=sin(t+pi/2); >>subplot(2,2,1) >>plot(t,y1) >>subplot(2,2,2) >>plot(t,y2) The commands >>subplot (2,2,3) and >>subplot (2,2,4) generate two more plots in 2 nd row

54 Graphics: plotting symbolic expressions The ezplot function will plot symbolic expressions. >> clear; syms x y >> ezplot( 1 / (5 + 4*cos(x)) ); >> hold on; axis equal >> g = x^2 + y^2-3; >> ezplot(g); use >>hold on For overlaying graphs

55 Graphics: 3D plots Plot a surface function of z= e -x2 -y 2 over the square [-2,2] x [-2,2] Compute a grid of points that can be plotted >> x=-2:.2:2; >> y=-2:.2:2; >> [X,Y]=meshgrid(x,y); >>z=exp(-x.^2-y.^2); >>mesh(z) >>xlabel( x ); >>ylabel( y ); >>zlabel( z ); >>titel( example of a 3D-meshplot) Example of a 3D-meshplot

56 Difference between ezplot and plot You do not need to use ezplot. You can also use the command plot The fundamental difference: The command ezplot uses intrinsically a useful stepsize to plot a graph If you use the plot command you must define the step size (increment) yourself Step size means: you define the number of points N that you want to plot You must define the increment, which is defined by h=1/n You must define the range for which you want to plot the graph: x from minimum value to maximum value

57 How to save plots use saveas(h,'filename.ext') to save a figure/file. >> f=figure; >> x=-5:0.1:5; >> h=plot(x,cos(2*x+pi/3)); >> title('figure 1'); >> xlabel('x'); >> saveas(h,'figure1.fig') >> saveas(h,'figure1.eps ) Useful extension types: bmp: Windows bitmap emf: Enhanced metafile eps: EPS Level 1 fig: MATLAB figure Useful extension types: jpg: JPEG image m: MATLAB M-file tif: TIFF image, compressed

58 4. Calculus: examples

59 Calculus Operations with MATLAB A digital computer operates on a numerical basis with addition, subtraction, multiplication, and division, along with memory storage and logic. True differentiation and integration can never be achieved exactly with numerical processes. However, there are two ways that differentiation and integration can be achieved on a practical level with MATLAB.

60 Two Types of Operations 1) MATLAB can be viewed as a very comprehensive "look-up" table in that numerous derivatives and integrals as character strings can be manipulated with software. The Symbolic Math Toolbox permits the exact formulas for derivatives and integrals to be extracted, manipulated, and plotted.

61 Two Types of Operations (Continued) 2) MATLAB can be used for calculus with numerical approximations to differentiation and integration. While such approximations are not exact, they can be used to provide extremely close approximations to the derivatives and integrals of experimental data, even when closed form solutions are impossible.

62 Symbolic Variables A symbolic variable is one that can be manipulated in the same manner as in an equation, and it may or may not ever take on any numerical values. To make x and a symbolic, the command is >> syms x a Alternately, the symbolic portion of a command may be enclosed with apostrophes ' '.

63 Symbolic Differentiation The command is diff( ) with the function to be differentiated in parentheses. All variables should have been established as symbolic variables or the function should be enclosed by apostrophes.

64 Determine the derivative of the function y >> syms x >> y = 4*x^5 y = 4*x^5 >> yprime = diff(y) >> yprime = 20*x^4

65 Continuation: second approach >> yprime = diff(4*x^5) >> yprime = 20*x^4

66 Continuation: third approach >> y = '4*x^5' y = 4*x^5 >> yprime = diff(y) >> yprime = 20*x^4

67 Continuation The third approach could also be performed in one step. >> yprime = diff('4*x^5') >> yprime = 20*x^4

68 Example: Use ezplot to plot y and yprime of the previous example The simplest form is ezplot(y), but the domain will be from -2 to 2. The domain can be changed by >> ezplot(y, [x1 x2]) In this example, ezplot(y, [-1 1]) ezplot(yprime,[-1 1]) The plots after labeling are shown on the next two slides.

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71 Symbolic Integration The command is int( ) with the function to be integrated in parentheses. All variables should have been established as symbolic variables or the function should be enclosed by apostrophes. Indefinite Integral: >> yint = int(y) Definite Integral: >> yint = int(y, a, b)

72 Determine the integral of the function y >> syms x >> y = x^2*exp(x) y = x^2*exp(x) >> z = int(y) z = x^2*exp(x)-2*x*exp(x)+2*exp(x)

73 Continuation We require that z(0) = 0, but the function obtained has a value of 2 at x = 0. Thus, >> z = z - 2 z = x^2*exp(x)-2*x*exp(x)+2*exp(x)-2 Plots of y and z are shown on the next two slides.

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76 Numerical Differentiation Generally, numerical differentiation is more prone to error than numerical integration due to the nature of sudden changes in differentiation. dy dx y x

77 Diff Command for Numerical Data Assume that x and y have been defined at N+1 points. A vector u with N points is defined as u y y u y y u y y u y y N N 1 N

78 Diff Command for Numerical Data, continuation The following command forms u: >> u = diff(y) The approximate derivative, >> yprime = diff(y)/delx Because x and y have one more point than yprime, they can be adjusted. >> x = x(1:n) >> y = y(1:n) >> plot(x, y, x, yprime)

79 For y = sin x, determine the numerical derivative based on 11 points in one cycle >> x = linspace(0, 2*pi, 11); >> y = sin(x); >> delx = 2*pi/10; >> yprime = diff(y)/delx; >> x = x(1:10); >> plot(x, yprime, x, cos(x), o ) The plots are shown on the next slide. The approximation is crude as expected.

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81 For y = sin x, determine the numerical derivative based on 101 points in one cycle >> x = linspace(0, 2*pi, 101); >> y = sin(x); >> delx = 2*pi/100; >> yprime = diff(y)/delx; >> x = x(1:100); >> plot(x, yprime, x, cos(x), o ) The plots are shown on the next slide. The approximation is much better.

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83 Numerical Integration Two types will be studied: 1. Zero-order integration 2. First-order integration, which is also known as the trapezoidal rule. Assume vectors x and y that have each been defined at N points.

84 Zero-Order Integration z z y x ( y y ) x z z y x ( y y y ) x z 1 y1 x z z y x y x k k 1 k n n 1 k

85 Two Zero-Order MATLAB Commands The two MATLAB commands for zero-order integration are sum() and cumsum(). >> area = delx*sum(y) >> z = delx*cumsum(y)

86 Two First-Order MATLAB Commands The two MATLAB commands for first-order integration are trapz() and cumtrapz(). >> area = delx*trapz(y) >> z = delx*cumtrapz(y)

87 Example: determine the exact area A for the following integral A x dx A 4 4x 4 2 x (2) 4 (0)

88 Determine the approximate area A1 with the zero-order integration algorithm and step-size of 0.05 >> delx = 0.05; >> x = 0:delx:2; >> y = 4*x.^3; >> A1 = delx*sum(y) A1 =

89 Determine the approximate area A2 with the first-order integration algorithm and step-size of 0.05 >> delx = 0.05; >> x = 0:delx:2; >> y = 4*x.^3; >> A2 = delx*trapz(y) A2 =

90 Determine the exact running integral for the following function z x sin 0 xdx z cos x x 0 cos x ( cos0) 1 cos x

91 Determine first-order approximations with 100 points per cycle and plot the two functions >> delx = 2*pi/100; >> x = 0:delx:2*pi; >> y = sin(x); >> z1 = delx*cumtrapz(y); >> plot(x, z1, x, 1 - cos(x), o ) The plots are shown on the next slide

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93 5. Matrices and Vectors

94 Basic functions Create matrices Matrix operations Matrix functions Matrix indexing Logical operations

95 Working with Matrices Matlab works with essentially only one kind of object, a rectangular numerical matrix A matrix is a collection of numerical values that are organized into a specific configuration of rows and columns The number of rows and columns can be any number Example 3 rows and 4 columns define a 3 x 4 matrix having 12 elements A scalar is a single number and is represented by a 1 x 1 matrix in matlab. A vector is a one dimensional array of numbers and is represented by an n x 1 column vector or a 1 x n row vector of n elements

96 Using Matlab Solving equations using variables Matlab is an expression language Expressions typed by the user are interpreted and evaluated by the Matlab system Variables are names used to store values Variable names allow stored values to be retrieved for calculations or permanently saved >> x = 6 >> x * y Variable = Expression Or Expression x = 6 >> y = 2 y = 2 >> x + y Ans = 12 >> x / y Ans = 3 >> x ^ y **Variable Names are Case Sensitive! Ans = 8 Ans = 36

97 Matrices & Vectors All (almost) entities in MATLAB are matrices Easy to define: >> A = [16 3; 5 10] A = Use, or to separate row elements -- use ; to separate rows

98 Order of Matrix - Matrices & Vectors m n m=no. of rows, n=no. of columns Vectors - special case n = 1 column vector m = 1 row vector

99 Creating Vectors and Matrices Define Transpose >> A = [16 3; 5 10] A = >> B = [ ] B = Vector : >> a=[1 2 3]; >> a' Matrix: >> A=[1 2; 3 4]; >> A' ans =

100 Variables Vectors and Matrices ALL variables are matrices e.g. 1 x 1 Variables 4 x 1 1 x 4 2 x They are case sensitive i.e x X Their names can contain up to 31 characters Must start with a letter Variables are stored in workspace

101 Some built-in functions mean(a) mean value of a vector max(a) maximum value of a vector element min(a) minimum value of a vector element sum(a) summation of the elements of A sort(a) sorted vector median(a) median value std(a) standard deviation det(a) determinant of a square matrix dot(a,b) dot product of two vectors cross(a,b) cross product of two vectors inv(a) inverse of a matrix A

102 Arithmetic Operators + addition - subtraction * multiplication / division \ left division. The operation A\B is effectively the same as INV(A)*B, although left division is calculated differently and is much quicker ^ exponentiation transponation complex conjugate transpose.* element wise multiplication./ element wise division.^ element wise power

103 Basic Operations: create matrices 1) column vector 2) row vector 3) scalar 4) matrix

104 Basic functions: matrix operators

105 Matrix indexing Basic functions

106 Matrix Operations Indexing Matrices [ A = [ The colon operator can be used to remove entire rows or columns Specify an element of a matrix >> A(:,3) = [ ] [ A = [ >> A(2,:) = [ ] [ A = [

107 Creating Matrices: built in functions zeros(m, n) matrix with all zeros ones(m, n) matrix with all ones. eye(m, n) the identity matrix rand(m, n) uniformly distributed random randn(m, n) normally distributed random magic(m) square matrix whose elements have the same sum, along the row, column and diagonal pascal(m) Pascal matrix length(m) computes the length of a vector size(m) determines the dimension of a matrix : automatic generation

108 How do we assign values to matrices? >>> A=[1 2 3;4 5 6;7 8 9] A = >>> Columns separated by space or a comma Rows separated by semi-colon

109 How do we access elements in a matrix or a vector? Try the followings: >>> A(2,3) ans = 6 >>> A(1,:) ans = >>> A(:,3) ans = >>> A(2,:) ans = 4 5 6

110 Vectors and Matrices Some special variables >>> 1/0 beep pi ( ) inf (e.g. 1/0) i, j ( 1 ) Warning: Divide by zero. ans = Inf >>> pi ans = >>> i ans = i

111 Arithmetic operations Matrices Performing operations to every entry in a matrix >>> A=[1 2 3;4 5 6;7 8 9] A = >>> Add and subtract >>> A+3 ans = >>> A-2 ans =

112 Arithmetic operations Matrices Performing operations to every entry in a matrix >>> A=[1 2 3;4 5 6;7 8 9] A = >>> Multiply and divide >>> A*2 ans = >>> A/3 ans =

113 Arithmetic operations Matrices Performing operations to every entry in a matrix >>> A=[1 2 3;4 5 6;7 8 9] A = >>> A^2 = A * A Power To square every element in A, use the element wise operator.^ >>> A.^2 ans = >>> A^2 ans =

114 Array Operations Evaluated element by element.' : array transpose (non-conjugated transpose).^ : array power.* : array multiplication./ : array division Very different from Matrix operations >> A=[1 2;3 4]; >> B=[5 6;7 8]; >> A*B But: >> A.*B

115 Vectors and Matrices Arithmetic operations Matrices Performing operations between matrices >>> A=[1 2 3;4 5 6;7 8 9] A = >>> B=[1 1 1;2 2 2;3 3 3] B = A*B = A.*B 1x1 4x2 7x3 2x1 5x2 8x3 3x1 6x2 9x3 =

116 Arithmetic operations Matrices Performing operations between matrices A/B? (matrices singular) A./B 1/1 4 / 2 7 / 3 2 /1 5 / 2 8 / 3 3 /1 6 / 2 9 / 3 =

117 Arithmetic operations Matrices Performing operations between matrices A^B??? Error using ==> ^ At least one operand must be scalar A.^B =

118 Matrix Operations Shortcut: Transposing Matrices The transpose of a matrix is the matrix formed by interchanging the rows and columns of a given matrix A = [ B = [ ] 7 3 3] >> transpose(a) >> B A = [1 6 B = [ ] ]

119 Built in functions (commands) Scalar functions used for scalars and operate element-wise when applied to a matrix or vector e.g. sin cos tan atan asin log abs angle sqrt round floor At any time you can use the command help to get help e.g. >>>help sin

120 Built in functions (commands) >>> a=linspace(0,(2*pi),10) a = Columns 1 through Columns 8 through >>> b=sin(a) b = Columns 1 through Columns 8 through >>>

121 Built in functions (commands) Vector functions operate on vectors returning scalar value e.g. max min mean prod sum length >>> a=linspace(0,(2*pi),10); >>> b=sin(a); >>> max(b) ans = >>> max(a) ans = >>> length(a) ans = >>> 10

122 Built in functions (commands) Matrix functions perform operations on matrices >>> help elmat >>> help matfun e.g. eye size inv det eig At any time you can use the command help to get help

123 Built in functions (commands) Matrix functions perform operations on matrices >>> x=rand(4,4) x = >>> xinv=inv(x) xinv = >>> x*xinv ans = >>>

124 Adding Elements to a Vector or a >> A=1:3 A= >> A(4:6)=5:2:9 A= >> B=1:2 B= 1 2 >> B(5)=7; B= Matrix >> C=[1 2; 3 4] C= >> C(3,:)=[5 6]; C= >> D=linspace(4,12,3); >> E=[C D ] E=

125 Long Array, Matrix t =1:10 t = k =2:-0.5:-1 k = B = [1:4; 5:8] x =

126 Creating Vectors Create vector with equally spaced intervals >> x=0:0.5:pi x = Create vector with n equally spaced intervals >> x=linspace(0, pi, 7) x = Equal spaced intervals in logarithm space >> x=logspace(1,2,7) x = Note: MATLAB uses pi to represent imaginary unit, uses i or j to represent

127 Generating Vectors from functions zeros(m,n) MxN matrix of zeros ones(m,n) MxN matrix of ones rand(m,n) MxN matrix of uniformly distributed random numbers on (0,1) x = zeros(1,3) x = x = ones(1,3) x = x = rand(1,3) x =

128 Concatenation of Matrices x = [1 2], y = [4 5], z=[ 0 0] A = [ x y] B = [x ; y] C = [x y ;z] Error:??? Error using ==> vertcat CAT arguments dimensions are not consistent.

129 The use of. Element A = [1 2 3; 5 1 4; 3 2 1] A = Operation x = A(1,:) x= y = A(3,:) y= b = x.* y b= c = x. / y c= d = x.^2 d= K= x^2 Erorr:??? Error using ==> mpower Matrix must be square. B=x*y Erorr:??? Error using ==> mtimes Inner matrix dimensions must agree.

130 6. m-files

131 Solution : use M-files M-files : Script and function files When problems become complicated and require re evaluation, entering command at MATLAB prompt is not practical Script Collections of commands Executed in sequence when called Saved with extension.m Function User defined commands Normally has input & output Saved with extension.m

132 There are script m-files and function m-files The difference is: Script m-files do exactly and only what you write in the script files. They are NOT interactive!!!

133 From function to m-file Function is a black box that communicates with workspace through input and output variables. INPUT FUNCTION Commands Functions Intermediate variables OUTPUT

134 From function to m-file Every function must begin with a header: function output=function_name(inputs) Output variable Must match the file name input variable

135 There are script m-files and function m-files Function m-files allow you to change, for example a parameter, and the function m file then executes the program by using this specific value of the parameter Every time you change this parameter you can generate a new set of results Function files thus are much more flexible and are used if you want to do the same computations over and over again, but with changing input conditions

136 Writing User Defined Functions Functions are m-files which can be executed by specifying some inputs and supply some desired outputs. The code telling the Matlab that an m-file is actually a function is function out1=functionname(in1) function out1=functionname(in1,in2,in3) function [out1,out2]=functionname(in1,in2) You should write this command at the beginning of the m-file and you should save the m-file with a file name same as the function name

137 Syntax for m files The name of the function file MUST be the same as the function name, for example: Function file logarithm.m Inside this function m file you must write function result=logarithm(x) result = log(x)

138 Writing User Defined Functions: Examples Write a function : out=squarer (A, ind) Which takes the square of the input matrix if the input indicator is equal to 1 And takes the element by element square of the input matrix if the input indicator is equal to 2 Same Name

139 Writing User Defined Functions Another function which takes an input array and returns the sum and product of its elements as outputs The function sumprod(.) can be called from command window or an m-file as

140 At Matlab prompt type in edit to invoke M-file editor Save this file as test1.m

141 Use of M-File Click to create a new M-File Extension.m A text file containing script or function or program to run

142 Use of M-File Save file as Denem430.m If you include ; at the end of each statement, result will not be shown immediately

143 To run the M-file, type in the name of the file at the prompt, e.g. >> Denem430.m It will be executed provided that the saved file is in the known path Type in matlabpath to check the list of directories listed in the path Use path editor to add the path: File Set path

144 M-files : script and function files (script) To run the M-file, type in the name of the file at the prompt e.g. >>> test1 It will be executed provided that the saved file is in the known path Type in matlabpath to check the list of directories listed in the path Use path editor to add the path: File Set path

145 M-files : script and function files (function) Function a simple example function y=react_c(c,f) %react_c calculates the reactance of a capacitor. %The inputs are: capacitor value and frequency in hz %The output is 1/(wC) and angular frequency in rad/s y(1)=2*pi*f; w=y(1); y(2)=1/(w*c); File must be saved to a known path with filename the same as the function name and with an extension.m Call function by its name and arguments help react_c will display comments after the header

146 M-files : script and function files (function) Function a more realistic example function x=impedance(r,c,l,w) %IMPEDANCE calculates Xc,Xl and Z(magnitude) and %Z(angle) of the RLC connected in series %IMPEDANCE(R,C,L,W) returns Xc, Xl and Z (mag) and %Z(angle) at W rad/s %Used as an example for IEEE student, UTM %introductory course on MATLAB if nargin <4 error('not enough input arguments') end; x(1) = 1/(w*c); x(2) = w*l; Zt = r + (x(2) - x(1))*i; x(3) = abs(zt); x(4)= angle(zt); impedance.m