1. M ATLAB Matrix Laboratory

2. Internet Resources: Dr. H-C Chen’s CVEN 302 Web Page http://ceprofs.tamu.edu/hchen/cven302.html For MATLAB, select chap01b.ppt, or click http://ceprofs.tamu.edu/hchen/cven302/chap01b.ppt

3. Programming structured The ideal style of programming is structured Break down a larger goal into tasks module Develop a module for each task A module has a single entrance and exit Modules are reusable

4. Use MATLAB to solve a problem State the problem clearly Describe the Input/Output (I/O) Algorithm - Numerical Method Develop a M ATLAB program Debugging and Testing Documentation If needed, work the problem by hand

5. Compiler and Program Execution Computer Program Compiler Machine language program Link/ Load Execute input output MATLAB Execution Interactive environment - does not require formal compilation, linking/loading, and execution Scripts - develop and execute m-files that contain MATLAB commands

6. comments Important! You must include comments in your programs you turn in - otherwise we will have great difficulty knowing what you are thinking and doing. You will have difficulty knowing what you were doing if you need to reuse the programs.

7. Command Window - enter commands and data - print results Graphics Window - display plots and graphs Edit Window - create and modify m-files MATLAB’s 3 Windows

8. Two simple MATLAB programs (also called scripts) Filename: lec2a.m % Find sum of 1 to b % Comments begin with “%” % cd y:\cven302\ clear all sum = 0; b = input('what is the value of b? ') for i = 1 : b sum = sum + i end Need extension.mcalled m-file

9. Filename: lec2b.m % plot 1/x v.s. tanh x clc a = linspace(0,5); f = 1./a; g = tanh(a); plot(a,f,'-', a, g,'--') axis([0 3 0 2]) xlabel ('x') ylabel ('y') title ('Plot 1/x and tanh x') text(0.7, 1.6, '1/x') text(2, 1.1, 'tanh x')

10. Now we have seen Command Window, Graphics Window, and Edit Window There is another window which is very useful: the Help Window You can also type help in Commend Window e.g., help input

11. » help input INPUT Prompt for user input. R = INPUT('How many apples') gives the user the prompt in the text string and then waits for input from the keyboard. The input can be any MATLAB expression, which is evaluated, using the variables in the current workspace, and the result returned in R. If the user presses the return key without entering anything, INPUT returns an empty matrix. R = INPUT('What is your name','s') gives the prompt in the text string and waits for character string input. The typed input is not evaluated; the characters are simply returned as a MATLAB string. See also KEYBOARD.

12. MATLAB’s basic component is a matrix i.e., all variables are treated as matrices All operations are vectorized (optimized for vector use) Loops run much slower in MATLAB than in Fortran (not a vector operation)

13. Scalar, vector, and matrix Scalar: k = 5 1  1 matrix Matrix: m  n matrix m rows (m = 3) n columns (n = 4) 1  4 matrix, or a row vector Vector: 3  1 matrix, or a column vector

14. Variable may consist up to 31 characters starting with a letter and followed by any combination of letters, digits, and underscores (Variable names must start with a letter) punctuation marks and spaces should not be included e.g., CVEN_302, TIME, Time, time, velocity, force_x, force_y

15. All numbers are double precision Text is stored as arrays of characters You don’t have to declare the type of data (defined when running) Variables are case sensitive Data Type e.g., CVEN_302, TIME, Time, time, velocity, force_x, force_y

16. Arithmetic operation of scalars Addition: a+bsubtraction: a-b multiplication: a*bdivision: a/b exponentiation: a^b= a b abs(x) : absolute value sqrt(x) : square root = x^(1/2) sin(x) : sine asin(x) : inverse sine sinh(x) : hyperbolic sine asinh(x) : inverse hyperbolic sine log(x) : natural logarithm log10(x) : common logarithm exp(x) : exponental: e x (e = 2.7183….) Elementary math functions

17. » a = 5; » b = 3; » a*b ans = 15 » d = ans * 3 d = 45 » e = ans * 3; » e e = 45 » a^2 ans = 25 » a/b ans = 1.6667 » format long » a/b ans = 1.66666666666667 » format short » a/b ans = 1.6667

18. » exp(2) ans = 7.3891 » 10^2 ans = 100 » log(100) ans = 4.6052 » log10(100) ans = 2 » pi ans = 3.1416 ; - result not displayed format long - 15 digits format short - 5 digits ans - result from previous calculation pi -  Ctrl-c : terminate a running program

19. Array Operations An array operation is performed element-by- element - Need “.” in front of the operatorAn array operation is performed element-by- element - Need “.” in front of the operator MATLAB: C = A.  B;

20. Arithmetic operation of Arrays Addition: a+bsubtraction: a-b multiplication: a.*bdivision: a./b exponentiation: a.^b = a b

21. MATLAB variables and matrix operation » b = [4 5 6] b = 4 5 6 » c = [1 3 2] c = 1 3 2 » c = [1; 3; 2] c = 1 3 2 » d = c' d = 1 3 2 - transpose ; - a new line

22. » b * c ans = 31 » b.*d ans = 4 15 12 » b./d ans = 4.0000 1.6667 3.0000 » f = [b; d; [3 6 8]] f = 4 5 6 1 3 2 3 6 8 b.*d = [4*1 5*3 6*2] b./d = [4/1 5/3 6/2] b d [3 6 8] » f(2,3) ans = 2 » size(f) size - size of a matrix

23. Matrix Operation (conti.) s = [1 2 3 4; 5 6 7 8] t = [1 2 3 4.... 5 6 7 8] continue

24. Colon Operator Creating new matrices from an existing matrix C = [1,2,5; -1,0,1; 3,2,-1; 0,1,4] F = C(:, 2:3) = [2,5; 0,1; 2,-1; 1,4] E = C(2:3,:) = [-1 0 1; 3 2 -1] G = C(3:4,1:2) = [3,2; 0,1] all columns rows 2 to 3

25. » x = 1:10 x = 1 2 3 4 5 6 7 8 9 10 » t = 1 : 2 : 10 t = 1 3 5 7 9 » k = 5:-1:-3 k = 5 4 3 2 1 0 -1 -2 -3 a : step : bgives number from a to b with the specified step between elements step = 1 if not specified

26. What if the step is not easy to calculate, or is an odd number? y = linspace (a, b, n_pts) » y = linspace (0, 1, 4) y = 0 0.3333 0.6667 1.0000 2 end points are included in n_pts. y = linspace (0,1) a number of 100 for n_pts is assigned.

27. Some useful special matrices » eye (3) ans = 1 0 0 0 1 0 0 0 1 » ones (3) ans = 1 1 1 » ones (1,4) ans = 1 1 1 1 » zeros (2,3) ans = 0 0 0 eye (m,n)eye (n) ones (m,n)ones (n) zeros (m,n)zeros (n)

28. Plotting plot (x, y)plot(x1, y1, x2, y2) plot (x, y, ‘color symbol line style’) » x = linspace(0, 2*pi); » y = sin (x); » z = cos (x); » plot (x, y) » plot (x, y, x, z) » figure (2) » plot (x, y, 'r o -'); grid on » hold on » plot (x, z, 'b x :') red, circle, solid blue, x-mark, dotted figure or figure (#) : open a figure try help plot add grids try grid off

29. » xlabel ('x') » ylabel ('y') » title ('sine and cosine') » text (2, 1, 'This is a sine fuction') » axis ([0 2*pi -2 2]) » hold off xlabel (‘ label ‘)ylabel (‘ label ‘) title (‘ title of the plot ‘) text ( x_location, y_location, ‘ text ‘) axis ([ x_min x_max y_min y_max ]) Plotting (continue) - text string hold on: hold the plots to avoid overwriting

30. subplot ( m, n, figure number ) - break the figure window into m by n small figures, and plot the specified figure Plotting (continue) » figure » subplot (3, 2, 1) » plot (x, y) » subplot (3, 2, 2) » plot (x, z) » subplot (3, 2, 4) » plot (x, y-z) 12 34 56

31. semilogx (x, y) logarithmic scales for the x axis. semilogy (x, y) logarithmic scales for the y axis. loglog (x, y) logarithmic scales for the x and y axes x = 10.^[-1:0.1:2]; y = exp(x); figure(1) semilogy(x,y,':^') figure(2) loglog(x,y,'-s') grid on

32. More control on plotting get(H)get object properties set(H,'PropertyName',PropertyValue,...) set object properties figure(2) h = loglog(x,y,'-sr') grid on get(h) set(h, 'LineWidth',2) set(h, 'MarkerFaceColor', [0 0 1])

33. who list of current variables clc clear the command window clf clear the graphical windows clear x clear the variable x clear all clear all variables close close the current figure close all close all figures cd y:\cven302\ change directory dir list all files what list all m-files CTRL-C Abort semicolons (;) at the end of line: Calculation results will not be displaced on the command window Some useful commands

34. Initializing Variables Explicitly list the values reads from a data file uses the colon “:” operator reads from the keyboard  

35. Input and output (I/O) a = input (‘ enter a value or string ‘) - wait for input from the keyboard load file_name - input an existing data file (ASCII format) diary file_name - save anything in the Command Window diary off save file_name variable(s) -ascii - save the data of the variable(s) to a file in ASCII format You may need to go to certain directory before loading and/or saving files

36. » diary tmp.dat » a = [ 1 2 3; 4 5 6] a = 1 2 3 4 5 6 » save a_file.dat a -ascii » load a_file.dat » b = a_file b = 1 2 3 4 5 6 » d = input ('give a value to d: ') give a value to d: 5 d = 5 » diary off Save in the tmp.dat file

37. M-Files: Scripts and Functions You can create and save code in text files using MATLAB Editor/Debugger or other text editors (called m-files since the ending must be.m) M-file is an ASCII text file similar to FORTRAN or C source codes ( computer programs) A script can be executed by typing the file name, or using the “run” command

38. Functions distinguished from the script that the first line is of the form function x = function_name (input arguments) function [x, y] = function_name (input arguments) A function has to be stored as a stand-along file ended with “.m”. The name of the function is usually (but not necessary) the same as the name of the file. Difference between scripts and functions Scripts share variables with the main workspace Functions do not

39. function t = trap_ex(a, b) t = (b - a) * (a^(-3) + b^(-3)) / 2; Approximate integral f(x) = 1/x 3 using basic trapezoid rule File name: trap_ex.m » y = trap_ex (1, 3) y = 1.0370 »

40. Trapezoidal rule ab f(a) f(a) f(b) f(b)

41. function t = trap_ex2(a, b, step) dx = (b - a)/step; x1 = a; t = 0; for i = 1 : step x1 = a + (i-1)*dx; x2 = a + i*dx; t = t + (x2 - x1) * (x1^(-3) + x2^(-3)) / 2; end File name: trap_ex2.m Approximate integral f(x) = 1/x 3 using basic trapezoid rule with n steps used

42. » y = trap_ex2 (1,3,1) y = 1.0370 » y = trap_ex2 (1,3,2) y = 0.6435 » y = trap_ex2 (1,3,100) y = 0.4445 » y = trap_ex2 (1,3,1000) y = 0.4444 » format long » y = trap_ex2 (1,3,1000) y = 0.44444543209743 Exact solution = 0.444444444444 …. » x1 ??? Undefined function or variable 'x1'. Why?

43. feval - evaluate function specified by string function y = my_func(x) % function 1/x^3 y = x.^(-3); function q = basic_trap(f, a, b) % trapezoid rule with input function f ya = feval(f, a); yb = feval(f, b); q = (b - a)* (ya + yb)/2; my_func.m basic_trap.m » y = basic_trap ('my_func', 1,3) y = 1.0370 » feval('my_func', 1) ans = 1

44. Decision Making: Control Flow (a)For Loops (b)While Loops (c)If-Else Structures

45. z = 0; for i = 1:10 y(11-i) = i; z = z+i; end For Loops for x = array commands end » y y = 10 9 8 7 6 5 4 3 2 1 » disp(z) 55 Without this line you will get: Warning: Reference to uninitialized variable z. disp (x) - display the value of x

46. Filename: lec2a.m % Find sum of 1 to b % Comments begin with “%” % cd y:\cven302\ clear all sum = 0; b = input('what is the value of b? ') for i = 1 : b sum = sum + i end

47. While Loops while expression commands end eps = 1; count = 0; while (1+eps) > 1 eps = eps/2; count = count + 1; end eps = eps*2 display(count-1) Floating point relative accuracy (is true) Determining machine epsilon 52 digits in double precision (64 bit) eps = 2.2204e-016 ans = 52

48. Floating point a number digitbase exponent d 1  0: normalized floating-point system Single precision: 32 bits (23 for digit, 8 for exponent, 1 for sign Double precision: 64 bits (52, 11, 1)

49. What if we omit this operation? Use “break” to terminate while and for loops prematurely (or set a limit in while loops) The loop becomes an “infinite loop” because the statement is always true. eps = 1; count = 0; while (1+eps) > 1 eps = eps/2; count = count + 1; end eps = eps*2 display(count-1) eps = 1; count = 0; while (1+eps) > 1 & count < 100 eps = eps/2; count = count + 1; end eps = eps*2 display(count)

50. If-Else Structures if expression (is true) commands end if expression 1 commands 1 elseif expression 2 commands 2 elseif expression 3 commands 3 : : else commands n end if expression commands 1 else commands 2 end

51. Relational Operators greater than >=greater than or equal to ==equal to ~=not equal to Logical Operators &and |or ~not

52. eps = 1; count = 0; while (1+eps) > 1 eps = eps/2; count = count + 1; if count > 100 break end eps = eps*2 display(count) eps = 1; count = 0; while (1+eps) > 1 & count < 100 eps = eps/2; count = count + 1; end eps = eps*2 display(count)

53. a = input('what is the value of input data? ') if a > 0 sign = 1; elseif a < 0 sign = -1; else sign = 0; end disp(sign)

54. print - print out current figure print -djpeg filename - save current figure in jpeg format print -djpeg# filename - save current figure in jpeg format with # (resolution level) between 0 and 100 (default 75) print -dtiff filename - save current figure in tiff format print -dpsc filename - save current figure in color PostScript format Print figures to image files See help print for more

55. Filename: lec2b.m % plot 1/x v.s. tanh x clc a = linspace(0,5); f = 1./a; g = tanh(a); plot(a,f,'-', a, g,'--') axis([0 3 0 2]) xlabel ('x') ylabel ('y') title ('Plot 1/x and tanh x') text(0.7, 1.6, '1/x') text(2, 1.1, 'tanh x') » cd y:\cven302 » print -djpeg myfigure1 » print -djpeg100 myfigure2