1. INTRODUCTION TO MATLAB MATLAB is a software package for computation in engineering, science, and applied mathemat-ics. It offers a powerful programming language, excellent graphics, and a wide range of expert knowledge.

2. A software environment for interactive numerical computations Examples: Matrix computations and linear algebra Solving nonlinear equations Numerical solution of differential equations Mathematical optimization INTRODUCTION TO MATLAB

3. Statistics and data analysis Signal processing Modelling of dynamical systems Solving partial differential equations Simulation of engineering systems ETC. INTRODUCTION TO MATLAB

4. Matlab Background Matlab = Mat rix Lab oratory Originally a user interface for numerical linear algebra INTRODUCTION TO MATLAB

5. Multipaneled MATLAB desktop command window optional windows workspace current directory type commands here INTRODUCTION TO MATLAB

6. Matlab’s help features type “help” at the command prompt and Matlab returns a list of help topics

7. Matlab’s help features >> help lang Matlab’s language constructs

8. Matlab’s help features >> help for how to use Matlab’s “for” statement

9. Matlab’s help features you can also access “on-line” help by clicking the question mark in the toolbar separate window

10. Basic Command and Syntax all variables are stored in 32bit floating point format no distinction between real and integer >>a = 3; >>a = 3.0; same assignment for “a” Matlab is case sensitive >>A=3; >>a=2; AaAa

11. If you type in a valid expression and press Enter, MATLAB will immediately execute it and return the result. >> 2+2 ans = 4 >> 4^2 ans = 16 >> sin(pi/2) ans = 1

12. MATLAB Variables can use numbers and underscore in variable names >>case34=6.45; names must start with a letter >>case_34=6.45; OK >>34case=23.45; results in a syntax error string (text) variables enclosed in single quotes. The variable is stored as array of characters >>title=‘This is the title’;

13. INTTRODUCTION >> x = sqrt(3) x = 1.7321 >> atan(x) ans = 1.0472 >> pi/ans ans = 3

14. MATLAB Variables to clear a variable from memory >>a=4 >>clear a if a variable is defined, typing the variable name returns its value >>a=45.57; a = 45.57 Matlab returns the value >>a

15. Arrays and Matrices The simplest way to construct a small array is by enclosing its elements in square brackets. >> A = [1 2 3; 4 5 6; 7 8 9] A = 1 2 3 4 5 6 7 8 9

16. MATLAB Variables Matlab will “echo” commands unless a semi-colon is used >>a=23.2; >> >>a=23.2 a = 23.2 >> Matlab echoes the command

17. >> b = [0;1;0] b = 0 1 0 >> [A b] ans = 1 2 3 0 4 5 6 1 7 8 9 0

18. MATLAB Variables Vectors column vectors row vectors >>a=[1;2;3]; >>a a = 1 2 3 >>a=[1,2,3]; >>a a = 1 2 3 use semi-colon to separate rows use comma to separate columns

19. MATLAB Variables Matrices 2-dimensional matrices >>a=[1,2,3;4,5,6]; >>a a = 1 2 3 4 5 6 again, separate columns with commas and rows with semi-colons

20. MATLAB Variables A vector is a special type of matrix row vector is a 1 x n matrix, 1 row n columns column vector is a n x 1 matrix, n rows 1 column Indexing Matrix elements >>a=[1,2,3]; >>a(2) ans = 2 could also reference by a(1,2) note, a(2,1) would produce an error because “a” only has one row

21. MATLAB Variables Indexing Matrix elements >>a=[1,2,3;4,5,6]; more examples >>a(2,3) ans = 6 >>a(2,2)=9; >>a a = 1 2 3 4 9 6 addressing assigning

22. USAGE Solutions using Matrix Operation 3x+2y-z=4 2x+4y-2z =-6 6x-7y+4z=8 >> D=[3 2 -1; 2 4 -2; 6 -7 4] D = 3 2 -1 2 4 -2 6 -7 4

23. >> E=[4; -6;8] E = 4 -6 8 >> X=D\E X = 3.5000 -39.0000 -71.5000

24. An especially important construct is the colon operator. >> 1:8 ans = 1 2 3 4 5 6 7 8 >> 0:2:10 ans = 0 2 4 6 8 10 >> 1:-.5:-1 ans = 1.0000 0.5000 0 -0.5000 -1.0000

25. MATLAB Variables complex-valued numbers Typically, the variable “i” or “j” is used to represent the complex variable; e.g. Then, a complex number is represented as z = a + ib Re(z) = a Im(z) = b

26. An especially important construct is the colon operator. >> 1:8 ans = 1 2 3 4 5 6 7 8

27. MATLAB Variables complex-valued numbers Unless i or j has been previously defined, Matlab assigns i and j the complex variable value In Matlab, a complex variable is represented in the following format (assuming all variables are cleared) >>z=23+i*56; >>z z = 23.00 + 56.00i >>z=23+j*56; >>z z = 23.00 + 56.00i Matlab always uses the symbol “i” to represent a complex number

28. MATLAB Variables complex-valued numbers What happens in this case? >>i=3; >> z=23+i*56; >>z z = What happens in this case? >>a=sqrt(-1); >>z=23+a*56; >>z z =

29. MATLAB Variables complex-valued numbers Note, a real-valued number is a special case of a complex-valued number assigning any element of a matrix as complex-valued makes the entire matrix complex-valued >>a=[1,2]; >>a a = 1 2 >>a(1)=1+i*5; >>a a = 1.00+5.00i 2.00+0.00i

30. MATLAB Variables Advanced data types n-dimensional arrays structures cell arrays

31. MATLAB Operations Basic operations addition + subtraction - multiplication * division right division / left division \ >>a=3;b=4; >>c1=a/b; >>c2=a\b; ? c1=0.75 c2=1.3333…. so, be careful!

32. MATLAB Operations Mixed Real and Complex valued Variables if both variables are real-valued, a real-valued result is obtained if one variable is complex-valued, Matlab recasts the real variable as complex and then performs the operation. The result is complex-valued however, the type casting is done internally, the real-valued variable remains real after the operation

33. MATLAB Operations Other (Scalar) Operations Math representation Matlab interpretation >>z=y^x; >>y=exp(x); >>y=log(x); >>y=log10(x) >>y=sin(x); >>y=cos(x); >>y=tan(x); >>y=asin(x); >>y=acos(x); >>y=atan(x);

34. MATLAB Operations Examples >>y=x^0.5; >>y=x^(1/2); >>y=sqrt(x); All variables in the preceding operations can be real or complex, negative or positive for x < 0, y is complex. Matlab assumes you allow complex valued numbers. If y is not to be complex, you must provide error checking.

35. MATLAB Operations Matrices Only matrices of the same dimension can be added and subtracted For multiplication, the inner dimensions must be the same No error Error >>D=A+B; >>D=A-B; >>D=A*C; >>D=C*A; >>D=A+C; >>D=A*B; >>D=B*A; Matrix multiplication not commutative

36. MATLAB Operations Left(\) and Right(/) Matrix “division” Math representation Matlab interpretation >>C=A\B; >>C=B/A; Remember, A must be square and full rank (linearly independent rows/columns)

37. MATLAB Operations Matrix Transpose Math representation Matlab interpretation >>C=A’; For complex-valued matrices, complex conjugate transpose >>B=A’; >>b=a’;

38. MATLAB m-files Two types of m-files script files collection of commands that Matlab executes when the script is “run” function files collection of commands which together represent a function, a procedure or a method Both types are separate files with a “.m” extension

39. MATLAB m-files To create an m-file, open the Matlab text editor Click on the “page” icon The Matlab text editor window will open

40. MATLAB m-files Script Files On the command line >>x=3.0; >>y=x^2; >>y y = 9.0 >> In the script file named test.m On the command line >>test y = 9.0 >>

41. MATLAB m-files Script Files script files share the workspace memory >>x=5.0; >>test >>y y = 25.0 >> test.m script

42. MATLAB m-files Script Files script files can call other script files >>outter y = 36.0 >> inner.m script outter.m script

43. MATLAB m-files Function Files Matlab identifies function files from script files by using the “function” and “return” keywords the name of the function file must be the same name as the function

44. The function file x2.m MATLAB m-files Function Files >>r=3; >>d=x2(r); >>d d = 9.0 >> >>h=x2(4.2); >>h h = 17.64 >>

45. MATLAB m-files Function Files Multiple Inputs and Outputs outputs in square brackets, [ ] inputs in parentheses ( )

46. variables created in the function are not retained in the workspace, except for the output variables the function does not have access to workspace variables, except for the inputs MATLAB m-files Function Files variables passed to the function are “copies” of the workspace variables. Changing their value inside the function has no effect on their value in the workspace.

47. MATLAB Flow Control The “while” and “if” statements if expression statements end if expression statements1 else statements2 end Matlab evaluates expression as logical “true” or “false” “false” equivalent to zero “true” equivalent to any non-zero number statements, any valid Matlab command while expression statements end

48. MATLAB Flow Control evaluating expression any valid equation a=4; b=5; c=5; if a+b if b-c “True” “False” watch out for round-off and word length error if sin(0) if sin(pi) sin(pi) = 1.22e-16 “False” “True” conditional operators == equal to < less than > greater than <= less than or equal to >= greater than or equal to ~= not equal to logical operators & and | or while(3<=a)&(a<=5)

49. MATLAB Flow Control The “for” statement for index = start : [increment :] end statements end index, start, increment, and end do not need to be integer valued increment is optional, if increment is not specified increment defaults to 1 index can be incremented positive (increment > 0) or negative (increment < 0) loop stops when index > end (or index < end)

50. MATLAB Flow Control example script file to cycle through x values function file to generate the y values

51. MATLAB Plotting Basic 2D plotting functions plot(x1,y1[,x2,y2,x3,y3.....]) xlabel(‘x axis name’) ylabel(‘y axis name’) title(‘graph name’) Additional functions grid on grid off axis([xmin,xmax,ymin,ymax])

52. MATLAB Plotting example y = sin(t) the “plot” function alone

53. MATLAB Plotting example y = sin(t) script file to generate a graph of y = sin(t)

54. function file to generate a graph of y = sin(t) >>graphsin >> MATLAB Plotting example y = sin(t)

55. “legend” remembers the order the graphs were plotted MATLAB Plotting Adding a Legend for multiple graphs