Introduction to MATLAB

Transcription

1 Sven K. Esche Department of Mechanical Engineering January,

2 Learning Objectives Using MATLAB, the student will be able to perform the following: Run MATLAB Manage MATLAB workspace Use MATLAB Editor to create m-files Input/output data from/to files Define constants, variables, vectors and matrices Perform simple calculations using numbers, vectors, matrices, operators and functions January, Sven K. Esche

3 Learning Objectives (cont.) Solve systems of linear equations Solve eigenvalue problems Generate D and 3D plots Solve systems of linear and nonlinear ODEs Solve unconstrained optimization problems Perform numerical integration Develop animations January, Sven K. Esche 3

4 MATLAB Background MATLAB (MATrix LABoratory) is highperformance language for technical computing MATLAB integrates computation, visualization, and programming in an easy-to-use environment Problems and solutions are expressed in familiar mathematical notation Basic data element is an array that does not require dimensioning demo MATLAB demonstration program January, Sven K. Esche 4

5 MATLAB Environment Prompt: (>>) indicates that MATLAB is ready to accept commands clc clears the Command Window % indicates a comment When a line ends with a semicolon (;), MATLAB performs the computation but does not display any output Three periods (...) followed by Return or Enter indicate that the statement continues on the next line format short/long/short e/long e formatting January, Sven K. Esche 5

6 MATLAB Workspace Command whos displays the name, size, and data type information for all variables defined in the workspace Command clear deletes all existing variables from the workspace January, Sven K. Esche 6

7 Computer Limits Command [ctype, maxsize] = computer displays type of computer ctype and max. array size maxsize Command intmax displays largest integer value Commands realmin /realmax display smallest / largest positive floating point number representable on the computer Command eps floating point accuracy January, Sven K. Esche 7

8 Example: Environment Perform the following operations: Clear workspace and command window Switch on echo function Evaluate expression - / + /3 - /4 + /5 - /6 + /7 - /8 + /9 - / + / - / using line continuation Suppress output when evaluating expression - / + /3 - /4 + /5 - /6 + /7; January, Sven K. Esche 8

9 Example: Environment (cont.) List name, size and data type of all variables in workspace Display current search path and current directory List all MATLAB files and all files in current directory January, Sven K. Esche 9

10 MATLAB Search Path When the name test is entered at the MATLAB prompt, the MATLAB interpreter Looks for test as a variable Checks for test as a built-in function Looks in the current directory for a file named test.m Searches the directories on the search path for test.m January, Sven K. Esche

11 MATLAB Search Path (cont.) Command path displays the current search path Command addpath /home/me64 appends the directory /home/me64 to the current path Command rmpath /home/me64 removes the directory /home/me64 from the current path Command pathtool starts the interactive Path Browser January, Sven K. Esche

12 MATLAB Search Path (cont.) Command cd (without any arguments) displays the current directory Command cd with a path changes the current directory Command what lists all MATLAB files in the current directory The exclamation point (!) is a shell escape (the rest of the input line is a command to the operating system) January, Sven K. Esche

13 M-Files Command type data.m displays existing m-file data.m Command edit launches the MATLAB Editor/Debugger with empty m-file Command edit data.m launches the MATLAB Editor/Debugger with existing m-file data.m Command dir lists all files in the current directory January, Sven K. Esche 3

14 Data Input / Output inputvar = input( input prompt ) displays string input prompt, waits for value to be entered from keyboard and assigns it to variable inputvar isempty(a) returns if array A is empty and otherwise save( filename, var, var, -ascii) save data in var and var to ASCII file inputvar = textread( filename', '%f') read floating point value from file January, Sven K. Esche 4

15 Program Flow Control if, else, elseif execution of statements if certain expression is TRUE switch, case, otherwise execution of statements depending on value of expressions for, while, continue, break loop for repeated execution of statements try, catch error control return program termination January, Sven K. Esche 5

16 MATLAB Expressions Building blocks of MATLAB expressions are: Constants Variables Operators Functions January, Sven K. Esche 6

17 MATLAB Data Types Scalars Vectors Matrices Strings Special values January, Sven K. Esche 7

18 Numbers Conventional decimal notation: optional decimal point and plus or minus sign Scientific notation: e to denote power-often factor Complex numbers: imaginary unit i or j Numbers are internally stored using long format specified by IEEE floating-point standard Special constants: pi, Inf, NaN January, Sven K. Esche 8

19 Variables MATLAB does not require type declarations and definition of dimensions (automatic variable creation and memory allocation) Variable names start with letter followed by letters, numbers and underscores (no punctuation characters) case sensitive 3 significant characters January, Sven K. Esche 9

20 Operators Arithmetic operators: +, -,.*,./,.\,.^ Relational operators: <, <=, >, >=, ==, ~= Logical operators: AND (&), OR ( ), NOT (~) Special operators: %,,., :, ; January, Sven K. Esche

21 Arithmetic Operators + Addition - Subtraction * Multiplication / Division \ Left division (described later) ^ Power ( ) Specification of evaluation order January, Sven K. Esche

22 Relational Operators < less than <= less than or equal than > greater than >= greater than or equal than == equal to ~= not equal to January, Sven K. Esche

23 Logical Operators AND (&) TRUE if all inputs are TRUE OR ( ) TRUE if at least one input is TRUE NOT (~) TRUE if the input is FALSE January, Sven K. Esche 3

24 Special Operators % comment Complex conjugate transpose. elementwise execution of vector/matrix operations : creation of equally spaced vectors ; output suppression January, Sven K. Esche 4

25 MATLAB Commands <command>enter <variable> = <command>enter [<variables>] = <command>enter January, Sven K. Esche 5

26 Example: Numbers Perform the following operations: Define integer numbers 3 and -99 Define real numbers., ,.6 -, Define complex numbers +i, j and 3 5 i Define Explore expressions / and / Define variables my_favorite_number = 3 and x = January, Sven K. Esche 6

27 Mathematical Functions sin(x), cos(x), tan(x), asin(x), acos(x), atan(x), atan(x, y) trigonometric functions sinh(x), cosh(x), tanh(x), asinh(x), acosh(x), atanh(x) hyperbolic functions exp(x), a^x, log(x), log(x) exponential and logarithmic functions January, Sven K. Esche 7

28 Mathematical Functions (cont.) abs(z), angle(z), conj(z), imag(z), real(z) complex functions abs(x) absolute value sign(x) signum sqrt(x) square root January, Sven K. Esche 8

29 Example: Functions Evaluate the following functions: sin(), cos(), tan(), sin - (.5), cos - (.5), tan - (.5), tan - (.5/) sinh(), cosh(), tanh(), sinh - (.5), cosh - (.5), tanh - (.5) e.5,.5, ln.5, log.5 +i, arg(+i), Re(+j), Im(+j), -.5, sign(-.5),.5 i January, Sven K. Esche 9

30 Simple Calculations Type commands directly at prompt Generate m-file using MATLAB Editor/Debugger (command edit, select Save As from Editor/Debugger File menu, select Run Script from MATLAB Command Window File menu) Calculator approach: +3 Variable approach: a=; b=3; a+b January, Sven K. Esche 3

31 Matrices Rectangular arrangement of numbers Special matrices: -by- matrix (scalar) -by-n or n-by- matrices (vectors) Defining a matrix in MATLAB: Enter an explicit list of elements Load matrices from external data files Generate matrices using built-in functions Create matrices with your own functions in M-files January, Sven K. Esche 3

32 Special Matrices zeros(m,n) generates m-by-n matrix of zeroes ones(m,n) generates m-by-n matrix of ones rand(m,n) generates m-by-n matrix of random numbers that are uniformly distributed in the interval (,) eye(n) generates n-by-n identity matrix January, Sven K. Esche 3

33 Basic Matrix Operations A (complex conjugate) transpose of A fliplr(a) flip A left to right size(a) size (dimensions) of A sum(a) sum of elements of each column of A diag(a) main diagonal of A trace(a) sum of diagonal elements (trace) of A January, Sven K. Esche 33

34 Advanced Matrix Operations det(a) determinant of matrix A rank(a) rank of matrix A rref(a) reduced row echelon form of A cond(a) condition number of matrix A inv(a) inverse of matrix A lu(a) LU factorization of matrix A qr(a) QR factorization of matrix A orth(a) orthonormal basis for range of A January, Sven K. Esche 34

35 Matrix Subscripts A(i, j) element in i th row and j th column A(i, :), A(:, j) i th row and j th column Reference to element outside of range leads to error message Index exceeds matrix dimensions. Upon storing a value in element outside of range, matrix size is automatically increased January, Sven K. Esche 35

36 Generation of Vectors i:j generates row vector containing integers from i to j with unit spacing i:j:k generates row vector containing integers from i to k with spacing of j a:b:c generates row vector containing real numbers from a to c with spacing of b linspace(a, b, n) generates row vector containing n real numbers from a to b January, Sven K. Esche 36

37 January, Sven K. Esche 37 Introduction to MATLAB Perform the following operations: Define matrices A, B, C, D and E E rand rand rand rand rand rand D C B A Example: Matrices

38 Example: Matrices (cont.) Determine the following A T dimensions of det A A condition number of A sum of columns of A main diagonal of A Refer to element A(, 3) Assign value to element A(, 4) January, Sven K. Esche 38

39 Example: Matrices (cont.) Define row and column vectors a, b, c, d and e: a b c d e T January, Sven K. Esche 39

40 Solving Systems of Equations Solution of system of linear equations A * x = b If b is zero vector, either det(a) is zero or x is zero vector x = null(a) solution of A * x = If b is nonzero, det(a) must be nonzero x = A \ b (more accurate and faster) x = inv(a) * b (less accurate and slower) January, Sven K. Esche 4

41 January, Sven K. Esche 4 Introduction to MATLAB Example: Singular System Given: Coefficient matrix A Right-hand-side vector b Note: A T b

42 Example: Singular System (cont.) Find: Determinant and condition number of A Inverse of A Check accuracy of inverse Solve system of equations A x = b using inverse of A Check solution accuracy Solve system of equations A x = b using backslash operator Check solution accuracy January, Sven K. Esche 4

43 Example: Singular System (cont.) Solution: Determinant of A: det A Condition number of A: cond 6 A 3.83 Inverse of A: A January, Sven K. Esche 43

44 January, Sven K. Esche 44 Introduction to MATLAB Example: Singular System (cont.) Check accuracy of inverse of A: Solve system of equations A x = b using inverse of A T T b A x A A AA

45 Example: Singular System (cont.) Check solution accuracy Ax b Solve system of equations A x = b using backslash operator x T T A \ b 7 Check solution accuracy Ax b T 3 T T T January, Sven K. Esche 45

46 Solving Eigenvalue Problems Solve eigenvalue problem A * v = * v [V, ] = eig(a) Matrix V containing eigenvectors v i as columns Matrix containing eigenvalues i on main diagonal poly(a) returns row vector with coefficients of characteristic polynomial of A in order of descending powers January, Sven K. Esche 46

47 Example: Eigenvalue Problem Given: Matrix A: Find: A Eigenvalues and eigenvectors using eig function Check solution accuracy Coefficients of characteristic polynomial Roots of characteristic polynomial Eigenvectors using null function 5 8 January, Sven K. Esche 47

48 Example: Eigenvalue Problem (cont.) Solution: Eigenvalues using eig function: Eigenvectors using eig function: v V v v January, Sven K. Esche 48

49 Example: Eigenvalue Problem (cont.) Check solution accuracy: 4 Av v T Av v Av3 3v T.53 4 A Iv T A Iv A Iv T 3 3 T T January, Sven K. Esche 49

50 Example: Eigenvalue Problem (cont.) Coefficients of characteristic polynomial 3 p(): p Roots i of characteristic polynomial p(): Eigenvectors using null function: v V v v January, Sven K. Esche

51 -D Plotting Functions plot(x, y) plot -D lines (graphs) feather(u, v) plot vectors with components (u, v) emanating from equally spaced points along a horizontal axis quiver(x, y, u, v) plot vectors as arrows with components (u, v) at points (x, y) contour(x, y, z) plot contour lines of surface z = f(x, y) = const. January, Sven K. Esche 5

52 3-D Plotting Functions plot3(x, y, z) plot 3-D lines (graphs) surf(x, y, z) plot shaded surface z = f(x, y) mesh(x, y, z) plot wireframe mesh of surface z = f(x, y) quiver3(x, y, z, u, v, w) plot vectors with components (u, v, w) at points (x, y, z) January, Sven K. Esche 5

53 Plot Options title( text ) put title text above plot xlabel( text ), ylabel( text ), zlabel( text ) put label text on x-, y- or z-axis legend( text ) put legend on graphs grid put grid on plot axis( equal ) ensure uniform scaling of x-axis and y-axis axis([xmin xmax ymin ymax]) specify plotting range for x-axis and y-axis January, Sven K. Esche 53

54 Plot Options (cont.) linespec set line style, width, color, marker type, marker size -, - -, :, -: solid, dashed, dotted, dash-dot line linewidth n set line width of graphs to n dots r, g, b, c, m, y, k, w red, green, blue, cyan, magenta, yellow, black, white line color January, Sven K. Esche 54

55 Plot Options (cont.) +, o, *,., x plus, circle, asterisk, dot, cross marker hold on/off add to/overwrite current graph zoom on/off zoom in/out interactively view(az, el) set azimuth and elevation viewing angle of plot figure create another figure window January, Sven K. Esche 55

56 Exporting Graphics Files Generate graphic using plot command Select Export from Figure Window File menu Select file name and file type (*.tif, *.jpeg, etc.) from Export Window January, Sven K. Esche 56

57 Example: Plotting a Graph Given: Function z(t): z f t x t y t t x t e y t sin t Find: Plot graphs of x(t), y(t) and z(t) in interval [, ] Add title, axis labels, legend and grid to figure Export figure as jpg file January, Sven K. Esche 57

58 Example: Plotting a Graph (cont.) Solution: January, Sven K. Esche 58

59 Example: Surface Plot Given: Function z: Find: Surface plot Mesh plot Contour plot z f 3 3 x, y.5x 6xy 4y January, Sven K. Esche 59

60 Example: Surface Plot (cont.) Solution: Surface plot January, Sven K. Esche 6

61 Example: Surface Plot (cont.) Mesh plot January, Sven K. Esche 6

62 Example: Surface Plot (cont.) Contour plot January, Sven K. Esche 6

63 Solving Linear ODEs ode45 Numerical solution of ordinary differential equation y = f(t, y) [t, y] = ode45( filename, t, y) t row vector with discrete time values y matrix with solutions for y and y in first and second columns filename name of m-file containing y = f(t, y) y row vector with initial conditions y() and y () in first and second columns January, Sven K. Esche 63

64 Example: Van der Pol s ODE Given: Van der Pol s differential equation: d x/dt -m ( -x ) dx/dt + x = Initial conditions: x() =, dx/dt() = Find: Numerical solution using ode45 function Time plot and phase plane plot January, Sven K. Esche 64

65 January, Sven K. Esche 65 Introduction to MATLAB Example: Van der Pol s ODE (cont.) Solution: Convert nd order ODE into system of two st order ODEs Define time vector t and initial condition vector y y y y m y x x y y x x y y x x x m x dt x d x dt dx x x dt dx x m dt x d y y T T o o, x, x y

66 Example: Van der Pol s ODE (cont.) Results: January, Sven K. Esche 66

67 Example: Double Pendulum Given: Spring-loaded double pendulum System parameters m = m =. kg d =.5 m, l =.5 m k = k 3 = N/m, k = N/m Initial conditions () =. rad, () =. rad = =. rad/s January, Sven K. Esche 67

68 Example: Double Pendulum (cont.) Find: Numerical solution using ode45 function Time plot Solution: Governing equations (for small angles): m l m l k k d mgl k d k k d m gl k d 3 January, Sven K. Esche 68

69 January, Sven K. Esche 69 Introduction to MATLAB Example: Double Pendulum (cont.) Convert ODEs T m l y m gl d k k y d k y m l y m gl d k k y d k y y y y y y y y y y y y

70 Example: Double Pendulum (cont.) Results: January, Sven K. Esche 7

71 Unconstrained Minimization Define (column) residue vector R = f(x) Define (scalar) objective function F(x) = R(x) T R(x) Minimize objective function using [x, fval] = fminsearch( filename, x, options) filename name of m-file containing F(x) x row vector with initial guess for x January, Sven K. Esche 7

72 Unconstrained Minimization (cont.) Options Display off, iter, final level of output display during iteration TolX termination tolerance for x TolFun termination tolerance for F(x) MaxIter maximum number of iterations allowed MaxFunEvals maximum number of function evaluations allowed January, Sven K. Esche 7

73 Example: Banana Function Given: Rosenbrock's banana function (objective function: f f x f x, y x y x T x R x R x x y x Initial vector: x T T x y. Find: x y x Surface and contour plots of f(x, y) Local minimum using fminsearch function January, Sven K. Esche 73

74 Example: Banana Function Solution: Result (minimum): f * * y * x x x * * * T x y T * x January, Sven K. Esche 74

75 Example: Four-bar Linkage Given: Link lengths: l = 3 in, l = in, l 3 = 4 in, l 4 = 3.5 in Link orientations: = º, = 45º Find: Assembled configuration Animation for full crank rotation January, Sven K. Esche 75

76 Example: Four-bar Linkage (cont.) Solution: Define residue vector R = f(x i, y i, i ) x i : x-coordinates of link centers of gravity y i : y-coordinates of link centers of gravity i : link orientations Define (scalar) objective function F = R(x i, y i, i ) T R(x i, y i, i ) Minimize objective function using fminsearch( filename, x, options) January, Sven K. Esche 76

77 Example: Four-bar Linkage (cont.) Initial and final configurations: January, Sven K. Esche 77

78 Numerical Integration Using quad quad( filename, a, b, tol) evaluate definite integral using adaptive Simpson quadrature filename name of m-file containing function f(x) to be integrated a, b finite integration limits tol absolute error tolerance January, Sven K. Esche 78

79 Example: Quadrature Given: Function f(x) = sin(x) Find: Euclidean norm of f(x) on interval [, ] Solution: Analytical solution: f f b x f x,f x f x dx sin x x sin x 4 x. 775 January, Sven K. Esche 79 a dx

80 Example: Quadrature (cont.) Main program: nsin = quad( matlab_quad_sub',, ( * pi)); nsin = sqrt(nsin) Subroutine: function nsin = matlab_quad_sub(x); nsin = sin(x).^ ; Numerical result: sin(x) =.775 January, Sven K. Esche 8

81 Numerical Integration Using trapz trapz(x, y) evaluate definite integral using trapezoidal numerical integration x (uniformly or nonuniformly spaced) integration grid (row vector) y function values y i = f(x i ) January, Sven K. Esche 8

82 Example: Trapezoidal Rule Given: Function f(x) = sin(x) Find: Definite integral of f(x) on interval [, ] Solution: Analytical solution: sin xdx cos x January, Sven K. Esche 8

83 Example: Trapezoidal Rule (cont.) MATLAB program: x =.:pi/8.:pi; y = sin(x); trapz(x, y) Numerical result: sin xdx.9999 January, Sven K. Esche 83