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EPSII

EPSII. 59:006 Spring 2004. Outline. Managing Your Session File Usage Saving Workspace Loading Data Files Creating M-files More on Matrices Review of Matrix Algebra Dot Product Matrix Multiplication Matrix Inverse Solutions to Systems of Linear Equations

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EPSII

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  1. EPSII 59:006 Spring 2004

  2. Outline • Managing Your Session • File Usage • Saving Workspace • Loading Data Files • Creating M-files • More on Matrices • Review of Matrix Algebra • Dot Product • Matrix Multiplication • Matrix Inverse • Solutions to Systems of Linear Equations • Solutions Using Matrix Inverse • Solutions Using Matrix Left Division • Plotting

  3. Managing Your Session

  4. Space Workspace File named myspace.mat • Step 1 – Type • save myspace at prompt or • Step 2 – choose File and the • Step 3 – Save Workspace As • To restore a workspace use load • load myspace

  5. Matlab M-Files • Can also create a Matlab file that contains a program or script of Matlab commands • Useful for using functions (Discuss later) • Step 1 – create a M-file defining function • Step 2 – make sure the M-file is in your MATLAB path (Use Current Directory menu or Use View and check Current DIerectory) • Step 3 – call function in your code

  6. Matlab M-Files Example • Calculate the integral • an M-file called g.m is created and contains the following code • function f=g(x) • f = x.^2 - x + 2 • at the Matlab prompt write • >> quad('g',0,10) • ans= • 303.3333 • help g - Displays the comments in the file g.m

  7. Data File Handling in MATLAB • Uses MAT-files or ASCII files • MAT-files Useful if Only Used by Matlab • save <fname> <vlist> -ascii (for 8-digit text format) • save <fname> <vlist> -ascii –double (for 16-digit text format) • save <fname> <vlist> -ascii –double –tabs (for tab delimited format) • Matlab can import Data Files • To read in a text file of numbers named file.dat: A = load(‘file.dat’); (dat recommended extension) • The file ‘file.dat’ should have array data in row, col format • Each row on a separate line

  8. More on Matrices • An m x n matrix has m rows and n columns • Aij refers to the element in the ith row and jth column. • A = [1 2 3; 4 5 6] • Instead of using numbers as element in the matrix, you can use other vectors or matrices • A’ is the transpose of A (its rows and columns are interchanged.

  9. Matrix Operations • r*A a scalar times an array • A+B array addition (arrays must be the same size) • A.*B element by element multiplication of arrays • A*B dot product multiplication • A./B element by element right division

  10. Matrix Transposition • The transpose operator (‘) switches rows and columns • Row vectors become column vectors and vice versa A(3,2) A(2,3)

  11. Array Addressing • V(:) means all the elements of V • V(2:5) means elements 2 through 5 • A(:,3) means all the elements of the 3rd column • A(:,2:5) means all the elements of the 2 through 5th columns

  12. Example • Given: • What is A(2:3,1:2)?

  13. Review of Matrix Algebra • Dot Product • Matrix Multiplication • Matrix Inverse • Solutions to Systems of Linear Equations • Solutions Using Matrix Inverse • Solutions Using Matrix Left Division

  14. Dot Products • The dot product is the sum of the products of corresponding elements in two vectors. • Dot product = A*B = • Matlab Commands (either will work) • sum(A.*B) • dot(A,B)

  15. Matrix Multiplication • A = [1 2 3; 0 1 0], B = [2 1; 0 0; 1 2] • C = A*B • c(1,1) = 1(2) + 2(0) + 3(1) = 5 = *

  16. What Size Matrices Multiply? • Write the matrix dimensions in normal form: • (A_rows, A_columns) (B_rows, B_columns) • (3,4)(4,16) -> (3,16) • (2,12)(12,19)->(2,19) columns If the inside values are the same, the matrices can be multiplied. They are conformable for multiplication. The result will have this size. rows

  17. The Identity Matrix and Powers • Let I bet the identity matrix • For any matrix X, X*I = X = I*X • I = [1 0 0; 0 1 0; 0 0 1], for a 3x3 matrix • In general I can be any square size with the diagonal elements = 1, all others = 0 • To multiply a matrix by itself, use X^2, or, in general, X^n

  18. Inverse Matrices • Let A-1 be the inverse matrix of A • Then, A-1A = AA-1 = I • Before computers, finding inverses was a pain in the neck • Not all matrices have inverses

  19. Determinants • Most of the techniques to find inverses, and many other techniques that use matrices, at some point or other require determinants. These are special functions performed on matrices • If A is a 2x2 matrix, then the determinant is: Det(A) = a(1,1)*a(2,2) – a(1,2)*a(2,1)

  20. Determinants for 3x3 Matrices • A [ 1 2 3; 4 5 6; 7 8 9] • Det(A) = • + a(1,1)*(a(2,2)*a(3,3) – a(2,3)*a(3,2)) • - a(1,2)*(a(2,1)*a(3,3) – a(2,3)*a(3,2)) • + a(1,3)*(a(2,1)*a(3,2) - a(2,2)*a(3,1)) =2 -0 -8=-6

  21. Solution of Linear Equations • Perhaps the best thing about matrices is how well they deal with systems of simultaneous linear equations A*X = B = *

  22. We Can Solve This In 2 Ways • A*X = B • One way is with A inverse • A-1A*X = A-1B => I*X = A-1B => X = A-1B • Another way is with Left Multiplication (which is similar, but uses a better numerical technique) • X = A\B’

  23. Why Do Some Matrices Not Have Inverses? X = -2y, No solution A = X = -2y, No solution A =

  24. An Example x1 – x2 – x3 – x4 = 5 x1 + 2x2 + 3x3 + x4 = -2 2x1 + 2x3 + 3x4 = 3 3x1 + x2 + 2x4 = 1 A = [1 –1 –1 –1;1 2 3 1; 2 0 2 3; 3 1 0 2] B = [5 –2 3 1] X = inv(A)*B’ or X = A\B’

  25. Some Useful Matrix Commands

  26. If A = 5 9 3 2 8 2 3 4 6 2 1 3 9 2 1 2 B = mean(A) = 7.0 3.75 2.0 3.75 C = max(A) = 9 9 3 4 D = min(A) = 5 2 1 2 E = sum(A) = 28 15 8 11

  27. If A = 5 9 3 2 8 2 3 4 6 2 1 3 9 2 1 2 S = sort(A) = 5 2 1 2 6 2 1 2 8 2 3 3 9 9 3 4

  28. Plotting

  29. Plotting Basics • 2D Plots (x,y) • y=f(x) • plot(x,y) • 3D Plots (x,y,z): a.k.a. “surface plots” • plot3(x,y,z)

  30. Basic Commands • plot(x,y) • xlabel(‘Distance (miles)’) • ylabel(‘Height (miles)’) • title(‘Rocket Height as a Function of Downrange distance’) • grid • axis([0 10 0 100]) • clf % clear current figure

  31. Subplots • Subplots • subplot(m,n,p) • m = the number of figure rows • n = the number of figure columns • Divides figure window into an array of rectangular frames • Comparing data plotted with different axis types

  32. Plot Customization • Data Markers/Line Types/Colors • Labeling Curves/Data • Hold Function • hold on • when a plot requires 2+ plot() commands

  33. Special Plot Types • Logarithmic Plots • loglog(x,y) • semilogx(x,y) • semilogy(x,y) • Tick Mark Spacing and Labels • set(gca,’XTick’,[xmin:dx:xmax], ’YTick’,[ymin:dy:ymax]) • set(gca,’Xticklabel’,[‘Jan’,’Feb’,’Mar’]) • Axis • axis( [ xmin, xmax, ymin, ymax ] ) • Stem, Stairs, Bar Plots • Polar Plots

  34. Plot bells & whistles Various line types, plot symbols and colors may be obtained with PLOT(X,Y,S) where S is a character string from any or all the following 3 columns: S = ‘ linetype color shape of data annotation ‘ Colors Data symbol Line type b blue . point - solid g green o circle : dotted r red x x-mark -. dashdot c cyan + plus -- dashed m magenta * star y yellow s square k black d diamond v triangle (down) ^ triangle (up) < triangle (left) > triangle (right) p pentagram h hexagram

  35. Plot(X,Y,S) examples PLOT(X,Y,'c+:') plots a cyan dotted line with a plus at each data point PLOT(X,Y,'bd') plots blue diamond at each data point but does not draw any line. PLOT(X1,Y1,S1,X2,Y2,S2,X3,Y3,S3,...) combines the plots defined by the (X,Y,S) triples, where the X's and Y's are vectors or matrices and the S's are strings. For example, PLOT(X,Y,'y-',X,Y,'go') plots the data twice, with a solid yellow line interpolating green circles at the data points.

  36. Function Discovery • The process of determining a function that can describe a particular set of data • LINEAR • POWER • EXPONENTIAL • polyfit() command

  37. 3D Plots • 3D Line Plots • plot3(x,y,z) • Surface Mesh Plots • meshgrid() • mesh(), meshc() • surf(), surfc() • Contour Plots • meshgrid() • contour()

  38. Data m-file y = 1:5; x = 1:12; for i = 1:5 for j = 1:12 z(i,j) = i^1.25 * j ; end; end;

  39. data x = 1 2 3 4 5 6 7 8 9 10 11 12 y = 1 2 3 4 5 z = 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 2.3784 4.7568 7.1352 9.5137 11.892 14.270 16.6489 3.9482 7.8964 11.844 15.792 19.741 23.689 27.6376 5.6569 11.313 16.970 22.627 28.284 33.941 39.5980 7.4767 14.953 22.430 29.907 37.383 44.860 52.3372 Columns 8 through 12 8.000 9.000 10.000 11.000 12.000 19.02 21.405 23.784 26.162 28.541 31.58 35.534 39.482 43.430 47.378 45.25 50.911 56.568 62.225 67.882 59.81 67.290 74.767 82.244 89.720

  40. 3-D demo m-file surf(x,y,z) pause mesh(x,y,z) pause waterfall(x,y,z) pause waterfall(y,x,z’) pause bar3(y,z) pause bar3h(y,z)

  41. Reduced mesh waterfall(x,y,z)

  42. waterfall(y,x,z')

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