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EPSII 59:006 Spring 2004

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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

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Managing Your Session CommandDescription casesen“Casesen off” turns off case sensitivity clcClears the command window clearRemoves variables from memory exist(‘name’)Does a file or variable have that name? help nameHelp on a topic lookfor nameKeyword search in help quitTerminates your session whoLists variables in memory whosLists variables and their sizes

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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

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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

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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

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Data File Handling in MATLAB Uses MAT-files or ASCII files MAT-files Useful if Only Used by Matlab save -ascii (for 8-digit text format) save -ascii –double (for 16-digit text format) save -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

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More on Matrices An m x n matrix has m rows and n columns A ij refers to the element in the i th row and j th 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.

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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

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

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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 3 rd column A(:,2:5) means all the elements of the 2 through 5 th columns

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

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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

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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)

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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) = 5 123 010 21 00 12 * = 57 00

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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) If the inside values are the same, the matrices can be multiplied. They are conformable for multiplication. The result will have this size. rows columns

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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

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

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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)

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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)) 102 121 201 =2-0-8=-6

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Solution of Linear Equations Perhaps the best thing about matrices is how well they deal with systems of simultaneous linear equations 3x+2y-z= 10 -x+3y+2z= 5 x-y-z= -1 32 32 1 x y z = * 10 5 A*X = B

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

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Why Do Some Matrices Not Have Inverses? X+2y= 0 X+2y= 0 A = 12 12 X = -2y, No solution X+2y= 0 2X+4y= 0 A = 12 24 X = -2y, No solution

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An Example x 1 – x 2 – x 3 – x 4 = 5 x 1 + 2x 2 + 3x 3 + x 4 = -2 2x 1 + 2x 3 + 3x 4 = 3 3x 1 + x 2 + 2x 4 = 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’

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Some Useful Matrix Commands CommandDescription mean(b)Finds average of each column of b max(b)Finds max of each column of b min(b)Finds min of each column of b sort(b)Sorts each column in b, ascending sum(b)Sums each column in b

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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

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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

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Plotting

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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)

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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

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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

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

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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

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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 ‘ ColorsData symbolLine 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

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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.

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

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

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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;

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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

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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)

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Reduced mesh waterfall(x,y,z)

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waterfall(y,x,z')

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