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# Engineering Math III Matrix Algebra 22M 033

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This 8 page Class Notes was uploaded by Virgil Wyman on Friday October 23, 2015. The Class Notes belongs to 22M 033 at University of Iowa taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/227994/22m-033-university-of-iowa in Mathematics (M) at University of Iowa.

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Date Created: 10/23/15

22M033 J Simon Approximate Solution of an OverDetermined Linear System i e Method of Least Squares Suppose we have a system of 5 equations in 2 unknowns Your experience tells you that in general there is no solution But how close can we come to a solution This is not just an quotacademicquot question Here is one reason to care about this question Example You are collecting data about the relative behavior of two variables x and y You observe the following data points XQ 12 23 33 44 145 with linalg with plots DataLiSt 12 I 23 I 33 I 4 4 I 5I45 DataList l 2 2 3 3 3 4 4 5 45 V V V Plot39I39hePoints plot DataLi st sty1epoint colorb1ue thickness4 1abe15xY Tlr lV Vl display PlotThePoints 45 o 4 o 35 Y I 3 O O 25 20 I I I I I I I I I I I I I I I I II 1 2 3 4 X Suppose you believe either because of some deeper understanding you have of how X and y are connected or just because you look at the plot of data points and make a geometrically inspired guess that the quantities X and y actually are related linearly e g ymXb for some m and b and that the fact that these observed points do not lie on a straight line is just an artifact the variables actually do lie on a line but there is experimental or observational error in the data recorded Can we nd a quotbest linequot that gives in some sense the best estimate of a linear relation between X and y For eXample here are three lines yXl y09X 09 y08 X 07 Which is a better fit to the data What would be the quotbestquot line gt gt LineEquation1x1 LineEquation x l gt LineEquationZ 0 9 x 0 9 LineEqualion2 09 x 09 igt LineEquation3 0 8x0 7 L LineEqualionj 08 x 07 gt Plot39I39heLines plot LineEquationl LineEquationZ LineEquation3 x0 5 colorred thickness2 gt display PlotThePoints PlotTheLines O Let39s approach this as a linear algebra problem We seek numbers mb such that if it were possible which it is not each of the 5 points lies on the line ymxb That means we wish mb was a solution of the linear system of 5 equations in 2 unknowns 2m lb 3m2b 3m3b 4m4b 45 m 5b Write this system in matrix form 7gt mmatrixltn1112113114115111gt l 2 1 Mi 3 1 4 l 5 l 7gt Ymatrix2334457 2 3 Y 3 4 45 7gt mbMatrixmatrix m b mbMatrix b gt LinearSystemevalm M evalm mbMatrix evalm Y l l 2 2 l 3 m LinearSystem 3 1 3 b 4 l 4 5 l 45 There is no point in trying to solve this system in the usual way adjoin the column of Y values to the coefficient matrix and then rowreduce the augmented coef cient matrix All you39ll get is that the system is inconsistent But just to be thorough lgt AconcatMY 2 1 3 A 3 1 3 4 1 4 5 1 45 7gt rrefA 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 The third row says the system is inconsistent ie there is no straight line that passes through all five points Now let39s do some magic 71F we had solutions to the system M unknown vector of mb Y vector then the mb vector would still satisfy any new system we get by multiplying both sides of this matrix equation by some other matrix That is Mcolumn mb Y gt WMcolumn mb WY In particular do this with MaransposeM gt Wtranspose M gt NewCoeffMatrix evalm WampM NewCoe atrix 15 5 gt NewYVectorevalmWampY 555 New Y Vector 165 gt NewLinearSystemeva1mNewCoeffMatrix evalmmbMatrixevalmNewYVector 15 m 5 5 5 NewLinearSystem 15 5 b 165 THIS system we can solve gt NewAugmentedCoeffMatrix concat NewCoeffMatrix NewYVector 5 5 l 5 5 55 NewAugm entedCoe llatrix 15 5 165 gt rrefNewAugmentedCoeffMatrix l 0 06000000016 0 l 1499999994 Assuming the decimal bits are due to computer arithmetic we have m06andbl5 Let39s see what that line looks like relative to the data points gt LeastSquaresLine 0 6x1 5 LeastSquaresLine 06 x 15 gt PlotLeastSquaresLine plot LeastSquaresLine x0 5 colormagenta thickness4 igt displayP1 quotquot 1 quotno P1 quotquot oints 3 01 0 N m gt When you have an overdetermined linear system multiplying both sides by the transpose of the coefficient matrix gives a system whose solution will be an approximate solution to the original problem In fact it will give you the quotbest possiblequot approximate solution in the sense that a measure of error will be smallest The line we just found is the same line as you get using the quotmethod of least squaresquot as perhaps taught in multivariable calculus class We consider hypothetical mb and compare the predicted yvalue ie mxb to the observed yvalue at each of the observation points For each pair of observed vs predicted data points we measure the error by taking the square of the difference between the predicted value mxb and the observed value y Then add up these squared errors to get ameasure of the total error for a given choice of mb This error function Emb is a just a quadratic polynomial in m and b so it is easy to use partial derivatives dEdm 0 dEdb 0 to find which mb gives the smallest error If you do this with the above example of 5 data points you ll get the same NewLinearSystem gt xlistconvertcolM1 1ist 7 Xlz39st 1 2 3 4 5 gt Ylist convert colY 1 list le39st 2 3 3 4 45 gt Predicted mx1ist1b mx1ist2 b mx1ist 3b mx1ist 4b mx1ist5b PredictedYmb2mb3mb4mb5mb f ErrorFunction mb gt sum Predicted i Ylist i quot2 i1 5 5 2 ErrorFunclion m b a E Predictele Yllsti i 1 gt EErrorFunction mb EF mb2f2mb3f3mb3f4mb4f5mb4 2 gt Eq1diffEm0 Eq110m30b11100 V Eq2 diffEb0 Eq230m10b3300 gt r L 1 L quotnnffini 39 39 39Ymatrix110 30 1113o 10 33H 110 30 111 30 10 33 Compare this to the augmented coef cient matrix we got from the quotunsolvablequot overdetermined system by multiplying both sides by transposeM gt evalm NewAugmentedCoe ffMat rix end of handout

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