solns6 - MATH 235/W08: Orthogonality; Least-Squares;...

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Unformatted text preview: MATH 235/W08: Orthogonality; Least-Squares; & Best Approximation SOLUTIONS to Assignment 6 Solutions to questions 1,2,6,8. Contents 1 Least Squares and the Normal Equations*** 2 1.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Best Polynomial Fit*** 7 2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Least Squares Solutions and Errors 10 4 Best Quadratic Polynomial Fit 10 5 Best Approximation in Continuous Function Space 10 6 Orthogonality*** 10 6.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 7 MATLAB; Best Line Fit 12 8 MATLAB; Best Quadratic Fit*** 12 8.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 9 BONUS: Eigenvalues and Best Approximation 14 1 1 Least Squares and the Normal Equations*** Find the least-squares approximation and the error to a solution of A x = b , by constructing the normal equations for x and then solving for x : (a) A = 3 5- 2- 6- 5- 11 , b = 3 3 3 (b) A = - 2 i 1- 1 3 i- 1 + 2 i- 1 + 3 i , b = - 3 + i- 4 + 2 i 2 + i Note that the normal equations over C for A x = b are A * A x = A * b where the operation * denotes complex conjugation plus transposition, that is, A * = A T . (c) A = - 2- 1- 8- 6- 4 2- 3 8 1- 9- 2- 3 1- 1- 1- 4- 7 3- 7- 3- 1 1 9 8 3- 4- 2- 7- 2- 2 9 4 20- 51- 1- 23 37 7 25 56 17 48- 71- 18 17- 56 42 7- 58 4- 54- 107- 174- 50 60 84 , b = - 83- 30- 3- 67- 49 191 375- 676- 558 734 Hint: The file tempoutput.txt at http://orion.math.uwaterloo.ca/hwolkowi/henry/teaching/w08/235.w08/miscfiles/tempoutput.txt and MATLAB may help. 1.1 Solution BEGIN SOLUTION: (a) Following is the solution first using \ in MATLAB and then using GE. The MATLAB output follows. format compact A=[ 3 5-2-6-5-11 ]; b=[3;3;3]; disp(construct matrix and RHS for normal equations) construct matrix and RHS for normal equations AtA=A*A;Atb=A*b; disp(solve for xhat using \) 2 solve for xhat using \ xhat=AtA\Atb xhat = 4.0000-2.0000 disp(the error ||A xhat -b|| is) the error ||A xhat -b|| is norm(A*xhat-b) ans = 1.7321 disp(solve for xhat using GE with a check col.) solve for xhat using GE with a check col. disp(First form the augmented matrix) First form the augmented matrix augM=[AtA Atb ] augM = 38 82-12 82 182-36 disp(Now add a check column) Now add a check column augM=[augM augM*ones(3,1)] augM = 38 82-12 108 82 182-36 228 tempA=sym(augM) tempA = [ 38, 82, -12, 108] [ 82, 182, -36, 228] disp(Now start to pivot) Now start to pivot tempA(1,:)=tempA(1,:)/tempA(1,1) tempA =...
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solns6 - MATH 235/W08: Orthogonality; Least-Squares;...

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