assign6 - MATH 235/W08: Orthogonality; Least-Squares;...

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Unformatted text preview: MATH 235/W08: Orthogonality; Least-Squares; & Best Approximation Assignment 6 Hand in questions 1,2,6,8 by 9:30 am on Wednesday March 19, 2008. Contents 1 Least Squares and the Normal Equations*** 2 2 Best Polynomial Fit*** 2 3 Least Squares Solutions and Errors 3 4 Best Quadratic Polynomial Fit 3 5 Best Approximation in Continuous Function Space 3 6 Orthogonality*** 3 7 MATLAB; Best Line Fit 4 8 MATLAB; Best Quadratic Fit*** 4 9 BONUS: Eigenvalues and Best Approximation 5 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 =...
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assign6 - MATH 235/W08: Orthogonality; Least-Squares;...

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