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# assign6 - MATH 235/W08 Orthogonality Least-Squares& Best...

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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 ˆ x and then solving for ˆ 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& Best...

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