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set12 - $ Set 12 Orthogonality and ApproximationPart 3 Kyle...

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a39 a38 a36 a37 Set 12: Orthogonality and Approximation- Part 3 Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 2 Spring 2011 1
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a39 a38 a36 a37 Discrete Least Squares Suppose x 0 < x 1 < · · · < x m are given and the metric m summationdisplay i =0 ω i ( f ( x i ) p n ( x i )) 2 with ω i > 0 is used to determine the polynomial, p * n ( x ) , of degree n that achieves the minimal value. Typically, m n . If m = n then the unique interpolating polynomial is the solution. 2
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a39 a38 a36 a37 Discrete Least Squares Suppose we have a basis of polynomials ( φ 0 ( x ) , . . . , φ n ( x ) ) and let p n ( x ) = n summationdisplay j =0 φ j ( x ) ξ j then the conditions are ρ 0 ρ 1 . . . ρ m = ω 1 / 2 0 f ( x 0 ) ω 1 / 2 1 f ( x 1 ) . . . ω 1 / 2 m f ( x m ) ω 1 / 2 0 φ 0 ( x 0 ) . . . ω 1 / 2 0 φ n ( x 0 ) ω 1 / 2 1 φ 0 ( x 1 ) . . . ω 1 / 2 1 φ n ( x 1 ) .
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