NumericalAnalysis-683-2010jan

# NumericalAnalysis-683-2010jan - MAT 683 Numerical Analysis...

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Unformatted text preview: MAT 683 Numerical Analysis January, 2010 NAME: 1. (a) Let S be a linear spline function that interpolates f at a sequence of nodes 0 = x < x 1 < ··· < x n = 1. Find an expression of R 1 S ( x ) dx in terms of x i and f ( x i ), i = 0 , 1 ,...,n . (b) Can a and b be defined so that the function ( x- 2) 3 + a ( x- 1) 2 , x ∈ (-∞ , 2]; ( x- 2) 3- ( x- 3) 2 , x ∈ [2 , 3]; ( x- 3) 3 + b ( x- 2) 2 , x ∈ [3 , ∞ ); is a natural cubic spline? Why or why not? 2. Let I = [0 , 1]. Let k ≥ 2. Let { ξ 1 ,...,ξ k- 1 } be the roots of L k , where L k is the Legendre polynomial of degree k . Recall that the Legendre polynomial are such that Z 1 L m ( x ) L n ( x ) dx = 1 2 m + 1 δ mn , ≤ m,m ≤ k and P n = span {L ,..., L n } . (a) Show that R 1 L n ( x ) q ( x ) dx = 0 for all q ∈ P n- 1 . (b) Let θ ,...,θ k be the Lagrange polynomials associated with the nodes { ξ ,ξ 1 ,...,ξ k- 1 ,ξ k } where ξ = 1 and ξ k = 1 (i.e., θ i ∈ P k and θ i ( ξ j ) = δ ij ). How should the weights)....
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NumericalAnalysis-683-2010jan - MAT 683 Numerical Analysis...

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