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hw3 - Homework 3 Foundations of Computational Math 2 Spring...

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Homework 3 Foundations of Computational Math 2 Spring 2011 Solutions will be posted Wednesday, 2/2/11 Problem 3.1 Show that given a set of points x 0 , x 1 , . . . , x n a Leja ordering can be computed in O ( n 2 ) operations. Problem 3.2 Consider a polynomial p n ( x ) = α 0 + α 1 x + · · · + α n x n p n ( x ) can be evaluated using Horner’s rule (written here with the dependence on the formal argument x more explicitly shown) c n ( x ) = α n for i = n - 1 : - 1 : 0 c i ( x ) = xc i +1 ( x ) + α i end p n ( x ) = c 0 ( x ) If the roots of the polynomial are known we can use a recurrence based on p n ( x ) = α n ( x - ρ 1 ) · · · ( x - ρ n ) given by: d 0 = α n for i = 1 : n d i = d i - 1 * ( x - ρ i ) end p n ( x ) = d n This algorithm can be shown to compute p n ( x ) to high relative accuracy. Specifically, d n = p n ( x )(1 + μ ) , | μ | ≤ γ 2 n u where γ k = ku/ (1 - ku ) and u is the unit roundoff of the floating point system used. 1
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3.2.a An error analysis of Horner’s rule shows that the computed value of the polynomial satisfies ˆ c 0 = (1 + θ 1 ) α 0 + (1 + θ 3 ) α 1 x + · · · + (1 + θ 2 n ) α n x n where | θ k | ≤ γ k .
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