solhw3 - Solutions for Homework 3 Foundations of...

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Solutions for Homework 3 Foundations of Computational Math 2 Spring 2011 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. Solution: The following MATLAB code due to Higham (see Higham 2002 Second Edition) shows how a Leja ordering can be computed in O ( n 2 ) operations. function [a,perm] = leja(a) n = max(size(a)); perm= (1:n)’; [t,i] = max(abs(a)); if i ~= 1 a([1 i]) = a([i 1]) ; perm([1 i]) = perm([i 1]) ; end p = ones(n,1); for k=2:n-1 for i=k:n p(i) = p(i) *(a(i)-a(k-1)); end [t, i]=max(abs(p(k:n))); i = i+k-1; if i ~=k a([k i]) = a([i k]); p([k i]) = p([i k]); perm([k i]) = perm([i k]); end end 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) 1
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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. 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 . Let ˜ p n ( x ) = | α 0 | + | α 1 | x + · · · + | α n | x n Show that | p n ( x ) ˆ c 0 | | p n ( x ) | γ 2 n ˜ p n ( | x | ) | p n ( x ) | (1) and therefore κ rel = ˜ p ( | x | ) | p ( x ) | is a relative condition number for perturbations to the coefficients bounded by γ 2 n . 2
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3.2.b Equation 1 also yields an a priori bound on the forward error | p n ( x ) ˆ c 0 | that can be computed along with evaluating p n ( x ) with Horner’s rule. Write a code that evaluates p n ( x ) and the forward error bound using Horner’s rule and p n ( x ) using the product form . Apply the code to the polynomial p 9 ( x ) = ( x 2) 9 = x 9 18 x 8 + 144 x 7 672 x 6 + 2016 x 5 4032 x 4 + 5376 x 3 4608 x 2 + 2304 x 512 to evaluate p 9 ( x ) in both forms and the a priori bound on forward error at several hundred points in the interval [1 . 91 , 2 . 1].
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