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Unformatted text preview: Solutions for Homework 3 Foundations of Computational Math 2 Spring 2011 Problem 3.1 Show that given a set of points x , 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:n1 for i=k:n p(i) = p(i) *(a(i)a(k1)); end [t, i]=max(abs(p(k:n))); i = i+k1; 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 + 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 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 ( 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 = 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 = (1 + θ 1 ) α + (1 + θ 3 ) α 1 x + ··· + (1 + θ 2 n ) α n x n where  θ k  ≤ γ k . Let ˜ p n ( x ) =  α  +  α 1  x + ··· +  α n  x n Show that  p n ( x ) − ˆ c   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 3.2.b Equation 1 also yields an a priori bound on the forward error  p n ( x ) − ˆ c  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...
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This note was uploaded on 11/10/2011 for the course MAD 5404 taught by Professor Gallivan during the Spring '11 term at FSU.
 Spring '11
 gallivan

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