solhw3

solhw3 - Solutions for Homework 3 Foundations of...

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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: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 ) = + 1 x + + n x n p n ( x ) can be evaluated using Horners rule (written here with the dependence on the formal argument x more explicitly shown) 1 c n ( x ) = n for i = n 1 : 1 : 0 c i ( x ) = xc i +1 ( x ) + 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 ) = n ( x 1 ) ( x n ) given by: d = n for i = 1 : n d i = d i 1 ( x 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 + ) , | | 2 n u where k = ku/ (1 ku ) and u is the unit roundoff of the floating point system used. 3.2.a An error analysis of Horners rule shows that the computed value of the polynomial satisfies c = (1 + 1 ) + (1 + 3 ) 1 x + + (1 + 2 n ) n x n where | k | k . Let p n ( x ) = | | + | 1 | x + + | n | x n Show that | p n ( x ) c | | p n ( x ) | 2 n p n ( | x | ) | p n ( x ) | (1) and therefore rel = p ( | x | ) | p ( x ) | is a relative condition number for perturbations to the coefficients bounded by 2 n . 2 3.2.b Equation 1 also yields an a priori bound on the forward error | p n ( x ) c | that can be computed along with evaluating p n ( x ) with Horners rule. Write a code that evaluates p n ( x ) and the forward error bound using Horners 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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solhw3 - Solutions for Homework 3 Foundations of...

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