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Unformatted text preview: 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 , x 1 , . . . , x n a Leja ordering can be computed in O ( n 2 ) operations. 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) 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....
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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