hw2 - e n the error of the n-th approximation ( e n = p n-x...

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Homework 2 – Math 104A, Winter 2009 Due on Wednesday, January 21st, 2009 Section 2.1: 6, 12, 14, 16, and 20. Section 2.2: 1, 2, 3, 4, 11.a, 11.b, 11.c, and 23. Additional problem: Consider the iteration in Newton’s method: p n +1 = g ( p n ) where g ( x ) = x - f ( x ) f 0 ( x ) Prove that if f ( x * ) = 0, and f 0 ( x * ) 6 = 0, then g 0 ( x * ) = 0. Deduce from this that the convergence of Newton’s method is quadratic, i.e., if we denote by
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Unformatted text preview: e n the error of the n-th approximation ( e n = p n-x * ), then e n +1 ≤ Ke 2 n for some K > 0, as long as the initial iterate p is sufficiently close to x * . You may assume that f has infinitely many continuous derivatives. 1...
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This note was uploaded on 12/26/2011 for the course MATH 104a taught by Professor Staff during the Fall '08 term at UCSB.

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