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Homework 2 due Thursday October 21 before 13:30 in corre- sponding group folder on door of office CM110 Let a > 0. Could the iterative formula p n +1 = a/p n be used to solve x 2 = a , i.e. to find the square root of a ? Give reasons for your answer. Let g ( x ) := a/x and note that g ( ± a ) = ± a . Let p 0 = α , then p 1 = a/α and p 2 = α . Therefore the sequence oscillates about the fixed point and will not converge unless you start at the fixed point p = ± a . As in the last assignment we wish to compute the cube root of 5, we could proceed by finding the zeros of f ( x ) = x 3 - 5. Let the initial interval be defined by a = 0 and b = 3 and calculate an approximation for 5 1 / 3 using the false position algorithm.
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Unformatted text preview: Compare your new results to those found using the bisection method. Which method is converging faster? False position — with values quoted correct too 4d.p. n 1 2 3 4 5 a n . 5556 0 . 9955 1 . 3048 1 . 4949 b n 3 3 3 3 3 p n . 5556 0 . 9955 1 . 3048 1 . 4949 1 . 6004 f ( a n )-----f ( b n ) + + + + + f ( p n )-----Our final false position estimate is p 6 = 1 . 6555 correct to 4d.p. which has a rather larger error! In this case bisection seems to be converging faster to the root p ≈ 1 . 7100 MATH2051 Homework 1 2010-10-18...
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This note was uploaded on 02/09/2011 for the course MATH 2051 taught by Professor Dr.a.wacher during the Fall '11 term at Durham.

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