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Unformatted text preview: Durham University EXAMINATION PAPER date May/June 2009 exam code MATH2051/01 description Numerical Analysis II Time allowed: 3 hours Examination material provided: None Instructions: Credit will be given for the best FOUR answers from Section A and the best THREE answers from Section B. Questions in Section B carry TWICE as many marks as those in Section A. Approved electronic calculators may be used. Visiting Erasmus students are permitted dictionaries. ED01/2009 Durham University Copyright continued page number 2 exam code MATH2051/01 SECTION A 1. (a) Consider the following iterative formula p n +1 = 1 2 p n + 1 p n , for n . Quoting any theorems that you use, prove that the iterative formula converges to 2 whenever p > 2. (b) Use the fact that 0 < ( p- 2) 2 whenever p 6 = 2 to show that if 0 < p < 2, then p 1 > 2. (c) Use the results of part (a) and (b) to show that the sequence in (a) converges to 2 whenever p > 0 and calculate the fixed point correct to 4 decimal places starting with p = 1. 2. (i) The equation 2 x- cosh( x ) = 0 has a solution s in [0 , 1]. Find the interpolation polynomial on x = 0, x 1 = 0 . 5 and x 2 = 1 for the function on the left side of the equation. By setting the interpolation polynomial equal to zero and solving the equation, find an approximation to s ....
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