Lecture20101028_1

Lecture20101028_1 - Numerical Analysis II Dr Abigail Wacher...

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Numerical Analysis II Dr Abigail Wacher Oct 28, 2010 1 Convergence of an iteration sequence can be very slow, i.e. many iterations may be required to reach the desired accuracy. When possible we would like to accelerate. We begin by understanding the order of convergence and then introduce an acceleration method for linearly converging sequences. Define the error of the itereation after n steps as e d p K where , e Z is the iteration sequence for g and is the fixed point of . We will use our knowledge of the error (rate of convergence) in order to reduce the error, leading to a sequence which converges more quickly to . Assuming that is a contraction mapping. and has as many continuous derivatices as required, we can use Taylors theorem. C 1 = C 1 K = K gp = C K = ' C 1 2 '' 2 C O 3 lim / N C 1 = lim / N ' = ' Since is a contraction mapping L ! 1 We consider 2 cases i) s 0 ii) = 0 (i) It can be shown that if s 0 c d lim / N Asymptotically p cL [[ p asymptotically like]]
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Lecture20101028_1 - Numerical Analysis II Dr Abigail Wacher...

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