Lecture20101028_2

Lecture20101028_2 - Numerical Analysis II Dr Abigail Wacher...

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Numerical Analysis II Dr Abigail Wacher Oct 28, 2010 1 Newton's Method gx = x K fx f ' Since g ' = '' ' 2 ' p = 0 as long as ' s 0 so care must be taken if has a double root or higher multiplicity root. Provided the starting value is "close" to and ' s 0, speed of convergence is quadratic. Motivating using the definition of the derviative of a function, replace ' by the secant. ' n z K K 1 K K 1 As the points get closer together secants method approaches the tangent Secant Method C 1 = K K K 1 K C 1 Note resemblance to Regula Falsi! The secant method is slower than the Newton method, but is still superlinear. It can be shown that for the secant method e C 1 z ce k K 1 where c is a constant = K From this one can derive C 1 z h where h = and the exponent satisfies 2 K K 1 = 0 taking the positive root = 1 2 1 C 5 z 1.61803398875 This number is related to the "Golden Section" popular in the ancient Greek civilization. There is also relation with the Fibonacci Sequence.
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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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Lecture20101028_2 - Numerical Analysis II Dr Abigail Wacher...

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