Lecture 8 Slides

Lecture 8 Slides - SecantMethod (x0,x1 ( Iterativeformula...

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10/6/2010 1 Secant Method Requires two initial values ( x 0 , x 1 ) Initial values do not have to lie on either side of root (not a bracketing method) Iterative formula: ) )( ( 1 i i i x x x f Like Newton Raphson but with numerically estimated derivative. ) ( ) ( 1 1 i i i i x f x f x x ) ( ) ( ) )( ( ) ( / ) ( ) ( ) ( : Raphson Newton 1 i i i i i i i f f f x f x f x x f x f x x ) ( / ) ( ) ( ) ( : Secant 1 1 1 1 1 i i i i i i i i i i i i i x x x x x f x x f x f x x x x x Approximate derivate Modified Secant Method Only one initial value required ( x 0 ) Iterative formula: ) ( ) ( ) ( 1 i i i i i i i x f x x f x f x x x where δ is a small perturbation fraction Same basic idea but derivative estimated in a slightly different way i i i i i i i i i i i i x f x x f x f x x f x x x x f x f x x f x f x x ) ( ) ( / ) ( ) ( : Secant Modified ) ( / ) ( ) ( ) ( : Raphson Newton 1 Approximate derivative i i i i i i i i x x f x x f ) ( ) ( 1 Choice of δ important: too small > numerical errors too large > inefficient, method may diverge

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