Lecture 3 Typeset Notes

Lecture 3 Typeset Notes - ME218 Engineering Computational...

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1 ME218 Engineering Computational Methods Lecture 3 Notes Methods for Solving Nonlinear Equations (Continued) Newton’Raphson ±s’ Method This technique is a root-finding algorithm that uses the first few terms of the Taylor series of a function f(x) in the vicinity of a suspected root, x 0 . .... ) ( 2 ) ( ) ( ) ( ) ( ) ( 0 2 0 0 0 0 ! " " # ! " # ! $ % x f x x x f x x x f x f higher terms Now, for x to be the root, f(x) = 0. Thus, taking only the linear terms in the Taylor series expansion, ) ( ) ( ) ( 0 ) ( 0 0 0 x f x x x f x f " # ! $ $ Rearranging, x = x 0 – f(x 0 )/f’(x 0 ) , where x 0 is the initial guess. Generalizing the iterative process: x N+1 = x N f(x N ) / f’(x N ) , till | x N+1 x N | < e This method converges faster than the Bisection and the Regula-Falsi methods, but suffers from the drawback that the process might hit a local extremum and make the derivative term go to zero. In that case, the method becomes unstable, as there is a division by 0!
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2 Secant’Method
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Lecture 3 Typeset Notes - ME218 Engineering Computational...

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