1 the roots satisfy 0 the roots of 0 are i

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Unformatted text preview: yn;i = h 0 f (tn yn) + h k X i=1 i fn;i (5.7.1) where 0 = 1 (as a normalization), 0 6= 0, and yn;i and fn;i, i = 1 2 : : : k, are assumed known. Equation (5.7.1) has a unique solution for h su ciently small that may be computed by functional iteration yn = ; () k X i=1 i y n; i + h 0 f (tn yn ( ;1) ) + h X k i=1 i fn;i = 1 2 ::: (5.7.2a) (0) with yn being an initial guess for yn. Convergence of the iteration occurs when h satis es hLj 0 j < 1 45 (5.7.2b) where L is a Lipschitz constant for f (t y). If f 2 C 1 then we may take L = max jfy (tn y)j y2I I = yn ; yn + ]: (5.7.2c) For non-sti problems, where L is small, the step size is usually determined by accuracy conditions rather than by (5.7.2b). Thus, functional iteration will give acceptable performance. However, for sti problems, where L is large, the step size is severely restricted by (5.7.2b). Newtons iteration is typically provides better performance in these cases. With Newton's method, we nd the roots of F (yn) = yn + k X i=1 i yn;i ; h 0 f (tn yn) ; h k X i=1 i fn;i : (5.7.3a) As noted in Section 2.2, Newton's method is obtained by linearizing (5.7.3a) about an ( iterate yn ;1) thus, ( ( ( ( ( 0 = F (yn )) = F (yn ;1)) + Fy (yn ;1))(yn ) ; yn ;1)): Using (5.7.3a) to calculate the derivative yields ( ( ( ( 1 ; h 0 fy (tn yn ;1))](yn ) ; yn ;1)) = ;F (yn ;1) ) = 1 2 ::: (5.7.3b) (0) Newton iteration converges when the initial guess yn is su ciently close to yn and/or h is su ciently small. Unfortunately, it is di cult to get bounds like (5.7.2b,c) for Newton's method however, typically h does not have to be as small as required for convergence with functional iteration (5.7.2). (Problem 1 contains a simple example.) Newton's iteration usually converges when the Jacobian fy is not reevaluated after each iteration. Thus, for example, we may use the same Jacobian for the entire iteration or even for iterations at several times. With either functional or Newton iteration, a function evaluation must be made at each iteration. For vector systems, it is important to minimize the number of function (0)...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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