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Unformatted text preview: CS205 - Class 7 Nonlinear Equations Continued 1. We usually solve nonlinear equations with iterative methods: 1 x , 2 x , 3 x … n x stopping when the error, k k exact e x x =- , is small, i.e. when k e ε < . a. The convergence rate is determined by looking at the equation 1 r k k e C e + = with C ≥ in the limit as k → ∞ . i. If 1 r = , then we need 1 C < to guarantee convergence. In this case we say that the convergence rate is linear . ii. If r > 1 we say that the convergence rate is superlinear. iii. If r = 2 we say that the convergence rate is quadratic. iv. These terms state how fast we converge once convergence is occurring. However, there is no guarantee of converging to the root we want, or to any root in general. 1. Moreover, recall that our nonlinear function might be an approximation to what we really want, so how many digits of accuracy we need is important in deciding how fast things need to converge. 2. fixed point iteration- iterate 1 ( ) k k x g x + = to find ( ) x g x = a. locally convergent if '( *) 1 g x < , i.e. if the initial guess o x is close enough to * x the method will converge i. 1 1 * ( ) ( *) '( )( *) '( ) k k k k k k k e x x g x g x g x x g e θ θ + + =- =- =- = for k θ between k x and * x as determined by the Mean Value Theorem. Show that if ( 29 * = ′ x g and ( 29 x g ′ ′ is bounded, then convergence is in fact quadratic ii. if all the '( ) 1 k g C θ ≤ < , then 2 1 2 k k k k o e C e C e C e-- ≤ ≤ ≤ ≤ L so that as k → ∞ , k C → and k e → b. only converges if the initial guess is close enough to the solution...
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- Fall '07
- Numerical Analysis, A. Newton, xk, xk − xk