roots - Finding Roots Strategy for finding f () = 0: x need...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Finding Roots Strategy for finding f () = 0: x need a good first guess (graph f (x)) bracket root if possible (guarantees convergence of bisection and false position) beware of vertical asymptotes! tune method to problem: bisection (to get near root) + Newton (for quadratic convergence near root) is a good combination Newton's Method 0 = f (x + x) f (x) + f (x)x f (x) , f (x) x = - f (x) = 0 Secant Method xn+1 - xn = - f (xn )(xn - xn-1 ) f (xn ) - f (xn-1 ) Slower than Newton, but more stable. If the root is always bracketed, the secant method becomes the method of false position, and convergence is guaranteed. Convergence of Newton's Method Suppose the true root is x. Define the error en = xn - x. Then en+1 = en - f (en + x) f () x en f () + e2 f ()/2 + x x n = en - e2 n f (en + x) f () + en f () + x x 2f () x Quadratic convergence! ...
View Full Document

This note was uploaded on 03/11/2012 for the course MAT 421 taught by Professor Staff during the Fall '11 term at ASU.

Ask a homework question - tutors are online