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