Lecture+17--Secant+_+fzero

Lecture 17-Secant_ - Newtons Method-Matlab code function[root,ea,iter]=newtraph(func,dfunc,xr,es,maxit,varargin if nargin<3,error'at least 3 input

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Newton’s Method--Matlab code >> [r,ea,it]=newtraph(@(x) 3*x+sin(x)-exp(x),@(x) 3+cos(x)-exp(x),1) r = 0.360421702960200 ea = 3.459226917839900e-011 it = 6 function [root,ea,iter]=newtraph(func,dfunc,xr,es,maxit,varargin) if nargin<3,error( 'at least 3 input arguments required' ), end if nargin<4|isempty(es),es=0.0001; end if nargin<5|isempty(maxit),maxit=50; end iter = 0; while (1) xrold = xr; xr = xr - func(xr)/dfunc(xr); iter = iter + 1; if xr ~= 0, ea = abs((xr - xrold)/xr) * 100; end if ea <= es | iter >= maxit, break , end end root = xr; x 1 = x 0 - f 0/ f 0 '
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Newton’s Method Example : f ( x ) = 3 x + sin( x ) - exp( x ) Iter. # x n f ( x n) e n= x n- r 1 0.333333333 -6.84E-02 2.71E-02 2 0.360170714 -6.28E-04 2.52E-04 0.346 3 0.360421681 -5.63E-08 2.25E-08 0.354 4 0.360421703 -6.66E-16 Comments: * It may not converge if x 0 is too far from x = r . * If x 0 is close to r , Newton's method converges very fast. x 0 = 0 doubling
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Newton’s Method Quadratic convergence: = constant Illustration of quadratic convergence: 0 2 4 6 8 10 12 0 0 0 0 0 0 0 0.01 1 100 parabola n Error=|xn-r| Linear convergence
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Newton-Raphson Method Examples of poor convergence or failure
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Secant Method Ø Use secant line instead of tangent line at f ( xi )
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Secant Method The formula for the secant method is Notice that this is very similar to the false position method in form Still requires two initial estimates But it doesn ' t bracket the root at all times -- there is no sign test ) ( ) ( ) )( ( i 1 i i 1 i i i 1 i x f x f x x x f x x - - - = - - +
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False-Position and Secant Methods
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Algorithm for Secant method Open Method . 1 Begin with any two endpoints [ a, b ] = [ x0 , x1 ] . 2 Calculate x2 using the secant method formula 3. Replace x0 with x1 , replace x1 with x2 and repeat from (2) until convergence is reached Use the two most recently generated points in subsequent iterations (not a bracket method!) ) ( ) ( ) )( ( i 1 i i 1 i i i 1 i x f x f x x x f x x - - - = - - +
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This note was uploaded on 11/07/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Spring '09 term at University of Florida.

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Lecture 17-Secant_ - Newtons Method-Matlab code function[root,ea,iter]=newtraph(func,dfunc,xr,es,maxit,varargin if nargin<3,error'at least 3 input

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