Rational Root-Finding Method, the approximation in this case is not linear but rational f(x)~ a+b/(x+c) (hypothesis: f'(x_k)*f''(x_k)\neq 0 )
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function [flag,xk1] = RationalCode(x0,f,df,d2f,TOL,MaxIt) | |
%[flag,xk] = RationalCode(x0,f,df,d2f,TOL,MaxIt) | |
% | |
% This code approximate a root of f using a rational fuction approx. | |
% Input: | |
% x0 := initial value. | |
% f := function whose roots will be approximated. | |
% df := derivative of f. | |
% d2f := second derivative of f. | |
% TOL := Stoping criteria: relative error<TOL. | |
% MaxIt:= Maximun number of iterations. | |
% | |
% Output: | |
% flag:= 0 if df(xk)~0 or d2f(xk)~0. | |
% xk1:= root approximation. | |
% Last modified: Feb. 11, 2018. | |
it = 0; % Number of iterations | |
eps1 = 10^(-14); % | |
xk1 = x0; % First candidate to root | |
xk = x0; | |
AbsEr= inf; % Absolute error | |
while((AbsEr>TOL*abs(xk)) && MaxIt>it) | |
xk=xk1; | |
if(abs(df(xk))>eps1 && abs(df(xk))>eps1) | |
xk1 = xk+2*(df(xk)/d2f(xk))*... | |
(1-(df(xk)^2/(df(xk)^2-0.5*f(xk)*d2f(xk)))); | |
AbsEr = abs(xk1-xk); | |
it = it+1; | |
else | |
flag = 0; % | |
xk1 = NaN; | |
break | |
end | |
flag=1; | |
end | |
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