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 )

RationalCode.m

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