Roots Finding: Open Methods

Here are the M-files provided to implement the Newton-Raphson, secant & modified secant method. 

Newton-Raphson M-file

function [root,ea,iter]=newtraph(func,dfunc,xr,es,maxit,varargin)

% newtraph: Newton-Raphson root location zeroes

% [root,ea,iter]=newtraph(func,dfunc,xr,es,maxit,p1,p2,...):

%   uses Newton-Raphson method to find the root of func

% input:

%   func = name of function

%   dfunc = name of derivative of function

%   xr = initial guess

%   es = desired relative error (default = 0.0001%)

%   maxit = maximum allowable iterations (default = 50)

%   p1,p2,... = additional parameters used by function

% output:

%   root = real root

%   ea = approximate relative error (%)

%   iter = number of iterations

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;

Secant M-file

function root = secant(func,xrold,xr,es,maxit)

% secant(func,xrold,xr,es,maxit):

% uses secant method to find the root of a function

% input:

%  func = name of function

%  xrold, xr = initial guesses

%  es = (optional) stopping criterion (%)

%  maxit = (optional) maximum allowable iterations

% output:

%  root = real root

%  if necessary, assign default values

if nargin<5, maxit=50; end %if maxit blank set to 50

if nargin<4, es=0.001; end %if es blank set to 0.001

% Secant method

iter = 0;

while (1)

xrn = xr - func(xr)*(xrold - xr)/(func(xrold) - func(xr));

iter = iter + 1;

if xrn ~= 0, ea = abs((xrn - xr)/xrn)*100; end

if ea <= es | iter >= maxit, break, end

xrold = xr;

xr = xrn;

end

root = xrn;

Modified Secant M-file

function root = modsec(func,xr,delta,es,maxit)

% secant(func,xrold,xr,es,maxit):

% uses the modified secant method

% to find the root of a function

% input:

%  func = name of function

%  xr = initial guess

%  delta = perturbation fraction

%  es = (optional) stopping criterion (%)

%  maxit = (optional) maximum allowable iterations

% output:

%  root = real root

%  if necessary, assign default values

if nargin<5, maxit=50; end %if maxit blank set to 50

if nargin<4, es=0.001; end %if es blank set to 0.001

if nargin<3, delta=1E-5; end %if delta blank set to 0.00001

% Secant method

iter = 0;

while (1)

xrold = xr;

xr = xr - delta*xr*func(xr)/(func(xr+delta*xr)-func(xr));

iter = iter + 1;

if xr ~= 0, ea = abs((xr - xrold)/xr)*100; end

if ea <= es | iter >= maxit, break, end

end

root = xr;