secant.m

nonlinear eq routines in matlab

function [x, ithist, iflag] = secant( f, x, xo, tolf, tolx, maxit )
%
%  function [x, ithist, iflag] = secant( f, x, xo, tolf, tolx, maxit )
%
%  secant attempts to compute a root of f.
%
%  Input parameters:
%    f       name of a matlab function that evaluates 
%            f and its derivative.
%    x       initial iterate
%    xo      additional point needed to start the second method
%    tolf    stopping tolerance (optional. Default tolf = 1.e-7)
%            Secant method stops if  |f(x)| < tolf
%    tolx    stopping tolerance (optional. Default tolx = 1.e-7)
%            Secant method stops if  |s| < tolx, 
%            where s = -f(x_k)*(x_{k-1} - x_k)/f(x_{k-1}) - f(x_k))
%            is the secant step.
%    maxit   maximum number of iterations (optional. Default maxit = 100)
%
%
%  Output parameters:
%    x       approximation of the solution. 
%    ithist  array with the iteration history
%            The i-th row of ithist contains  [it, x, fx, s]
%    ifag    return flag
%            iflag =  0  |f(x)| <= tolf 
%            iflag =  1  iteration terminated because maximum number of 
%                        iterations was reached. |f(x)| > tolf 
%
%  Matthias Heinkenschloss
%  Department of Computational and Applied Mathematics
%  Rice University
%  Jan 17, 2002
%

% set tolerances if necessary
if( nargin <= 3 ) tolf = 1.e-7; tolx = 1.e-7; maxit = 100; end
if( nargin <= 4 ) tolx = 1.e-7; maxit = 100; end
if( nargin <= 5 ) maxit = 100; end

   
it    = 0;
iflag = 0;
[fx]  = feval(f, x);
[fo]  = feval(f, xo);
s     = - fx*(xo-x)/(fo-fx);

while( it < maxit & abs(s) > tolx & abs(fx) > tolf )
   s  = - fx*(xo-x)/(fo-fx);
   
   ithist(it+1,:) = [it, x, fx, s];
   
   xo = x;
   fo = fx;
   x  = x+s;
   it = it+1;
   [fx] = feval(f, x);
end


% check why the Secant method truncated and set iflag
if( abs(fx) > tolf )
    % Secant method truncated because the maximum number of iterations
    % was reached
    iflag = 1;
    return
else
    % Secant method truncated because abs(fx) <= tolf
    % print info for last iteration
    ithist(it+1,:) = [it, x, fx, 0];
end