%% Optimal Fit of a Non-linear Function
% This is a demonstration of the optimal fitting of a non-linear function to a
% set of data.  It uses FMINSEARCH, an implementation of the Nelder-Mead simplex
% (direct search) algorithm, to minimize a nonlinear function of several
% variables.
%
% Copyright 1984-2004 The MathWorks, Inc.
% $Revision: 5.15.4.3 $ $Date: 2004/11/29 23:30:49 $

%%
% First, create some sample data and plot it.

t = (0:.1:2)';
y = [5.8955 3.5639 2.5173 1.9790 1.8990 1.3938 1.1359 1.0096 1.0343 ...
     0.8435 0.6856 0.6100 0.5392 0.3946 0.3903 0.5474 0.3459 0.1370 ...
     0.2211 0.1704 0.2636]';
plot(t,y,'ro'); hold on; h = plot(t,y,'b'); hold off;
title('Input data'); ylim([0 6])

%%
% The goal is to fit the following function with two linear parameters and two
% nonlinear parameters to the data:
%
%     y =  C(1)*exp(-lambda(1)*t) + C(2)*exp(-lambda(2)*t)
% 
% To fit this function, we've create a function FITFUN.  Given the nonlinear
% parameter (lambda) and the data (t and y), FITFUN calculates the error in the
% fit for this equation and updates the line (h).

type fitfun

%%
% Make a guess for initial estimate of lambda (start) and invoke FMINSEARCH.  It
% minimizes the error returned from FITFUN by adjusting lambda.  It returns the
% final value of lambda. Use an output function to plot intermediate fits.

start = [1;0];
% We use annonymous function to pass additional parameters t, y, h to the
% output function. 
outputFcn = @(x,optimvalues,state) fitoutputfun(x,optimvalues,state,t,y,h);
options = optimset('OutputFcn',outputFcn,'TolX',0.1);
estimated_lambda = fminsearch(@(x)fitfun(x,t,y),start,options)