% USAGE [alpha,c,rms] = expfit( deg, x1, h, y )
%
% Prony's method for non-linear exponential fitting
%
% Fit function: \sum_1^{deg} c(i)*exp(alpha(i)*x)
%
% Elements of data vector y must correspond to
% equidistant x-values starting at x1 with stepsize h
%
% The method is fully compatible with complex linear
% coefficients c, complex nonlinear coefficients alpha
% and complex input arguments y, x1, non-zero h .
% Fit-order deg must be a real positive integer.
%
% Returns linear coefficients c, nonlinear coefficients
% alpha and root mean square error rms. This method is
% known to be more stable than 'brute-force' non-linear
% least squares fitting.
%
% Example
% x0 = 0; step = 0.05; xend = 5; x = x0:step:xend;
% y = 2*exp(1.3*x)-0.5*exp(2*x);
% error = (rand(1,length(y))-0.5)*1e-4;
% [alpha,c,rms] = expfit(2,x0,step,y+error)
%
% alpha =
% 2.0000
% 1.3000
% c =
% -0.50000
% 2.00000
% rms = 0.00028461
%
% The fit is very sensitive to the number of data points.
% It doesn't perform very well for small data sets.
% Theoretically, you need at least 2*deg data points, but
% if there are errors on the data, you certainly need more.
%
% Be aware that this is a very (very,very) ill-posed problem.
% By the way, this algorithm relies heavily on computing the
% roots of a polynomial. I used 'roots.m', if there is
% something better please use that code.
%