Exponential fit, without start-guess

Fits 1) f=s1+s2*exp(-t/s3) or 2) f=s1+s2*exp(-t/s3)+s4*exp(-t/s5) to numerics, without startguess

exp2fit solves the non-linear least squares problem exact
and using it as a start guess in a least square method
in cases with noise, of the specific exponential functions:
--- caseval = 1 ----
f=s1+s2*exp(-t/s3)
--- caseval = 2 (general case, two exponentials) ----
f=s1+s2*exp(-t/s3)+s4*exp(-t/s5)
--- caseval = 3 ----
f=s1*(1-exp(-t/s2)) %i.e., constraints between s1 and s2

Syntax: s=exp2fit(t,f,caseval) gives the parameters in the fitting function specified by the choice of caseval (1,2,3). t and f are (normally) vectors of the same size, containing the data to be fitted.
s=exp2fit(t,f,caseval,lsq_val,options), using lsq_val='no' gives the analytic solution, without least square approach (faster), where options (optional or []) are produced by optimset, as used in lsqcurvefit.

This algorithm is using analytic formulas using multiple integrals. Integral estimations are used as start guess in lsqcurvefit. Note: For infinite lengths of t, and f, without noise the result is exact.

%--- Example 1: (see also help exp2fit)
t=linspace(1,4,100)*1e-9;
noise=0.02;
f=0.1+2*exp(-t/3e-9)+noise*randn(size(t));

%--- solve without startguess
s=exp2fit(t,f,1)

%--- plot and compare
fun = @(s,t) s(1)+s(2)*exp(-t/s(3));
tt=linspace(0,4*s(3),200);
ff=fun(s,tt);
figure(1), clf;plot(t,f,'.',tt,ff);

%--- Example 2, Damped Harmonic oscillator:
%--- Note: sin(x)=(exp(ix)-exp(-ix))/2i
t=linspace(1,12,100)*1e-9;
w=1e9;
f=1+3*exp(-t/5e-9).*sin(w*(t-2e-9));

%--- solve without startguess
s=exp2fit(t,f,2,'no')

%--- plot and compare
fun = @(s,t) s(1)+s(2)*exp(-t/s(3))+s(4)*exp(-t/s(5));
tt=linspace(0,20,200)*1e-9;
ff=fun(s,tt);
figure(1), clf;plot(t,f,'.',tt,real(ff));

%% By Per Sundqvist january 2009.