optim_doc: expfit

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### 2.9 Prony’s method for non-linear exponential fitting

#### Helptext:

    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.

    Demo for a complex fit-function:
    deg= 2; N= 20; x1= -(1+i), x= linspace(x1,1+i/2,N).';
    h = x(2) - x(1)
    y= (2+i)*exp( (-1-2i)*x ) + (-1+3i)*exp( (2+3i)*x );
    A= 5e-2; y+= A*(randn(N,1)+randn(N,1)*i); % add complex noise
    [alpha,c,rms]= expfit( deg, x1, h, y )