MATLAB: Lsqnonlin – Use levenberg-marquardt with real parameters and complex function

levenberg marquardtlsqnonlinminimizing

I want to minimize the term f(x)-v (where v is a constant complex vector number). The Function f(x) take an complex input vector and returns an complex vector. For my computations, i want that x has a certain curve (Equation 2 in the picture, x is sigma). So, each element in the x-vector should have 3 parameters x_infinity, x_zero and tau that are real (!) numbers. The problem is: if i use a complex function in lsqnonlin (levenberg-marquardt), the all the parameters are directly complex. How can i command to the solver to just consider real parameters?

Best Answer

Break the function into real and imaginary parts. In other words, modify your objective function from an N x 1 vector to 2N x 1 consisting of the real and imaginary parts,

   f_new(x) = [real(f(x)); imag(f(x))]

Related Solutions

MATLAB: How to fit a complex valued function to the data

The Curve Fitting Toolbox does not accept complex valued functions. To carry out the fit, use the LSQNONLIN function or the FMINSEARCH function. The objective function passed to LSQNONLIN or FMINSEARCH should take the real and complex parts of the parameters separately and return the real and complex parts of the residuals separately.

For example, create the function 'myfun.m'.

function F = myfun(c,xdata,ydata)
% c is a two-element vector whose components are the real
% and imaginary parts of the parameter respectively
errors = (c(1) + c(2)*1i)*xdata*1i + xdata.^2 - ydata;
F = [real(errors(:)); imag(errors(:))];

Note that here, xdata and ydata may be complex, but the input and output are real.

To use this with LSQNONLIN:

[x,resnorm] = lsqnonlin(@(c)myfun(c,xdata,ydata),[1  3])

To use this with FMINSEARCH, add

F = sum(F.^2);

to the bottom of 'myfunc.m' and replace 'lsqnonlin' with 'fminsearch' in the above syntax.

MATLAB: FSOLVE seems to find complex roots correctly (although it shouldn’t)

The description of when fsolve works with complex functions is in Complex Numbers in Optimization Toolbox Solvers. In short, the fsolve algorithms have not changed, but MathWorks has only relatively recently qualified its algorithms to work with complex numbers under the condition that the objective functions are analytic, and the algorithms often work better using a complex initial point.

Alan Weiss

MATLAB mathematical toolbox documentation

Best Answer

Related Solutions

MATLAB: How to fit a complex valued function to the data

MATLAB: FSOLVE seems to find complex roots correctly (although it shouldn’t)

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