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Package levmar (in levmar.i) -

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DOCUMENT a = levmar(y, x, f, a0, avar, acovar) perform a Levenberg-Marquardt non-linear least squares fit to data values Y, which are functions of X. The dimensions of X need bear no particular relationship to the dimensions of Y, but Y must be a 1D array. The function F maps X into Y, as a function of some additional parameters A, according to Y = F(X, A) Again, the dimensions of A bear no particular relation to the dimensions of either X or Y. The input A0 is the initial estimate of the parameter values, which must be made with some care for the algorithm to converge. Note that it may not converge to the expected relative minimum in many situations. The parameters Y and A must be 1D arrays, X can be anything. AVAR and ACOVAR are return arguments; ACOVAR is the covariance matrix of the fit parameters A, and AVAR is its diagonal. Hence AVAR has the same dimensions as A0, while ACOVAR is a 2D symmetric matrix, whose diagonal is AVAR. Multiply AVAR or ACOVAR by levmar_chi2 if you did not use wgt= and want to use the quality of the fit to estimate the variances of the parameters. ACOVAR is what is returned by regress_cov, see help,regress_cov. You may be able to improve the performance of levmar by supplying analytic derivatives. To support this improvement, levmar permits three different "prototypes" for the function F: func F(x, a) if you cannot compute analytic dfda approximate dfda will be computed by levmar_partial (see help) func F(x, a, &dfda) if you can return analytic dfda if levmar will use dfda, dfda=1 on input if levmar will not use dfda, dfda=[] on input func F(x, a, &dfda, deriv=) if levmar will use dfda, it sets deriv=1, else deriv=0 (this form is for backward compatibility with original lmfit) The levmar function automatically detects the prototype of F. (!!) In all cases, dfda is defined as: dfda(i,j) = partial[Y(i)] / partial[A(j)] Note that the external variable fit (the fit= keyword to levmar) is available to F, indicating that only a subset of the partial derivatives must be computed. Set unused j indices to zero. EXTERNAL VARIABLES: outputs: levmar_chi2 final value of chi2 levmar_chi20 initial value of chi2 (for a0) levmar_lambda final value of lambda levmar_neval number of calls to F inputs: levmar_itmax = 100 maximum number of gradient recalculations levmar_tol = 1.e-7 stop when chi2 changes by less than this levmar_lambda0 = 0.001 initial value of lambda levmar_lambda1 = 1.e12 maximum permitted value of lambda levmar_gain = 10. factor by which to change lambda levmar_aabs, levmar_arel, levmar_ada -- see levmar_partial KEYWORDS: fit= index list into a0 of parameters to be varied the returned model will equal a0 for parameters not varied, avar and acovar will be 0.0 for parameters not varied amin= minimum values for parameters, same size as a0 amax= maximum parameter values, same size as a0 the function F will not be called with parameters outside these specified ranges wgt= same size as Y, weightings for each point if sigma_y(i) is standard deviation of i-th point, then wgt=1./sigma_y^2 is the appropriate weight. lu= set non-zero to use LUsolve instead of SVsolve LUsolve will be faster, which will only be an issue if the number of parameters is large

SEE ALSO: regress, levmar_partial

DOCUMENT y = levmar_partial(f, x, a, dfda) return y=F(X,A) and DFDA(i,j) = partial[F(X,A)(i)] / partial[A(j)] by finite differences. Accepts fit=, amin=, amax= keywords with same meaning as levmar. DFDA is only computed if DFDA=1 on input; if DFDA=[] or 0, the (expensive) DFDA calculation is skipped. Uses levmar_arel and levmar_aabs to compute the step sizes use to compute the patial derivatives as follows: da = levmar_aabs + levmar_arel*abs(A) (Hence levmar_aabs, levmar_arel can be arrays with the same length as the parameter array A.) By default, levmar_arel = 1.e-6, and levmar_aabs = 0. If levmar_aabs = 0 and A(i)=0, then da(i)=1.e-9. You can also supply a function, levmar_ada(A) returning da>0 by whatever formula you wish.

SEE ALSO: levmar