GNU Scientific Library – Reference Manual: Nonlinear Least-Squares Testing for Convergence

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### 39.8 Testing for Convergence

A minimization procedure should stop when one of the following conditions is true:

- A minimum has been found to within the user-specified precision.
- A user-specified maximum number of iterations has been reached.
- An error has occurred.

The handling of these conditions is under user control. The functions below allow the user to test the current estimate of the best-fit parameters in several standard ways.

Function: int gsl_multifit_nlinear_test (const double xtol, const double gtol, const double ftol, int * info, const gsl_multifit_nlinear_workspace * w)
Function: int gsl_multilarge_nlinear_test (const double xtol, const double gtol, const double ftol, int * info, const gsl_multilarge_nlinear_workspace * w)

  These functions test for convergence of the minimization method using the following criteria:

  - Testing for a small step size relative to the current parameter vector

        |\delta_i| <= xtol (|x_i| + xtol)

    for each 0 <= i < p. Each element of the step vector \delta is tested individually in case the different parameters have widely different scales. Adding xtol to |x_i| helps the test avoid breaking down in situations where the true solution value x_i = 0. If this test succeeds, info is set to 1 and the function returns `GSL_SUCCESS`.

    A general guideline for selecting the step tolerance is to choose xtol = 10^{-d} where d is the number of accurate decimal digits desired in the solution x. See Dennis and Schnabel for more information.

  - Testing for a small gradient (g = \nabla \Phi(x) = J^T f) indicating a local function minimum:

        ||g||_inf <= gtol

    This expression tests whether the ratio (\nabla \Phi)_i x_i / \Phi is small. Testing this scaled gradient is a better than \nabla \Phi alone since it is a dimensionless quantity and so independent of the scale of the problem. The `max` arguments help ensure the test doesn’t break down in regions where x_i or \Phi(x) are close to 0. If this test succeeds, info is set to 2 and the function returns `GSL_SUCCESS`.

    A general guideline for choosing the gradient tolerance is to set `gtol = GSL_DBL_EPSILON^(1/3)`. See Dennis and Schnabel for more information.

  If none of the tests succeed, info is set to 0 and the function returns `GSL_CONTINUE`, indicating further iterations are required.

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