Convergence in MIGRAD, and Positive-definiteness.

MIGRAD uses its current estimate of the covariance matrix of the function to determine the current search direction, since this is the optimal strategy for quadratic functions and ``physical'' functions should be quadratic in the neighbourhood of the minimum at least. The search directions determined by MIGRAD are guaranteed to be downhill only if the covariance matrix is positive-definite, so in case this is not true, it makes a positive-definite approximation by adding an appropriate constant along the diagonal as determined by the eigenvalues of the matrix. Theoretically, the covariance matrix for a ``physical'' function must be positive-definite at the minimum, although it may not be so for all points far away from the minimum, even for a well-determined physical problem. Therefore, if MIGRAD reports that it has found a non-positive-definite covariance matrix, this may be a sign of one or more of the following:

• A non-physical region. On its way to the minimum, MIGRAD may have traversed a region which has unphysical behaviour, which is of course not a serious problem as long as it recovers and leaves such a region.
• An underdetermined problem. If the matrix is not positive-definite even at the minimum, this may mean that the solution is not well-defined, for example that there are more unknowns than there are data points, or that the parametrization of the fit contains a linear dependence. If this is the case, then Minuit (or any other program) cannot solve your problem uniquely, and the error matrix will necessarily be largely meaningless, so the user must remove the underdeterminedness by reformulating the parametrization. Minuit cannot do this itself, but it can provide some hints (contours, global correlation coefficients, eigenvalues) which can help the clever user to find out what is wrong.
• Numerical inaccuracies. It is possible that the apparent lack of positive-definiteness is in fact only due to excessive roundoff errors in numerical calculations, either in FCN or in Minuit. This is unlikely in general, but becomes more likely if the number of free parameters is very large, or if the parameters are badly scaled (not all of the same order of magnitude), and correlations are also large. In any case, whether the non-positive-definiteness is real or only numerical is largely irrelevant, since in both cases the error matrix will be unreliable and the minimum suspicious.