A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to pr...

Newton--Krylov methods are variants of inexact Newton methods where the approximate Newton direction is taken from a subspace of small dimension. Here we introduce a new hybrid Newton-GMRES method where a global strategy restricted to a low-dimensional subspace generated by GMRES is performed. The obtained process is consistent with preconditioning and with matrix-free implementation. Computational results indicate that our proposal enhances the classical backtracking inexact method.

In this work, we consider computing the real logarithm of a real matrix. We pay attention to general conditioning issues, provide careful implementation for several techniques including scaling issues, and finally test and compare the techniques on a number of problems. All things considered, our recommendation for a general purpose method goes to the Schur decomposition approach with eigenvalue grouping, followed by square roots and diagonal Padé approximants of the diagonal blocks. Nonetheless, in some cases, a well-implemented series expansion technique outperformed the other methods. We have also analyzed and implemented a novel method to estimate the Frechet derivative of the log, which proved very successful for condition estimation.