Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi met...

Block Krylov subspace spectral (KSS) methods are a “best-of-both-worlds” compromise between explicit and implicit time-stepping methods for variable-coefficient PDE, in that they combine the efficiency of explicit methods and the stability of implicit methods, while also achieving spectral accuracy in space and high-order accuracy in time. Block KSS methods compute each Fourier coefficient of t...

Recent advances in the eld of iterative methods for solving large linear systems are reviewed. The main focus is on developments in the area of conjugate gradient-type algorithms and Krylov subspace methods for non-Hermitian matrices .

In the paper we show how, based on the preconditioned Krylov subspace methods, to compute the covariance matrix of parameter estimates, which is crucial for efficient methods of optimum experimental design. Mathematics Subject Classification (2000). Primary 65K10; Secondary 15A09, 65F30.

The Lanczos process constructs a sequence of orthonormal vectors vm spanning a nested sequence of Krylov subspaces generated by a hermitian matrix A and some starting vector b. In this paper we show how to cheaply recover a secondary Lanczos process, starting at an arbitrary Lanczos vector vm and how to use this secondary process to efficiently obtain computable error estimates and error bounds...

The meshless method plays an important role in solving problems in computational mechanics where conventional computational methods are not well suited. In this paper, we examine the property of the kernel matrix and investigate the convergence and timing performance of some well-known Krylov subspace methods on solving linear systems from meshless discretizations.