We study the roundoff error propagation in an algorithm which computes the orthonormal basis of a Krylov subspace with Householder orthonormal matrices . Moreover, we analyze special implementations of the classical GMRES algorithm, and of the Full Orthogonalization Method . These techniques approximate the solution of a large sparse linear system of equations on a sequence of Krylov subspaces ...

We present a Krylov-W-code ROWMAP for the integration of stii initial value problems. It is based on the ROW-methods of the code ROS4 of Hairer and Wanner and uses Krylov techniques for the solution of linear systems. A special multiple Arnoldi process ensures order p = 4 already for fairly low dimensions of the Krylov subspaces independently of the dimension of the diierential equations. Numer...

The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation (WR) method based on block Krylov subspaces. Second, we compare this new WR–Krylov implementation against Krylov subspace methods combined with the shift and invert (SAI) technique. Some analysis and numerical experiments are presented. Since the WR–Krylov and SAI–Krylov m...

In this note, we extend the Vandermonde with Arnoldi method recently advocated by Brubeck et al. (2021) to dealing confluent matrix. To apply process, it is critical find a Krylov subspace which generates column space of A theorem established for such subspaces any order derivatives. This enables us compute derivatives high degree polynomials precision. It also makes many applications involving...

Residual norm estimates are derived for a general class of methods based on projection techniques on subspaces of the form K m + W, where K m is the standard Krylov subspace associated with the original linear system, and W is some other subspace. Thesèaugmented Krylov subspace methods' include eigenvalue deeation techniques as well as block-Krylov methods. Residual bounds are established which...

The predominant technique for computing the transient distribution of a Continuous Time Markov Chain (CTMC) exploits uniformization, which is known to be stable and efficient for non-stiff to mildly-stiff CTMCs. On stiff CTMCs however, uniformization suffers from severe performance degradation. In this paper, we report on our observations and analysis of an alternative technique using Krylov su...

Recently, we proposed an algebraic difference scheme, with extended stability properties, for linear boundary value problems involving stiff differential equations of first order. Here, an efficient approximation scheme is presented for matrix square roots, which provides the stabilization of that scheme in case of stiffness. It combines the use of low-rank matrix approximations from projection...

Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov in which is augmented with fixed spanned by vectors deemed be helpful accelerating convergence or conveying knowledge of the solution. Recently, survey was published, framework describing vast majority such proposed [Soodhalter et al., GAMM-Mitt., 43 (2020), Art. e202000016]. In man...

A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising finite elements, multigrid, spectral, and pseudospectral methods for pde; they arise the form Krylov subspaces matrix computations, as multiresolution analysis wavelets constructions. They common statistics too—principal component, canonical correlation, correspondence analyses may all be viewed extrac...

‎The global FOM and GMRES algorithms are among the effective‎ ‎methods to solve Sylvester matrix equations‎. ‎In this paper‎, ‎we‎ ‎study these algorithms in the case that the coefficient matrices‎ ‎are real symmetric (real symmetric positive definite) and extract‎ ‎two CG-type algorithms for solving generalized Sylvester matrix‎ ‎equations‎. ‎The proposed methods are iterative projection metho...