It has been shown that approximate extended Krylov subspaces can be computed –under certain assumptions– without any explicit inversion or system solves. Instead the necessary products A−1v are obtained in an implicit way retrieved from an enlarged Krylov subspace. In this paper this approach is generalized to rational Krylov subspaces, which contain besides poles at infinite and zero also fini...

The aim of the paper is to compile and compare basic theoretical facts on Krylov subspaces and block Krylov subspaces. Many Krylov (sub)space methods for solving a linear system Ax = b have the property that in exact computer arithmetic the true solution is found after ν iterations, where ν is the dimension of the largest Krylov subspace generated by A from r0, the residual of the initial appro...

In this paper we study numerical methods for solving Sylvester matrix equations of the form AX +XB +CD = 0. A new projection method is proposed. The union of Krylov subspaces in A and its inverse and the union of Krylov subspaces in B and its inverse are used as the right and left projection subspaces, respectively. The Arnoldi-like process for constructing the orthonormal basis of the projecti...

The FEAST eigenvalue algorithm is a subspace iteration algorithm that uses contour integration in the complex plane to obtain the eigenvectors of a matrix for the eigenvalues that are located in any user-defined search interval. By computing small numbers of eigenvalues in specific regions of the complex plane, FEAST is able to naturally parallelize the solution of eigenvalue problems by solvin...

We propose the use of a multishift bi-conjugate gradient method (BiCG) in combination with a suitable chosen polynomial preconditioning, to efficiently solve the two sets of multiple shifted linear systems arising at each iteration of the iterative rational Krylov algorithm (IRKA, [Gugerkin, Antoulas, and Beattie, 2008]) for H2-optimal model reduction of linear systems. The idea is to construct...

This article presents an algorithm to achieve stable reduced models using Krylov-based model order reduction for discrete time systems while matching a certain number of Markov parameters. By using the invariance properties of the controllability matrix suitable input and output Krylov subspaces are derived. The method is illustrated performing model reduction of four well-known benchmark probl...

We propose algorithms for Tucker approximation of 3-tensor, that use it only through tensor-by-vector-by-vector multiplication subroutine. In matrix case, Krylov methods are methods of choice to approximate dominant column and row subspace of sparse or structured matrix given through matrix-by-vector subroutine. However, direct generalization to tensor case, proposed recently by Eldén and Savas...

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 methods onto matrix Kr...

For large scale problems, an effective approach for solving the algebraic Lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation. Although Krylov subspaces have been used for long time, only more recent developments have shown that rational Krylov subspaces can be a competitive alternative to the classical and v...

In this paper, structure preserving order reduction of proportionally damped and undamped second order systems is presented. The discussion is based on Second Order Krylov Subspace method and it is shown that for systems with a proportional damping, the damping matrix does not contribute to the projection matrices and the reduction can be carried out using the classical Krylov subspaces. As a r...