This paper presents a factorization of the error system that arises in model reduction of linear time invariant systems by Krylov subspace methods. The factorization is introduced for reduced models that match moments and/or Markov parameters of the original system with multiple inputs and outputs. Furthermore, dual results are given for the reduction with input and output Krylov subspaces. To ...

The Generalized Minimal Residual method (GMRES) seeks optimal approximate solutions of linear system Ax = b from Krylov subspaces by minimizing the residual norm ‖Ax − b‖2 over all x in the subspaces. Its main cost is computing and storing basis vectors of the subspaces. For difficult systems, Krylov subspaces of very high dimensions are necessary for obtaining approximate solutions with desire...

An application of the Generalized Minimal Residual (GMRES) algorithm to the solution of sequentially multiple nearby systems of equations through the reuse of Krylov subspaces is presented. The main focus is on the case when only the right-hand side vector changes. However, the case in which both the matrix and the right-hand side change is also addressed. Applications of these formulations inc...

Aggressive early deflation has proven to significantly enhance the convergence of the QR algorithm for computing the eigenvalues of a nonsymmetric matrix. One purpose of this paper is to point out that this deflation strategy is equivalent to extracting converged Ritz vectors from certain Krylov subspaces. As a special case, the single-shift QR algorithm enhanced with aggressive early deflation...

The Progressive GMRES algorithm, introduced by Beckermann and Reichel in 2008, is a residual-minimizing short-recurrence Krylov subspace method for solving a linear system in which the coefficient matrix has a low-rank skew-Hermitian part. We analyze this algorithm, observing a critical instability that makes the method unsuitable for some problems. To work around this issue we introduce a diff...

Moment matching theorems for Krylov subspace based model reduction of higherorder linear dynamical systems are presented in the context of higher-order Krylov subspaces. We introduce the definition of a nth-order Krylov subspace Kn k ({Ai} n i=1;u) based on a sequence of n square matrices {Ai}i=1 and vector u. This subspace is a generalization of Krylov subspaces for higher-order systems, incor...

Abstract. The evaluation of matrix functions of the form f(A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small prob...

We investigate two iterative methods for solving nonsingular linear systems P(A)x = b; () where P(A) = P p i=0 i A i denotes a matrix polynomial and A is hermitian. The key idea of the methods is to choose the approximate solutions x j from the Krylov subspaces with respect to the matrix A instead of P(A). Thus, only one matrix vector multiplication (Gemv) is necessary to extend the current Kry...

In recent years, Krylov subspace methods have become popular tools for computing reduced order models of high order linear time invariant systems. The reduction can be done by applying a projection from high order to lower order space using the bases of some subspaces called input and output Krylov subspaces. The aim of this paper is to give an introduction into the principles of Krylov subspac...

The IDR method of Sonneveld and van Gijzen [SIAM J. Sci. Comput., 31:1035–1062, 2008] is shown to be a Petrov-Galerkin (projection) method with a particular choice of left Krylov subspaces; these left subspaces are rational Krylov spaces. Consequently, other methods, such as BiCGStab and ML(s)BiCGStab, which are mathematically equivalent to some versions of IDR, can also be interpreted as Petro...