On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations

Abstract

Abstract:In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term AXB − X + EF T = 0. These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments.

Extended Krylov subspace methods for solving Sylvester and Stein tensor equations

“…For more details about the EBA and EGA processes and the construction of the restriction matrices T m and T m , we refer the reader to [11,3,18] and the references therein.…”

“…where A (i) , i = 1, 2, 3, are the same as in example 2. The correponding tensor equation is given as follows Z − Z × 1 A (1) − Z × 2 A (2) − Z × 3 A (3) .…”

The extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations

“…For more details on extended block Krylov projection method for solving large matrix equations see [3][4][5]10]. When we apply the ENBL algorithm 2 on the triple A, B, B B F , we get two biorthonormal matrices…”

“…Some methods have been proposed for solving large matrix equation, see, e.g. [10]. The main idea employed in these methods is to use an extended Krylov subspace and then apply the Galerkin-type orthogonality condition.…”

The Block Rational Arnoldi Method

“…Block Krylov methods were introduced in the 1970s, starting with the block Lanczos algorithm for linear eigenproblems with repeated eigenvalues [16,27,36]. More recently, block Krylov methods have found applications in model order reduction [2,22,23], for the solution of matrix equations [6,19,31,33], matrix function approximation [24,37,38,40], including multisource electromagnetic modeling [13,15,42,43], and solving linear systems with multiple right-hand sides [12,14,18,21,28,41,48]. While the theory of single-vector rational Krylov spaces is well developed [9,10,44,45,46,47], the block case has only been explored to a limited extent [4,24,29].…”