Recursive blocked algorithms for linear systems with Kronecker product structure

Abstract

Recursive blocked algorithms have proven to be highly efficient at the numerical solution of the Sylvester matrix equation and its generalizations. In this work, we show that these algorithms extend in a seamless fashion to higher-dimensional variants of generalized Sylvester matrix equations, as they arise from the discretization of PDEs with separable coefficients or the approximation of certain models in macroeconomics. By combining recursions with a mechanism for merging dimensions, an efficient algorithm is derived that outperforms existing approaches based on Sylvester solvers.

Extended Krylov subspace methods for solving Sylvester and Stein tensor equations

“…In this section, we present four numerical examples to show the effectiveness of our methods for solving (1) and (2). The low dimensional tensor equations ( 12), ( 13), ( 16) and (19) will be solved by the recursive blocked algorithms presented in [5]. The numerical results were performed on a 2.8 GHz Intel Core i5 and 4 Go of RAM with Matlab R2016a.…”

“…. , N , and we obtain low-dimensional equations that are solved by recursive blocked algorithms presented in [5]. Numerical examples show that projecting onto extended Krylov subspaces is more efficient than standard Krylov subspaces.…”

Fast global spectral methods for three-dimensional partial differential equations

“…We need to incorporate the discretized boundary conditions (10) into the discretized PDE ( 9) to obtain the unique solution U. Following the ideas of [57, Section 6], we compute U by substituting (10) into (9). Since B 1 , B 2 , B 3 have linearly independent rows, we can assume without loss of generality that…”

“…To solve Laplace-like equations ( 17), we apply the recursive blocked algorithm developed in [9]. It transforms the matrices U, V, W into quasi-triangular form by computing Schur decompositions.…”

“…For instance, Poisson's equation with homogeneous Dirichlet boundary conditions leads to a Laplace-like equation [4,10,37,44]. Laplace-like equations can be solved efficiently using the blocked recursive algorithm in cite [9]. This algorithm is asymptotically slower than the nested alternating direction implicit method in [22], but our numerical experiments in Section 5 demonstrate that the recursive blocked algorithm is significantly faster in practice.…”