An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Abstract

Summary.We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB . This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm. Mathematics Subject Classification (1991): 65F15

System-theoretical algorithmic solution to waiting times in semi-Markov queues

“…For this reason, the Reference [32] recommends not to use the Jordan decomposition whenever it is possible and use instead the more reliable Schur form, which we do in the current paper. We note that this view is also shared by [27] in which the standard serial algorithm for the SDC problem is proposed to be the ordered Schur decomposition due to its well-established numerical stability. The ordered Schur form implementations are available in various platforms in LAPACK [34], OCTAVE [35], MATLAB 7.0 [36], and as a public add-on to MATLAB [31].…”

“…where the eigenvalues of A DD are exactly the same as the eigenvalues of A in D. This problem is called the ordinary SDC problem [28]. A real square matrix A of size n can be transformed via an orthogonal transformation U into the so-called real Schur form by writing U T AU = R where R is quasi-upper triangular, which means that the matrix R has either 1-by-1 or 2-by-2 diagonal blocks on the diagonal corresponding to the real and complex eigenvalues, respectively, of the matrix A [29].…”

“…The ordered Schur form implementations are available in various platforms in LAPACK [34], OCTAVE [35], MATLAB 7.0 [36], and as a public add-on to MATLAB [31]. Although there are other proposed algorithms for the decomposition given by (5) (see [28] for an elaborate discussion of various SDC-solvers), we will focus on the ordered Schur decomposition approach for obtaining numerical results for this paper.…”

“…• As the numerical engine, we propose to use the ordered Schur decomposition which is known to be the standard serial algorithm for solving the SDC problem in the numerical linear algebra literature due to its well-established numerical stability and computational efficiency [27]. The stability of the Schur decomposition approach stems from its ability to avoid the calculation of all eigenvectors.…”

Fast linear algebra is stable

“…To show how to solve eigenvalue problems quickly and stably, we use an algorithm from [5], modified slightly to use only the randomized rank revealing decomposition from the last section. As described in Section 6.1, it can compute either an invariant subspace of a matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB, using only QRR, RURV and matrix multiplication.…”