Parallel Robust Computation of Generalized Eigenvectors of Matrix Pencils

Abstract

In this paper we consider the problem of computing generalized eigenvectors of a matrix pencil in real Schur form. In exact arithmetic, this problem can be solved using substitution. In practice, substitution is vulnerable to floating-point overflow. The robust solvers xTGEVC in LAPACK prevent overflow by dynamically scaling the eigenvectors. These subroutines are sequential scalar codes which compute the eigenvectors one by one. In this paper we discuss how to derive robust blocked algorithms. The new StarNEig library contains a robust task-parallel solver Zazamoukh which runs on top of StarPU. Our numerical experiments show that Zazamoukh achieves a super-linear speedup compared with DTGEVC for sufficiently large matrices.

Introduction to StarNEig—A Task-Based Library for Solving Nonsymmetric Eigenvalue Problems

“…Here, the data dependences are comparatively simple but the computations must be protected against floating-point overflow. This is a nontrivial issue to address in a parallel setting; see [7,8,9]. Furthermore, the Schur reduction and eigenvalue reordering steps apply a series of overlapping local transformations to the matrices.…”

Task‐based, GPU‐accelerated and robust library for solving dense nonsymmetric eigenvalue problems

“…In particular, the fundamental principles for solving triangular linear systems in parallel without suffering from overflow are discussed in References 32,34 . The StarNeig solvers for computing standard and generalized eigenvectors are the subjects of separate papers 35,36 …”

Task-based, GPU-accelerated and Robust Library for Solving Dense Nonsymmetric Eigenvalue Problems

“…In particular, the fundamental principles for solving triangular linear systems in parallel without suffering from overflow are discussed in 29,30 . The StarNeig solver for computing generalized eigenvectors is the subject of a separate paper 31 .…”