1: Carl Christian Kjelgaard Mikkelsen and Mirko Myllykoski: Parallel Robust Computation of Generalized Eigenvectors of Matrix Pencils
https://link.springer.com/chapter/10.1007/978-3-030-43229-4_6
https://arxiv.org/abs/2003.04776

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 scalar and sequential codes which compute the eigenvectors one by one. In this paper, we discuss how to derive robust algorithms which are blocked and parallel. 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.

2: Mirko Myllykoski and Carl Christian Kjelgaard Mikkelsen: Introduction to StarNEig – A Task-based Library for Solving Nonsymmetric Eigenvalue Problems.
https://link.springer.com/chapter/10.1007/978-3-030-43229-4_7
https://arxiv.org/abs/2002.05024

In this paper, we present the StarNEig library for solving dense nonsymmetric (generalized) eigenvalue problems. The library is built on top of the StarPU runtime system and targets both shared and distributed memory machines. Some components of the library support GPUs. The library is currently in an early beta state and only real arithmetic is supported. Support for complex data types is planned for a future release. This paper is aimed at potential users of the library. We describe the design choices and capabilities of the library, and contrast them to existing software such as ScaLAPACK. StarNEig implements a ScaLAPACK compatibility layer that should make it easy for new users to transition to StarNEig. We demonstrate the performance of the library with a small set of computational experiments.

3: Angelika Beatrix Schwarz and Carl Christian Kjelgaard Mikkelsen: Robust Task-Parallel Solution of the Triangular Sylvester Equation.
https://link.springer.com/chapter/10.1007/978-3-030-43229-4_8
https://arxiv.org/abs/1905.10574

The Bartels-Stewart algorithm is a standard approach to solving the dense Sylvester equation. It reduces the problem to the solution of the triangular Sylvester equation. The triangular Sylvester equation is solved with a variant of backward substitution. Backward substitution is prone to overflow. Overflow can be avoided by dynamic scaling of the solution matrix. An algorithm which prevents overflow is said to be robust. The standard library LAPACK contains the robust scalar sequential solver dtrsyl. This paper derives a robust, level-3 BLAS-based task-parallel solver. By adding overflow protection, our robust solver closes the gap between problems solvable by LAPACK and problems solvable by existing non-robust task-parallel solvers. We demonstrate that our robust solver achieves a performance similar to non-robust solvers.

Robust algorithms do not suffer from overflow and always return a valid result. In LAPACK eigenvectors (standard and generalized) are computed using robust algorithms. The existing algorithms are scalar and sequential. This new work presents algorithms which are blocked and parallel. The analysis is supported by parallel software running on top of StarPU. Further improvements are possible, but the new software is already orders of magnitude faster than the existing software.

The authors are Carl Christian Kjelgaard Mikkelsen, Angelika Beatrix Schwarz and Lars Karlsson