https://nlafet.github.io/StarNEig/

StarNEig is either comparable to LAPACK and ScaLAPACK or significantly faster depending on the computational step. Moreover, StarNEig realizes new parallel and blocked algorithms for computing eigenvectors without suffering from floating point overflow. In LAPACK the corresponding solvers are sequential scalar codes which compute eigenvectors one by one. In ScaLAPACK the corresponding solvers are vulnerable to overflow.

Eigenvalue problems can be found in every field of natural science. Clear examples are supplied by the analysis of systems of ordinary differential equations. The stability analysis of first order systems produces standard eigenvalue problems which are not necessarily symmetric and the analysis of second order systems produce quadratic eigenvalue problems which are equivalent to nonsymmetric generalized eigenvalue problems. Without the ability to solve eigenvalue problems rapidly and accurately, we would be unable to complete the calculations needed to maintain and advance our civilization. Therefore, it is important that we continue to develop new algorithms and software which maximizes both the performance and the accuracy using existing and emerging hardware

StarNEig is one of very few libraries to offer support for nonsymmetric eigenvalue problems. It is built on top of the runtime system StarPU which is used to schedule the tasks. Currently, StarNEig applies to real problems which have real or complex eigenvalues and eigenvectors. By design, StarNEig applies to both shared and distributed memory machines and it has experimental support for GPU accelerators.

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.