Blocked Algorithms for Robust Solution of Triangular Linear Systems

Mikkelsen, Carl Christian Kjelgaard; Karlsson, Lars

doi:10.1007/978-3-319-78024-5_7

Cited by 8 publications

(14 citation statements)

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“…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.…”

Section: Novelty In Starneigmentioning

confidence: 99%

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

Myllykoski

Mikkelsen

2020

Parallel Processing and Applied Mathematics

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In this paper, we present the StarNEig library for solving dense non-symmetric (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 for 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 a new user to transition to StarNEig. We demonstrate the performance of the library with a small set of computational experiments.

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Section: Novelty In Starneigmentioning

confidence: 99%

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

Myllykoski

Mikkelsen

2020

Parallel Processing and Applied Mathematics

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“…1. If dim(S jj ) = 1, then x ∈ R m is a real eigenvector corresponding to the real eigenvalue λ ∈ R if and only if V = x has rank 1 and solves equation (7). 2.…”

Section: Computing a Single Eigenvectormentioning

confidence: 99%

Parallel Robust Computation of Generalized Eigenvectors of Matrix Pencils

Mikkelsen

Myllykoski

2020

Parallel Processing and Applied Mathematics

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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.

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“…xLATRS keeps the solution y consistently scaled throughout the computation, frequent scaling events incur significant overhead. The overhead of scaling events can be reduced as shown in [10,11]. There, the single right-hand side is partitioned into segments.…”

Section: Robust Solution Of Triangular Linear Systemsmentioning

confidence: 99%

“…The local scaling factors are reduced to a vector of global scaling factors ξ. We extend this approach to the eigenvector computation and use augmented vectors, introduced in [11,10], as a means to represent scaled vectors.…”

Section: Robust Solution Of Triangular Linear Systemsmentioning

confidence: 99%

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Robust parallel eigenvector computation for the non-symmetric eigenvalue problem

2020

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A standard approach for computing eigenvectors of a non-symmetric matrix reduced to real Schur form relies on a variant of backward substitution. Backward substitution is prone to overflow. To avoid overflow, the LAPACK eigenvector routine DTREVC3 associates every eigenvector with a scaling factor and dynamically rescales an entire eigenvector during the backward substitution such that overflow cannot occur. When many eigenvectors are computed, DTREVC3 applies backward substitution successively for every eigenvector. This corresponds to level-2 BLAS operations and constitutes a bottleneck. This paper redesigns the backward substitution such that the entire computation is cast as tile operations (level-3 BLAS). By replacing LAPACK's scaling factor with tile-local scaling factors, our solver decouples the tiles and sustains parallel scalability even when a lot of numerical scaling is necessary.

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Blocked Algorithms for Robust Solution of Triangular Linear Systems

Cited by 8 publications

References 1 publication

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

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

Parallel Robust Computation of Generalized Eigenvectors of Matrix Pencils

Robust parallel eigenvector computation for the non-symmetric eigenvalue problem

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