Replacing Pivoting in Distributed Gaussian Elimination with Randomized Techniques

Lindquist, Neil; Łuszczek, Piotr; Dongarra, Jack

doi:10.1109/scala51936.2020.00010

Cited by 5 publications

(3 citation statements)

References 15 publications

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“…In spite of the benefits one can gain in reducing computation time, one could lose stability when removing pivoting with GE. This idea had been explored previously in [13], where the authors ran experiments using a combination of butterfly transformations and one step of iterative refinement in a mixed-precision setup to establish how much stability was regained. Our results for both the naïve and worst-case models align with theirs, in that one step of iterative refinement seems necessary to stabilize computed solutions when using GENP.…”

Section: Discussionmentioning

confidence: 99%

“…), although this is not the upper bound of ρ ∞ . 13 See [3] for additional discussions relating to explicit relationships and bounds between different growth factors.…”

Section: ]mentioning

confidence: 99%

“…In particular, we want to further explore the question as to whether removing pivoting is a good idea in practice in spite of the saved computational time. Additionally, in line with [13], we will explore combining butterfly transformations with iterative refinement. Our main contribution is to measure success of this combination against other random transformations taken from randomized numerical linear algebra, as well as comparing performance metrics against other pivoting schemes used with GE.…”

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confidence: 99%

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Growth factors of random butterfly matrices and the stability of avoiding pivoting

John¹,

Trogdon²

2022

Preprint

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Random butterfly matrices were introduced by Parker in 1995 to remove the need for pivoting when using Gaussian elimination. The growing applications of butterfly matrices have often eclipsed the mathematical understanding of how or why butterfly matrices are able to accomplish these given tasks. To help begin to close this gap using theoretical and numerical approaches, we explore the impact on the growth factor of preconditioning a linear system by butterfly matrices. These results are compared to other common methods found in randomized numerical linear algebra. In these experiments, we show preconditioning using butterfly matrices has a more significant dampening impact on large growth factors than other common preconditioners and a smaller increase to minimal growth factor systems. Moreover, we are able to determine the full distribution of the growth factors for a subclass of random butterfly matrices. Previous results by Trefethen and Schreiber relating to the distribution of random growth factors were limited to empirical estimates of the first moment for Ginibre matrices.

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Section: Discussionmentioning

confidence: 99%

“…), although this is not the upper bound of ρ ∞ . 13 See [3] for additional discussions relating to explicit relationships and bounds between different growth factors.…”

Section: ]mentioning

confidence: 99%

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confidence: 99%

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