Author ORCID Identifier

Neil Lindquist https://orcid.org/0000-0001-9404-3121

Piotr Luszczek https://orcid.org/0000-0002-0089-6965

Jack Dongarra https://orcid.org/0000-0003-3247-1782

Document Type

Article

Publication Date

12-2024

DOI

https://doi.org/10.1145/3699714

Abstract

Parker and Lê introduced random butterfly transforms (RBTs) as a preprocessing technique to replace pivoting in dense LU factorization. Unfortunately, their FFT-like recursive structure restricts the dimensions of the matrix. Furthermore, on multinode systems, efficient management of the communication overheads restricts the matrix’s distribution even more. To remove these limitations, we have generalized the RBT to arbitrary matrix sizes by truncating the dimensions of each layer in the transform. We expanded Parker’s theoretical analysis to generalized RBT, specifically that in exact arithmetic, Gaussian elimination with no pivoting will succeed with probability 1 after transforming a matrix with full-depth RBTs. Furthermore, we experimentally show that these generalized transforms improve performance over Parker’s formulation by up to 62% while retaining the ability to replace pivoting. This generalized RBT is available in the SLATE numerical software library.

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