Exact solution of sparse linear systems via left-looking roundoff-error-free LU factorization in time proportional to arithmetic work

Christopher Lourenco, Adolfo Escobedo, Erick Moreno-Centeno, Timothy A. Davis

Research output: Contribution to journalArticle

Abstract

The roundoff-error-free (REF) LU factorization, along with the REF forward and backward substitution algorithms, allows a rational system of linear equations to be solved exactly and efficiently. The REF LU factorization framework has two key properties: all operations are integral, and the size of each entry is bounded polynomially-a bound that rational arithmetic Gaussian elimination achieves only via the use of computationally expensive greatest common divisor operations. This paper develops a sparse version of REF LU, termed the Sparse Left-looking Integer-Preserving (SLIP) LU factorization, which exploits sparsity while maintaining integrality of all operations. In addition, this paper derives a tighter polynomial bound on the size of entries in L and U and shows that the time complexity of SLIP LU is proportional to the cost of the arithmetic work performed. Last, SLIP LU is shown to significantly outperform a modern full-precision rational arithmetic LU factorization approach on a set of real world instances. In all, SLIP LU is a framework to efficiently and exactly solve sparse linear systems.

Original languageEnglish (US)
Pages (from-to)609-638
Number of pages30
JournalSIAM Journal on Matrix Analysis and Applications
Volume40
Issue number2
DOIs
StatePublished - Jan 1 2019

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Keywords

  • Exact linear solutions
  • LU factorizations
  • Roundoff errors
  • Solving linear systems
  • Sparse IPGE word length
  • Sparse linear systems
  • Sparse matrix algorithms

ASJC Scopus subject areas

  • Analysis

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