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

Fingerprint

LU Factorization
Sparse Linear Systems
Rounding error
Exact Solution
Directly proportional
Integer
Highest common factor
Gaussian elimination
Integrality
System of Linear Equations
Sparsity
Time Complexity
Substitution
Polynomial
Costs

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

Cite this

Exact solution of sparse linear systems via left-looking roundoff-error-free LU factorization in time proportional to arithmetic work. / Lourenco, Christopher; Escobedo, Adolfo; Moreno-Centeno, Erick; Davis, Timothy A.

In: SIAM Journal on Matrix Analysis and Applications, Vol. 40, No. 2, 01.01.2019, p. 609-638.

Research output: Contribution to journalArticle

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