Solution of dense linear systems via roundoff-error-free factorization algorithms

Theoretical connections and computational comparisons

Adolfo Escobedo, Erick Moreno-Centeno, Christopher Lourenco

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Exact solving of systems of linear equations (SLEs) is a fundamental subroutine within number theory, formal verification of mathematical proofs, and exact-precision mathematical programming. Moreover, efficient exact SLE solution methods could be valuable for a growing body of science and engineering applications where current fixed-precision standards have been deemed inadequate. This article contains key derivations relating, and computational tests comparing, two exact direct solution frameworks: roundoff-error-free (REF) LU factorization and rational arithmetic LU factorization. Specifically, both approaches solve the linear system Ax = b by factoring the matrix A into the product of a lower triangular (L) and upper triangular (U) matrix, A = LU . Most significantly, the featured findings reveal that the integer-preserving REF factorization framework solves dense SLEs one order of magnitude faster than the exact rational arithmetic approach while requiring half the memory. Since rational LU is utilized for basic solution validation in exact linear and mixed-integer programming, these results offer preliminary evidence of the potential of the REF factorization framework to be utilized within this specific context. Additionally, this article develops and analyzes an efficient streamlined version of Edmonds's Q-matrix approach that can be implemented as another basic solution validation approach. Further experiments demonstrate that the REF factorization framework also outperforms this alternative integer-preserving approach in terms of memory requirements and computational effort. General purpose codes to solve dense SLEs exactly via any of the aforementioned methods have been made available to the research and academic communities.

Original languageEnglish (US)
Article number40
JournalACM Transactions on Mathematical Software
Volume44
Issue number4
DOIs
StatePublished - Jun 1 2018
Externally publishedYes

Fingerprint

Rounding error
Factorization
Linear systems
Linear equations
Linear Systems
System of Linear Equations
LU Factorization
Triangular
Number theory
Data storage equipment
Subroutines
Mathematical programming
Integer programming
Q-matrix
Integer
Mixed Integer Programming
Factoring
Formal Verification
Engineering Application
Mathematical Programming

Keywords

  • Dense linear systems
  • Exact linear programming
  • Roundoff errors

ASJC Scopus subject areas

  • Software
  • Applied Mathematics

Cite this

Solution of dense linear systems via roundoff-error-free factorization algorithms : Theoretical connections and computational comparisons. / Escobedo, Adolfo; Moreno-Centeno, Erick; Lourenco, Christopher.

In: ACM Transactions on Mathematical Software, Vol. 44, No. 4, 40, 01.06.2018.

Research output: Contribution to journalArticle

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