### 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 language | English (US) |
---|---|

Article number | 40 |

Journal | ACM Transactions on Mathematical Software |

Volume | 44 |

Issue number | 4 |

DOIs | |

State | Published - Jun 1 2018 |

Externally published | Yes |

### Fingerprint

### Keywords

- Dense linear systems
- Exact linear programming
- Roundoff errors

### ASJC Scopus subject areas

- Software
- Applied Mathematics

### Cite this

*ACM Transactions on Mathematical Software*,

*44*(4), [40]. https://doi.org/10.1145/3199571

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

Research output: Contribution to journal › Article

*ACM Transactions on Mathematical Software*, vol. 44, no. 4, 40. https://doi.org/10.1145/3199571

}

TY - JOUR

T1 - Solution of dense linear systems via roundoff-error-free factorization algorithms

T2 - Theoretical connections and computational comparisons

AU - Escobedo, Adolfo

AU - Moreno-Centeno, Erick

AU - Lourenco, Christopher

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

KW - Dense linear systems

KW - Exact linear programming

KW - Roundoff errors

UR - http://www.scopus.com/inward/record.url?scp=85060552334&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060552334&partnerID=8YFLogxK

U2 - 10.1145/3199571

DO - 10.1145/3199571

M3 - Article

VL - 44

JO - ACM Transactions on Mathematical Software

JF - ACM Transactions on Mathematical Software

SN - 0098-3500

IS - 4

M1 - 40

ER -