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

Pages (from-to) | 609-638 |

Number of pages | 30 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 40 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### 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

*SIAM Journal on Matrix Analysis and Applications*,

*40*(2), 609-638. https://doi.org/10.1137/18M1202499

**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.

Research output: Contribution to journal › Article

*SIAM Journal on Matrix Analysis and Applications*, vol. 40, no. 2, pp. 609-638. https://doi.org/10.1137/18M1202499

}

TY - JOUR

T1 - Exact solution of sparse linear systems via left-looking roundoff-error-free LU factorization in time proportional to arithmetic work

AU - Lourenco, Christopher

AU - Escobedo, Adolfo

AU - Moreno-Centeno, Erick

AU - Davis, Timothy A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Exact linear solutions

KW - LU factorizations

KW - Roundoff errors

KW - Solving linear systems

KW - Sparse IPGE word length

KW - Sparse linear systems

KW - Sparse matrix algorithms

UR - http://www.scopus.com/inward/record.url?scp=85070855826&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85070855826&partnerID=8YFLogxK

U2 - 10.1137/18M1202499

DO - 10.1137/18M1202499

M3 - Article

AN - SCOPUS:85070855826

VL - 40

SP - 609

EP - 638

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 2

ER -