### 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) |
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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 |

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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

### Cite this

*SIAM Journal on Matrix Analysis and Applications*,

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