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.

Keywords

  1. exact linear solutions
  2. LU factorizations
  3. roundoff errors
  4. solving linear systems
  5. sparse matrix algorithms
  6. sparse IPGE word length
  7. sparse linear systems

MSC codes

  1. 15A23
  2. 90C05
  3. 65F50
  4. 65F05
  5. 65G50
  6. 15A15

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Information & Authors

Information

Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 609 - 638
ISSN (online): 1095-7162

History

Submitted: 24 July 2018
Accepted: 4 March 2019
Published online: 14 May 2019

Keywords

  1. exact linear solutions
  2. LU factorizations
  3. roundoff errors
  4. solving linear systems
  5. sparse matrix algorithms
  6. sparse IPGE word length
  7. sparse linear systems

MSC codes

  1. 15A23
  2. 90C05
  3. 65F50
  4. 65F05
  5. 65G50
  6. 15A15

Authors

Affiliations

Funding Information

Texas A&M Graduate Merit Fellowship
National Science Foundation https://doi.org/10.13039/100000001 : CMMI-1252456

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