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 R.

AU - Moreno-Centeno, Erick

AU - Davis, Timothy A.

N1 - Publisher Copyright: © 2019 Society for Industrial and Applied Mathematics

PY - 2019

Y1 - 2019

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

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U2 - 10.1137/18M1202499

DO - 10.1137/18M1202499

M3 - Article

SN - 0895-4798

VL - 40

SP - 609

EP - 638

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 2

ER -