Numerically safe Gaussian elimination with no pivoting

Abstract

Gaussian elimination with no pivoting and block Gaussian elimination are attractive alternatives to the customary but communication intensive Gaussian elimination with partial pivoting 1 provided that the computations proceed safely and numerically safely, that is, run into neither division by 0 nor numerical problems. Empirically, safety and numerical safety of GENP have been consistently observed in a number of papers where an input matrix was pre-processed with various structured multipliers chosen ad hoc. Our present paper provides missing formal support for this empirical observation and explains why it was elusive so far. Namely we prove that GENP is numerically unsafe for a specific class of input matrices in spite of its pre-processing with some well-known and well-tested structured multipliers, but we also prove that GENP and BGE are safe and numerically safe for the average input matrix pre-processed with any nonsingular and well-conditioned multiplier. This should embolden search for sparse and structured multipliers, and we list and test some new classes of them. We also seek randomized pre-processing that universally (that is, for all input matrices) supports (i) safe GENP and BGE with probability 1 and/or (ii) numerically safe GENP and BGE with a probability close to 1. We achieve goal (i) with a Gaussian structured multiplier and goal (ii) with a Gaussian unstructured multiplier and alternatively with Gaussian structured augmentation. We consistently confirm all these formal results with our tests of GENP for benchmark inputs. We have extended our approach to other fundamental matrix computations and keep working on further extensions.2000 Math. Subject Classification: 15A06, 15A52, 15A12, 65F05, 65F35

CUR Low Rank Approximation of a Matrix at Sublinear Cost

“…This continues our earlier study of dual problems of matrix computations with random input, in particular Gaussian elimination where randomization replaces pivoting (see [PQY15], [PZ17a], and [PZ17b]). We further advance this approach in [PLSZa], [PLa], [PLb], and [LPSa].…”

Randomized numerical linear algebra: Foundations and algorithms

“…Parker also observed that structured random matrices (such as the randomized trigonometric transforms from Section 9.3) allow us to perform the preconditioning step at lower cost than the subsequent Gaussian elimination procedure. Parker (1995) inspired many subsequent papers, including (Baboulin, Li and Rouet 2014, Trogdon 2017, Demmel et al 2012, Baboulin, Dongarra, Rémy, Tomov and Yamazaki 2017 and (Pan and Zhao 2017). Another related direction concerns the smoothed analysis of Gaussian elimination undertaken in (Sankar, Spielman and Teng 2006).…”

“…§17. General linear solvers Parker (1995) inspired many subsequent papers, including (Baboulin, Li and Rouet 2014, Trogdon 2017, Demmel et al 2012, Baboulin, Dongarra, Rémy, Tomov and Yamazaki 2017, Pan and Zhao 2017. Another related direction concerns the smoothed analysis of Gaussian elimination undertaken in (Sankar, Spielman and Teng 2006).…”

Application of the Concept of Linear Equation Systems in Balancing Chemical Reaction Equations

“…Several methods that are often used to solve systems of linear equations are the Gaussian and Gauss-Jordan elimination methods. Research on this method has been carried out, including research on (Baier et al, 2020;Ding and Schroeder, 2020;Schork and Gondzio, 2020;Pan and Zhao, 2017;Alonso et al, 2010;Geng et al, 2013). Then some research on Gauss-Jordan elimination can be seen in (David, 2016;Ma and Li, 2017;Anzt et al, 2018;Sheng, 2018), Based on the explanation and research, this paper focuses on balancing chemical reactions using a system of linear equations with the Cramer method and Gauss-Jordan elimination to see that a system of linear equations can be used to find the coefficient of each compound in chemical reaction equations.…”