Numerically safe Gaussian elimination with no pivoting

Gaussian elimination with no pivoting and block Gaussian elimination are attractive alternatives to the customary but communication intensive Gaussian elimination with partial pivoting1provided that the computations proceed safely and numerically safely, that is, run into neither division by 0 nor n...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Linear algebra and its applications Jg. 527; S. 349 - 383
Hauptverfasser: Pan, Victor Y., Zhao, Liang
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Amsterdam Elsevier Inc 15.08.2017
American Elsevier Company, Inc
Schlagworte:
Block Gaussian elimination
Conditioning
Dividing (mathematics)
Gaussian elimination
Linear algebra
Linear systems of equations
Mathematics
Matrix
Multipliers
Normal distribution
Pivoting
Preconditioning
Random matrix algorithms
15A52
15A12
Gaussian elimination
15A06
Block Gaussian elimination
Linear systems of equations
Pivoting
Random matrix algorithms
65F35
Preconditioning
65F05
ISSN:0024-3795, 1873-1856
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:Gaussian elimination with no pivoting and block Gaussian elimination are attractive alternatives to the customary but communication intensive Gaussian elimination with partial pivoting1provided 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. 1Hereafter we use the acronyms GENP, BGE, and GEPP.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2017.04.007