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...
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| Vydáno v: | Linear algebra and its applications Ročník 527; s. 349 - 383 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Amsterdam
Elsevier Inc
15.08.2017
American Elsevier Company, Inc |
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| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | 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. |
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| AbstractList | 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. 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. |
| Author | Pan, Victor Y. Zhao, Liang |
| Author_xml | – sequence: 1 givenname: Victor Y. surname: Pan fullname: Pan, Victor Y. email: victor.pan@lehman.cuny.edu organization: Department of Mathematics and Computer Science, Lehman College of the City University of New York, Bronx, NY 10468, USA – sequence: 2 givenname: Liang surname: Zhao fullname: Zhao, Liang email: lzhao1@gc.cuny.edu organization: Ph.D. Programs in Mathematics and Computer Science, The Graduate Center of the City University of New York, New York, NY 10036, USA |
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| Keywords | 15A52 15A12 Gaussian elimination 15A06 Block Gaussian elimination Linear systems of equations Pivoting Random matrix algorithms 65F35 Preconditioning 65F05 |
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| SubjectTerms | Block Gaussian elimination Conditioning Dividing (mathematics) Gaussian elimination Linear algebra Linear systems of equations Mathematics Matrix Multipliers Normal distribution Pivoting Preconditioning Random matrix algorithms |
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