Smoothed analysis of Gaussian elimination
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
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| Format: | Thesis |
| Language: | en_US |
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Massachusetts Institute of Technology
2005
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| Online Access: | http://hdl.handle.net/1721.1/28311 |
| _version_ | 1811070496026918912 |
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| author | Sankar, Arvind, 1976- |
| author2 | Daniel A. Spielman. |
| author_facet | Daniel A. Spielman. Sankar, Arvind, 1976- |
| author_sort | Sankar, Arvind, 1976- |
| collection | MIT |
| description | Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. |
| first_indexed | 2024-09-23T08:36:57Z |
| format | Thesis |
| id | mit-1721.1/28311 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T08:36:57Z |
| publishDate | 2005 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/283112019-04-09T18:46:49Z Smoothed analysis of Gaussian elimination Sankar, Arvind, 1976- Daniel A. Spielman. Massachusetts Institute of Technology. Dept. of Mathematics. Massachusetts Institute of Technology. Dept. of Mathematics. Mathematics. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. Includes bibliographical references (p. 59-60). We present a smoothed analysis of Gaussian elimination, both with partial pivoting and without pivoting. Let A be any matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the smoothed precision needed to perform Gaussian elimination without pivoting. Our results improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997). We then extend the result on the growth factor to the case of partial pivoting, and present the first analysis of partial pivoting that gives a sub-exponential bound on the growth factor. In particular, we show that if the random perturbation is Gaussian with variance [sigma]², then the growth factor is bounded by (n/[sigma])[to the power of] (o log n) with very high probability. by Arvind Sankar. Ph.D. 2005-09-26T19:42:49Z 2005-09-26T19:42:49Z 2004 2004 Thesis http://hdl.handle.net/1721.1/28311 55635777 en_US M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 60 p. 2613986 bytes 2619272 bytes application/pdf application/pdf application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Sankar, Arvind, 1976- Smoothed analysis of Gaussian elimination |
| title | Smoothed analysis of Gaussian elimination |
| title_full | Smoothed analysis of Gaussian elimination |
| title_fullStr | Smoothed analysis of Gaussian elimination |
| title_full_unstemmed | Smoothed analysis of Gaussian elimination |
| title_short | Smoothed analysis of Gaussian elimination |
| title_sort | smoothed analysis of gaussian elimination |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/28311 |
| work_keys_str_mv | AT sankararvind1976 smoothedanalysisofgaussianelimination |