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Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
Open accessAbstract
Let Å be an arbitrary matrix and let A be a slight random perturbation of Å. We prove that it is unlikely that A has a large condition number. Using this result, we prove that it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we show that the smoothed precision necessary to solve Ax = b, for any b, using Gaussian elimination without pivoting is logarithmic. Moreover, when Å is an all-zero square matrix, our results significantly improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 18 (1997), pp. 499-517).
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Cited by
Quantum Algorithm for Linear Systems of Equations
Harrow, A. W., Hassidim, A., & Lloyd, S. (2009) · Physical Review Letters · doi:10.1103/physrevlett.103.150502 · Cited by 3,325
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Spielman, D. A., & Teng, S.-H. (2004) · Journal of the ACM · doi:10.1145/990308.990310 · Cited by 825
Self-citationRungger, I., & Sanvito, S. (2008) · Physical Review B · doi:10.1103/physrevb.78.035407 · Cited by 316
Metadata
- Authors
- Sankar, A., Spielman, D. A., & Teng, S.-H.
- Venue
- SIAM Journal on Matrix Analysis and Applications
- Published
- 2006-01
- Volume
- 28
- Issue
- 2
- Pages
- 446-476
- ISSN
- 0895-4798 (ISSN-L)
- 1095-7162
- OpenAlex
- W2170078634
- Cited by
- 207
- References
- 0
- FWCI
- 6.83
- Type
- journal article
- Language
- en
- Open access
- green
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- Sankar, A., Spielman, D. A., & Teng, S.-H. (2006). Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices. SIAM Journal on Matrix Analysis and Applications. https://doi.org/10.1137/s0895479803436202
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