Smoothed analysis of the condition numbers and growth factors of matrices

SIAM Journal on Matrix Analysis and Applications

Volumn 28, Issue 2, 2006, Pages 446-476

  Sankar, Arvind ^a   Spielman, Daniel A ^b   Teng, Shang Hua ^c  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b Yale University   (United States)

^c BOSTON UNIVERSITY   (United States)

Author keywords

Condition number; Gaussian elimination; Growth factor; Smoothed analysis

Indexed keywords

CONDITION NUMBERS; GAUSSIAN ELIMINATION; GROWTH FACTOR; SMOOTHED ANALYSIS;

COMPUTATIONAL COMPLEXITY; MATRIX ALGEBRA; PERTURBATION TECHNIQUES; PROBLEM SOLVING;

NUMBER THEORY;

EID: 34247360062     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/S0895479803436202     Document Type: Article

References (27)

1. K. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. DU CROZ, A. GREENBAUM, S. HAMMARLING. A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, 3rd ed., SIAM, Philadelphia, 1999.
2. M. ABRAMOWITZ AND I. A. STEGUN, EDS., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, DC, 1964. Tenth printing, with corrections, 1972.
3. A. BLUM AND J. DUNAGAN, Smoothed analysis of the perceptron algorithm for linear programming, in Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, SIAM, 2002, pp. 905-911.
4. L. BLUM, Lectures on a theory of computation and complexity over the reals (or an arbitrary ring), in Proceedings of the Complex Systems Summer School, Santa Fe. NM, 1989, pp. 1-47.
5. J. W. DEMMEL, The probability that a numerical analysis problem is difficult, Math. Comput., (1988), pp. 449-480.
6. J. W. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
7. K. R. DAVIDSON AND S. J. SZAREK, Local operator theory, random matrices, and Banach spaces in Handbook on the Geometry of Banach Spaces, W. B. Johnson and J. Lindenstrauss, eds., North Holland, Amsterdam, 2001, pp. 317-366.
8. J. DUNAGAN, D. A. SPIELMAN, AND S.-H. TENG, Smoothed analysis of Renegar's condition number for linear programming, available at http://arxiv.org/abs/cs.DS/0302011, 2003.
9. A. EDELMAN, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 543-560.
10. A. EDELMAN, Eigenvalue roulette and random test matrices, in Linear Algebra for Large Scale and Real-Time Applications, Marc S. Moonen, Gene H. Golub, and Bart L. R. De Moor, eds., NATO ASI Series, 1992, pp. 365-368.
11. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, Johns Hopkins Series in the Mathematical Sciences, The Johns Hopkins University Press, Baltimore, 1983.
12. M. GRÖTSCHEL, L. LOVASZ, AND A. SCHRIJVER, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1991.
13. N. HIGHAM, How accurate is Gaussian elimination? in Proceedings of the 13th Dundee Conference, Dundee, Australia, Pitman Res. Notes Math. Ser., Longman Sci. Tech. Harlow, UK, 1990, pp. 137-154.
14. N. JOHNSON, S. KOTZ, AND N. BALAKRISHNAN, Continuous Univariate Distributions. Vol. 2, John Wiley & Sons, New York, 1995.
15. S. KOTZ AND N. L. JOHNSON, EDS., Encyclopedia of Statistical Sciences, Volume 6, John Wiley & Sons, New York, 1982.
16. M. LEDOUX AND M. TALAGRAND, Probability in Banach Spaces, Springer-Verlag, Berlin, 1991.
17. J. RENEGAR, Incorporating condition measures into the complexity theory of linear programming, SIAM J. Optim., 5 (1995), pp. 506-524.
18. Y. SEGINER, The expected norm of random matrices, Combin. Probab. Comput., 9 (2000), pp. 149-166.
19. S. SMALE, Complexity theory and numerical analysis, Acta Numer., 6 (1997), pp. 523-551.
20. D. SPIELMAN, http://www.cs.yale.edu/homes/spielman/ SmoothedAnalysis.
21. D. SPIELMAN AND S. H. TENG, Smoothed analysis of termination of linear programming algorithms, Math. Program., 97 (2003), pp. 375-404.
22. D. A. SPIELMAN AND S.-H. TENG, Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time, J. ACM, 51 (2004), pp. 385-463.
23. L. N. TREFETHEN AND D. BAU, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
24. L. N. TREFETHEN AND R. S. SCHREIBER, Average-case stability of Gaussian elimination, SIAM J. Matrix Anal. Appl., 11 (1990), pp 335-360.
25. J. H. WILKINSON, Error analysis of direct methods of matrix inversion, J. Assoc. Comput. Mach., 8 (1961), pp. 281-330.
26. M. WSCHEBOR, Smoothed analysis of κ(a), J. of Complexity, 20 (2004), pp. 97-107.
27. M.-C. YEUNG AND T. F. CHAN, Probabilistic analysis of Gaussian elimination without pivoting, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 499-517.