Upper and lower bounds for the tails of the distribution of the condition number of a gaussian matrix

SIAM Journal on Matrix Analysis and Applications

Volumn 26, Issue 2, 2005, Pages 426-440

  Azaïs, Jean Marc ^a   Wschebor, Mario ^b  

^a CNRS   (France)

^b UNIVERSIDAD DE LA REPÚBLICA   (Uruguay)

Author keywords

Condition number; Eigenvalue distribution; Random matrices; Rice formulae

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL MODELS; NUMERICAL ANALYSIS; PROBABILITY DISTRIBUTIONS; RANDOM PROCESSES; THEOREM PROVING;

CONDITION NUMBER; EIGENVALUE DISTRIBUTION; RANDOM MATRICES; RICE FORMULAE;

MATRIX ALGEBRA;

EID: 17444415657     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479803429764     Document Type: Article

References (22)

1. M. ABRAMOWITZ AND I. A. STEGUN, EDS., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, John Wiley & Sons, New York, 1984
2. R. J. ADLER, The Geometry of Random Fields, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Chichester, UK, 1981.
3. J.-M. AZAÏS AND M. WSCHEBOR, On the distribution of the maximum of a Gaussian field with d parameters, Ann. Appl. Probab., to appear.
4. J. CUESTA-ALBERTOS AND M. WSCHEBOR, Condition numbers and extrema of random fields, in Seminar on Stochastic Analysis, Random Fields and Applications IV, Progr. Probab. 58, Birkhäuser, Basel, 2004, pp. 69-82.
5. K. R. DAVIDSON AND S. J. SZAREK, Local operator theory, random matrices and Banach spaces, in Handbook of the Geometry of Banach Spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 317-366.
6. J. W. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
7. A. EDELMAN, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 543-560.
8. A. EDELMAN, On the distribution of a scaled condition number, Math. Comp., 58 (1992), pp. 185-190.
9. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
10. M. KENDALL, A. STUART, AND J. K. ORD, The Advanced Theory of Statistics. Vol. 3, 4th ed., Macmillan, New York, 1983.
11. P. R. KRISHNAIAH AND T. C. CHANG, On the exact distributions of the extreme roots of the Wishart and MANOVA matrices, J. Multivariate Anal., 1 (1971), pp. 108-117.
12. M. LEDOUX, A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices, Séminaire de Probabilités XXXVII, Lecture Notes in Math. 1832, Springer-Verlag, Berlin, 2003, pp. 360-369.
13. A. SANKAR, D. A. SPIELMAN, AND S. H. TENG, Smoothed Analysis of the Condition Number and Growth Factors of Matrices, preprint, 2002.
14. S. SMALE, On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (N.S.), 13 (1985), pp. 87-121.
15. S. J. SZAREK, Condition numbers of random matrices, J. Complexity, 7 (1991), pp. 131-149.
16. J. E. TAYLOR AND R. J. ADLER, Euler characteristics for Gaussian fields on manifolds, Ann. Probab., 31 (2003), pp. 533-563.
17. A. M. TURING, Rounding-off errors in matrix processes, Quart. J. Mech. Appl. Math., 1 (1948), pp. 287-308.
18. J. VON NEUMANN AND H. H. GOLDSTINE, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc., 53 (1947), pp. 1021-1099.
19. E. P. WIGNER, Random matrices in physics, SIAM Rev., 9 (1967), pp. 1-23.
20. J. H. WILKINSON, Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs, NJ, 1963.
21. S. S. WILKS, Mathematical Statistics, John Wiley & Sons, New York, 1962.
22. M. WSCHEBOR, Smoothed analysis of κ(a), J. Complexity, 20 (2004), pp. 97-107.