Probabilistic Bounds on the Extremal Eigenvalues and Condition Number by the Lanczos Algorithm
- 1 April 1994
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Matrix Analysis and Applications
- Vol. 15 (2) , 672-691
- https://doi.org/10.1137/s0895479892230456
Abstract
The authors analyze the Lanczos algorithm with a random start for approximating the extremal eigenvalues of a symmetric positive definite matrix. They present some bounds on the Lebesgue measure (probability) of the sets of these starting vectors for which the Lanczos algorithm gives at the kth step satisfactory approximations to the largest and smallest eigenvalues. Combining these bounds gets similar estimates for the condition number of a matrix.Keywords
This publication has 8 references indexed in Scilit:
- Estimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random StartSIAM Journal on Matrix Analysis and Applications, 1992
- Adaptive Lanczos methods for recursive condition estimationNumerical Algorithms, 1991
- Incremental Condition EstimationSIAM Journal on Matrix Analysis and Applications, 1990
- Estimating Extremal Eigenvalues and Condition Numbers of MatricesSIAM Journal on Numerical Analysis, 1983
- On the Rates of Convergence of the Lanczos and the Block-Lanczos MethodsSIAM Journal on Numerical Analysis, 1980
- HOW FAR SHOULD YOU GO WITH THE LANCZOS PROCESS?††The authors are pleased to acknowledge partial support from Office of Naval Research Contract N00014-69-A-0200-1017.Published by Elsevier ,1976
- Computational Variants of the Lanczos Method for the EigenproblemIMA Journal of Applied Mathematics, 1972
- Estimates for some computational techniques in linear algebraMathematics of Computation, 1966