Probabilistic Bounds on the Extremal Eigenvalues and Condition Number by the Lanczos Algorithm

Authors:
J. Kuczynski;H. Wozniakowski
Affiliations:
-;-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
1994

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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 $k$th step satisfactory approximations to the largest and smallest eigenvalues. Combining these bounds gets similar estimates for the condition number of a matrix.