Abstract
Several extreme eigenvalues and vectors of large symmetric matrices can often be found to machine accuracy by carrying out far less than the full number of steps of the Lanczos process. To prove the accuracy of such results a rounding error analysis of the process with re-orthogonalization is given here. It is found that a stopping criterion has to be introduced to obtain a priori error bounds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Lanczos,An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Stand.45 (1950), 255–282.
N. J. Lehmann,Zur Verwendung optimaler Eigenwerteingrenzungen bei der Lösung symmetrischer Matrizenaufgaben, Num. Math.8 (1966), 42–55.
S. Kaniel,Estimates for some computational techniques in Linear Algebra, Mathematics of Computation 20 (1966), 369–378.
J. H. Wilkinson,The Algebraic Eigenvalue Problem, Oxford: Clarendon Press, 1965.
C. C. Paige,Error analysis of the symmetric Lanczos process for the eigenproblem, London Univ. Inst. of Computer Science, Tech. Note ICSI 209, 1969.
G. E. Forsythe and C. B. Moler,Computer Solution of Linear Algebraic Systems, Prentice-Hall, Inc. Englewood Cliffs, N.J., 1967.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Paige, C.C. Practical use of the symmetric Lanczos process with re-orthogonalization. BIT 10, 183–195 (1970). https://doi.org/10.1007/BF01936866
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01936866