Inclusion of eigenvalues of general eigenvalue problems for matrices. (English) Zbl 0663.65034
Scientific computation with automatic result verification, Proc. Conf. Comput. Arith. Sci. Comput., Karlsruhe/FRG 1987, Comput. Suppl. 6, 69-78 (1988).
[For the entire collection see Zbl 0646.00013.]
The paper is concerned with inclusion of eigenvalues of matrices. Let the eigenvalues be ordered with respect to their absolute values. A procedure for calculating sharp bounds for a fixed eigenvalue \(\lambda_ j\) is given. It consists of three steps: calculation of approximations \({\tilde \lambda}{}_{j-1}\), \({\tilde \lambda}{}_{j+1}\) for the corresponding eigenvalues and \(\tilde x_ j\) for a corresponding eigenvector, determination of rough upper and lower bounds and then calculation of accurate bounds for the eigenvalue by means of Temple quotients and their generalization by N. H. Lehmann [Z. Angew. Math. Mech. 29, 341-356 (1949) and ibid. 30, 1-16 (1956; Zbl 0034.375)], using interval arithmetic. The procedure also can be used in case of multiple or clustered eigenvalues. Some numerical examples are given which illustrate this property.
The paper is concerned with inclusion of eigenvalues of matrices. Let the eigenvalues be ordered with respect to their absolute values. A procedure for calculating sharp bounds for a fixed eigenvalue \(\lambda_ j\) is given. It consists of three steps: calculation of approximations \({\tilde \lambda}{}_{j-1}\), \({\tilde \lambda}{}_{j+1}\) for the corresponding eigenvalues and \(\tilde x_ j\) for a corresponding eigenvector, determination of rough upper and lower bounds and then calculation of accurate bounds for the eigenvalue by means of Temple quotients and their generalization by N. H. Lehmann [Z. Angew. Math. Mech. 29, 341-356 (1949) and ibid. 30, 1-16 (1956; Zbl 0034.375)], using interval arithmetic. The procedure also can be used in case of multiple or clustered eigenvalues. Some numerical examples are given which illustrate this property.
Reviewer: G.Schröder
MSC:
65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
15A42 | Inequalities involving eigenvalues and eigenvectors |
65G30 | Interval and finite arithmetic |