[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.