Practical methods for evaluating the accuracy of the eigenelements of a symmetric matrix

The results of the Householder and QL algorithms for determining the eigenelements of a symmetric matrix, provided by a computer, always contain the errors resulting from floating-point arithmetic round-off error propagation. The Permutation-Perturbation method is a very efficient practical method for evaluating these errors and consequently for estimating the exact significant figures of the eigenelements. But, in the cases of: eigenvalues very close to zero, eigenvalues of widely varying range, and multiple eigenvalues, the Permutation-Perturbation method is not complete. In this paper we propose an algorithm which is able to complete this method.