Improving the Accuracy of Inverse Iteration
Elizabeth R. Jessup, Ilse C. F. Ipsen · SIAM Journal on Scientific and Statistical Computing · 1992
The EISPACK routine TINVIT is an implementation of inverse iteration for computing eigenvectors of real symmetric tridiagonal matrices. Experiments have shown that the eigenvectors computed with TINVIT are numerically less accurate than those from implementations of Cuppen’s divide-and-conquer method (TREEQL) and of the QL method (TQL2). The loss of accuracy can be attributed to TINVIT’s choice of starting vectors and to its iteration stopping criterion. This paper introduces a new implementation of TINVIT that computes each eigenvector from a different random starting vector and performs an additional iteration after the stopping criterion is satisfied. A statistical analysis and the results of numerical experiments with matrices of order up to 525 are presented to show that the numerical accuracy of this new implementation is competitive with that of the implementations of the divide-and-conquer and QL methods. The extension of this implementation to larger order matrices is discussed, albeit in less detail.