Fernando's solution to Wilkinson's problem: An application of double factorization
An algorithm for the computation of eigenvectors to a real symmetric tridiagonal matrix, when eigenvalues are already computed, is studied. It is built up on a recursion, used already by \textit{W. Givens} (J. Assoc. Comput. Mach. 4, 298-307 (1957; MR 19.10806). Following a suggestion by \textit{K. V. Fernando} [On computing an eigenvector of a tridiagonal matrix. I: Basic results, SIAM J. Matrix Anal. Appl. 18, No. 4, 1013-1034 (1997)], it is shown at which coordinate this recursion should be started, in order to be stable and yield orthogonal eigenvectors, also when the eigenvalue approximations are multiple to the accuracy used. These starting coordinates are found by running Gaussian elimination both forwards and backwards. Several numerical examples are included.
scientific article; zbMATH DE number 1089500
scientific article