What Hadamard Missed,
Beresford Ν. Parlett · 1996
Abstract : Consider the task of finding all the eigenvalues of a dense matrix. We show how Hadamard's procedure (1891) can be organized into Aitken's H-table (1925) and how the H-table may be transformed into Rutishauser's qd-array (1953) with the help of the Lanczos algorithm. We show how the qd algorithm can be interpreted as defining the LR algorithm (1958). Finally we show how the original qd algorithm may be transformed into the shifted differential qd algorithm dqds developed by Fernando and Parlett (1993/94). The Lanczos algorithm takes a dense matrix into tridiagonal form and then dqds is a fast and accurate procedure for extracting the eigenvalues.