The QR Transformation--Part 2

The QR transformation is an analogue to the LR transformation (Rutishauser, 1958) based on unitary transformations. Both these transformations are global iterative methods for finding the eigenvalues of a matrix, the matrix converging in general to triangular form. In Par t1 of this paper the QR transformation was briefly described and we were then principally concerned with proving convergence, the main result being expressed in theorem 3. We also showed that if the matrix is first reduced to almost triangular form important advantages are gained (further advantages will become apparent) and we gave in outline a way in which convergence could be improved. In this part of the paper we consider the practical application of the QR transformation. Two versions of the algorithm have been programmed for the Pegasus computer; these are described and an attempt is made to evaluate the method. Some results and detailed algorithms are given in appendices. Part 1 was published on pp. 265–71 of this volume (Oct. 61).