The Q R Algorithm for Band Symmetric Matrices

Abstract

The Q R algorithm with shifts of origin may be used to determine the eigenvalues of a band symmetric matrix A. The algorithm is described by the relations

$$\matrix{ {{A_s} - {k_s}I = {Q_s}{R_s},} & {{R_s}{Q_s} = {A_{s + 1}}} & {(s = 1,2, \ldots )} \cr } $$

((1))

where the Q _s are orthogonal and the R _s are upper triangular. It has been described in some detail by Wilkinson [5, pp. 557–561]. An essential feature is that all the A _s remain of band symmetric form and R _s is also a band matrix, so that when the width of the band is small compared with the order of the A _s , the volume of computation in each step is quite modest. If the shifts ks are appropriately chosen the off-diagonal elements in the last row and column tend rapidly to zero, thereby giving an eigenvalue.

Prepublished in Numer. Math. 16, 85–92 (1970).