The Implicit QL Algorithm

Abstract

In [1] an algorithm was described for carrying out the QL algorithm for a real symmetric matrix using shifts of origin. This algorithm is described by the relations

$$\matrix{ {{Q_s}({A_s} - {k_s}I) = {L_s},} & {{A_{s + 1}} = {L_s}Q_s^T + {k_s}I,} & {{\rm{giving}}} & {{A_{s + 1}} = {Q_s}{A_s}Q_s^T,} \cr } $$

((1))

where Q _s is orthogonal, L _s is lower triangular and k _s is the shift of origin determined from the leading 2×2 matrix of A _s .

Prepublished in Numer. Math. 12, 377–383 (1968).