Global Convergence of the Basic QR Algorithm

0. Introduction. The QR algorithm was developed by Francis (1960) to find the eigenvalues (or roots) of real or complex matrices. We shall consider it here in the context of exact arithmetic. Sufficient conditions for convergence, listed in order of increasing generality have been given by Francis [1], Kublanovskaja [3], Parlett [4], and Wilkinson [8]. It seems that necessary and sufficient conditions would be very complicated for a general matrix. One of the many merits of Francis' paper was the observation that the Hessenberg form ion = 0, i > j + 1) is invariant under the QR transformation and the algorithm is usually applied to Hessenberg matrices which are unreduced, that is Oij 9a 0, i = j + 1. The properties of this form combine with those of the algorithm in such a way that a complete convergence theory can be stated quite simply. The aim is to produce a sequence of unitarily similar matrices whose limit is upper triangular. Elementwise convergence to a particular triangular matrix is not necessary for determining eigenvalues; block triangular form with 1X1 and 2X2 blocks on the diagonal is sufficient. Definition. A sequence {77(s) = (A(*yO> s = 1, 2, • • • } of n X n Hessenberg matrices is said to "converge" whenever hf+xjh^'/j-x —* 0, for each,/ = 2, ■ • -, n — 1. Theorem 1. The basic QR algorithm applied to an unreduced Hessenberg matrix 77 produces a sequence of Hessenberg matrices which "converges" if, and only if, among each set of H's eigenvalues with equal magnitude, there are at most two of even and two of odd multiplicity. This is a special case, tailored to computer programs, of the main theorem. In general let coi > o>2 > • • • > o>r > 0 be the distinct nonzero magnitudes occurring among the roots of H. Of the roots of magnitude o>¿ let pQ) have even multiplicities