Inverse Iteration on Defective Matrices

Very often, inverse iteration is used with shifts to accelerate convergence to an eigenvector. In this paper, it is shown that, if the eigenvalue belongs to a nonlinear elementary divisor, the vector sequences may diverge even when the shift sequences converge to the eigenvalue. The local behavior is discussed through a 2 X 2 example, and a sufficient condition for the convergence of the vector sequence is given. Introduction. If an accurate approximation a to an eigenvalue X of a matrix B is available, then inverse iteration is an attractive technique for computing the associated eigenvector. We choose an arbitrary unit vector vQ and a fixed shift a. Then for / = 1, 2, ... we solve (,) (BofyWj = Vj_v Vj = wy/Uwyll, where || • || is the users' preferred vector norm. Semisimple case. If X is a simple eigenvalue with unit eigenvector x, if a is close enough to X, and if v0 is not an unfortunate choice, then the vector sequence {u} converges linearly to x and the convergence factor is very favorable. This well-known result holds also for multiple eigenvalue X provided that: (i) the dimension of X's eigenspace is equal to X's algebraic multiplicity (i.e. linear elementary divisors), (ii) the spectral projection of v0 into X's eigenspace is not zero (i.e. the starting vector is not deficient in x). If a is known to equal X to within working precision of the computer, then only one or two steps of the iteration are necessary. Defective Case. Wilkinson [3] and Varah [2] pointed out that the situation is not so nice if X has generalized eigenvectors of grade higher than one, i.e., when X belongs to a nonlinear elementary divisor. In exact arithmetic the iteration converges not linearly, but harmonically like 1 // as /' —► °°. Even worse is the fact that except for very special choices of u0, the vectors v2 and v3 will be poorer approximations than Uj ! Variable Shifts. Inverse iteration can also be used with variable shifts. If the sequence of shifts {a} converges to X as / —► °°, then the vector sequence generated Received January 20, 1976; revised June 21, 1976. AMS (MOS) subject classifications (1970). Primary 65F15; Secondary 15A18.