Rayleigh quotient iteration fails for nonsymmetric matrices

08939659189 83.00 + 0.00 Copyright@ 1989 Pergamon Press plc

Appl. Math. Lett. Vol. 2, No. 1, pp. 19-20,1989 Printed in Great Britain @ Ah rights reserved

RAYLEIGH QUOTIENT FOR NONSYMMETRIC STEVE BATTERSON’

’ Department 2 Department

ITERATION FAILS MATRICES and JOHN SMILLIE~

of Mathematics, of Mathematics,

Emory Cornell

University University

Abstract. Examples are given of nonsymmetric matrices for which the Bayleigh Quotient Iteration fails to converge to an eigenspace for a large set of initial vectors.

Rayleigh Quotient Iteration is a method for finding an approximate eigenvector of a For an n x n real valued matrix A the algorithm produces a function FA : matrix. RP”-’ --i Rp”-l. To produce an approximate eigenvector one chooses an initial vector (we will assume at random) and applies FA to the initial vector until the desired degree of approximation is achieved. Corresponding to each real eigenvalue or conjugate pair of complex eigenvalues of A there is a primary invariant subspace. The probability of success of the algorithm is the measure of the set of initial vectors v for which F;(v) converges to a primary subspace of A. We say that Rayleigh Quotient Iteration succeeds for a given matrix A if F:(v) converges to a primary subspace of A for a set of initial vectors v of full measure. If this is not the case we say that the algorithm fails for A. by the following map FA defined on R”. The map FA is induced on RP”-’ FL(c) = (A - pA(z)I where pA = (Az,z)/(z,z). If A is a symmetric matrix then Rayleigh Quotient Iteration succeeds (see [5] and [2]). It is an important problem in numerical analysis to find analogues of Rayleigh Quotient Iteration which succeed for nonsymmetric matrices. In this note we will examine the behavior of the algorithm corresponding to FA in the nonsymmetric case. It is shown in [7] that this algorithm succeeds in the rr = 2 case. We will show that this algorithm fails for a substantial set of nonsymmetric matrices. For each n 2 3 there is a non-empty Iteration fails.

THEOREM.

Quotient

open set of matrices

for which Rayleigh

Note. This is the most straightforward generalization of the Rayleigh Quotient Iteration to nonsymmetric matrices. There are other generalizations however (see [6] and [4]). See also [3]. The proof of this result

will appear

in [I]. We give an outline

here.

matrix A,the map FA has a hyperbolic sink which is not contained in a primary subspace then, for every initial vector v in the basin of the sink, F:(v) fails to converge to a primary subspace of A. Since the basin of a sink is a non-empty open set we conclude that Rayleigh Quotient Iteration fails for A. To prove the theorem it suffices to find in each dimension greater than or equal to 3 a matrix A for which FA has such a sink. It then follows from the persistence of hyperbolic sinks that for any matrix A’ sufficiently close to A the map FANwill have a hyperbolic sink as above and the algorithm will fail for A’. OUTLINE OF PROOF:

If for a given

The second author was partiahy supported by the Mathematical Sciences Institute and the NSF. Typeset by A,@-TEX 19

S. BATTERSON AND J. SMILLIE

20

We begin by describing the construction in dimension 3. Let

A =

17, ( 0

ii’ 0

: d1

Let VI be the subspace of R3 spanned by the first two coordinate vectors. Let Vz be the subspace spanned by the last coordinate vector. The spaces VI and V2 are the primary subspaces of A. The subset of RP2 corresponding to VI is invariant under F. This set is a circle and we denote it by C. The dynamics of FIG depend only on c. At c = 1 FIG’ is a rotation of period two. For c > 0 there is an orbit of period two corresponding to the coordinate axes which we denote by p, and py . The stability of this orbit is determined by the derivative of F21C at p,. This number is 1 when c = 1 and decreases as c decreases. The period 2 orbit is attracting until the derivative becomes -1 when it undergoes a period doubling bifurcation which produces a sink z, of period 4. These results are proved by using a one-dimensional bifurcation argument. The normal derivative to C is controlled by the relative norms of the eigenvalues of the first block and the second block of A. The eigenvalues of the first block are fi and the eigenvalue of the second block is d. Our objective is to adjust c in order to create z, and then adjust d to make z, normally repelling. This causes two period 4 sinks to bifurcate from ze. To make this rigorous we use the center stable manifold theorem and a one-dimensional bifurcation argument on the curve produced by the center stable manifold theorem. These sinks are not contained in either primary subspace of A. This completes the outline of the construction for n = 3. For n > 3 we construct a matrix by adjoining to A a block of the form ICI where h: is chosen to be sufficiently large. Our computer calculations suggest that in the n = 3 case when c = 0.56 and is in the basin of a period 4 sink. This is an empirical result. These are not necessarily numbers which would be produced by our proof if we were to make it quantitative. Note.

d = 1.1 the point (-0.104,-0.995,0.107)

Note. The formula for Fi defines a map on CP”-l. This map may produce a better algorithm than the one considered here. We do not know whether this algorithm succeeds In any case the construction above does not seem to yield a counterexample on CP”-l. in this setting. REFERENCES 1. S. Batterson and J. SmiIIie, Rayleigh quotient iteration for nonsymmetric matrices. 2. S. Batterson and J. SmilIie, The dynamics of Rayleigh quotient iteration, SIAM J. Numerical Analysis (to appear). 3. P. Hriljac, The dynamics of dome birational maps on rational varieties with an application to computalional linear algebra I and II, preprint. 4. A. M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the eompulation of charaelerislic roota and vector8 I- VI, Arch. Rational Mech. Anal. l-4 (1958/59), 233-241, 423-428, 325-340, 341-347, 472-481, 153-165. 5. B. N. Parlett, “The Symmetric Eigenvalue Problem,” Prentice HalI, Endewood Cliffs, New Jersey, 1980. 6. B. N. Parlett, The Rayleigh quotient iteration and come generalizations for nonnormal matrices, Math. Comp. 28 (1974), 679-693. 7. M. Shub, Some remarks on dynamical systems and numerical analyaia, in “Dynamical Systems and Partial Differential Equations: Proceedings of the VII ELAM,” Equinoccio, Universidad Simon Bolivar, Caracas, 1986, pp. 69-92. ’ Department of Mathematics, Emory University, Atlanta, GA 36322 2 Department of Mathematics, Cornell University, Ithaca, NY 14850