Perturbation Splitting for More Accurate Eigenvalues

Let T be a symmetric tridiagonal matrix with entries and eigenvalues of di¤erent magnitudes. For some T , small entry-wise relative perturbations induce small errors in the eigenvalues, independently of the size of the entries of the matrix; this is certainly true when the perturbed matrix can be written as e T = XTTX, with small jjXTX Ijj. Even if it is not possible to express in this way the perturbations in every entry of T , a lot can be gained by doing so for as many as possible entries of larger magnitude. We propose a technique which consists upon splitting multiplicative and additive perturbations to produce new error bounds which, for some matrices, are much sharper than the usual ones. Such bounds may be useful in the development of improved software for the tridiagonal eigenvalue problem and we describe their role in the context of a mixed precision bisection-like procedure. Using the very same idea of splitting perturbations (multiplicative and additive), we show that, when T de…nes well its eigenvalues, the numerical values of the pivots in the usual decomposition T I = LDLT may be used to compute approximations with high relative precision. Key words. symmetric tridiagonal matrices, eigenvalues, perturbation theory. AMS subject classi…cations. 15A15, 15A09, 15A23 1. Introduction. Let A and E be n-by-n symmetric matrices. Let 1 n and e 1 e n be the eigenvalues of A and e A = A + E, respectively. Then j k e kj kEk2. This is a classical result in the perturbation theory (see [44], pp. 101–102) which is usually referred to as the Weyl’s theorem (see, for instance, [9], p. 198). Weyl’s theorem can be used to get error bounds for the eigenvalues computed by any backward stable algorithm since such an algorithm computes eigenvalues e k that are the exact eigenvalues of e A = A+E where kEk2 = O( ) kAk2 (here and throughout the paper we will use to denote the rounding error unit). This is a very satisfactory error bound for large eigenvalues, especially those of magnitude close to kAk2, but eigenvalues much smaller than kAk2 will have fewer correct digits (eventually none in extreme cases). The decade starting in 1990 was fertile in new results on bounds for relative errors of eigenvalues and several authors have contributed to this ([1], [4], [5], [16], [17], [22], [32], [33], [34], [35], [42]). In [22], Ipsen presents a good survey of the work done until 1998. Not surprisingly, many of the published results are for the Hermitian positive de…nite case. For an Hermitian inde…nite matrix A and, more generally, for normal matrices, the Hermitian positive-semide…nite factor H in the polar decomposition A = HU , with U unitary, may be used to derive bounds for the eigenvalues of A (see Theorems 2.4 and 2.10 in [22] and references therein). The …rst relative perturbation bound for eigenvalues is due to Ostrowski. Let b A = XAX , with X nonsingular, be a multiplicative perturbation of an Hermitian matrix A; for the eigenvalues k and b k, of A and b A, respectively, we have (Theorem 4.5.9 Departamento de Matemática, Universidade do Minho, 4710-057 Braga, Portugal ([email protected]). This research was supported by the Portuguese Foundation for Science and Technology through the research program POCI 2010. The author is indebted to Beresford Parlett, Zlatko Drmaμc and the anonymous referees for several criticisms and suggestions that improved the paper signi…cantly. The author also expresses his gratitude to Froilan Dopico for fruitful discussions and acknowledges that he proved Theorem 3.1 in an independent manner.