How to implement the spectral transformation

Abstract

Abstract. The general, linear eigenvalue equations (H -XM)z = 0, where H and M are real symmetric matrices with M positive semidefimte, must be transformed if the Lanczos algorithm is to be used to compute eigenpairs (X,z). When the matrices are large and sparse (but not diagonal) some factorization must be performed as part of the transformation step. If we are interested in only a few eigenvalues a near a specified shift, then the spectral transformation of Ericsson and Ruhe [1] proved itself much superior to traditional methods of reduction.The purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects. Perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it. It arises in the use of inverse iteration.A second goal is to introduce a systematic modification of the computed Ritz vectors, which improves the accuracy when M is ill-conditioned or singular.We confine our attention to the simple Lanczos algorithm, although the first two sections apply directly to the block algorithms as well. Overview. This contribution is an addendum to the paper by Ericsson and Ruhe[1] and also [7]. The value of the spectral transformation is reiterated in a later section. Here we outline our implementation of this transformation.The equation to be solved, for an eigenvalue À and eigenvector z, is(1) (H -AM)z = 0, H and M are real symmetric n X n matrices, and M is positive semidefinite. A practical instance of (1) occurs in dynamic analysis of structures, where H and M are the stiffness and mass matrices, respectively. We assume that a linear combination of H and M is positive definite. It then follows that all eigenvalues X are real. In addition, one has a real scalar a, distinct from any eigenvalue, and we seek a few eigenvalues X close to a, together with their eigenvectors z. Ericsson and Ruhe replace (1) by a standard eigenvalue equation (2) [c(H-aM)"1CT-rl]y = 0, where C is the Choleski factor of M; M = CTC and y = Cz. If M is singular then so is C, but fortunately the eigenvector z can be recovered from y via z = (H -oM)"'CTy. Of course, there is no intention to invert (H -oM) explicitly. The

Implicitly restarted Arnoldi with purification for the shift-invert transformation

“…This cannot occur when S is nondefective, which is the case considered by Nour-Omid, Parlett, Ericsson and Jensen [10]. The Ritz vectors are more severely affected, being corrupted by round-off errors in both the N and G subspaces, but Ericsson shows that these errors may be eliminated by two applications of S, though in practice only one is in fact needed by use of a clever trick, called 'purification' in [10], but also discussed in [6] and [5]. In fact, we find it convenient to use the expression 'purification' in a more general way, so that it refers to the general operation of forming Sx from x ∈ C N .…”

“…These approximations are known as 'spurious' eigenvalues and are sometimes hard to distinguish from approximations to wanted eigenvalues. Several techniques have been proposed to reduce the risk of computing spurious eigenvalues for the symmetric nondefective problem [10,5] and the defective problem [5]. In this paper we shall concentrate on the nonsymmetric defective case, as exemplified by S derived from (2), because of its importance in applications.…”

“…For the symmetric problem, both Ericsson [5] and Nour-Omid, Parlett, Ericsson and Jensen [10] use the Lanczos method with B-orthogonalisation, i.e. use the B semi-inner product x H By instead of the classical x H y.…”

“…This expression was used by Ericsson and Ruhe [6] as a means of improving accuracy in eigenvectors, but Nour-Omid, Parlett, Ericsson and Jensen [10] considered especially the case B singular and called the correction of V k z 'purification', because it has the potential for removing significant round-off error components in the N space. Since this is a rather important result for our method, we describe why this is so in Section 4.…”

“…Since B is positive semidefinite there exists a Cholesky like factorisation B = P T P with P ∈ R N ×N singular. Nour-Omid, Parlett, Ericsson and Jensen [10] point out to an equivalence result between B-orthogonal Arnoldi applied to A −1 B and standard Arnoldi applied to P A −1 P T . It appears that with appropriate initial vectors, the eigenvalues computed by both methods are equal and that one needs a purification step to find the eigenvectors of A −1 B in the B-orthogonal Arnoldi method.…”