A Generalised Eigenvalue Problem and The Lanczos Algorithm

Implicitly restarted Arnoldi with purification for the shift-invert transformation

“…Let n = 2, m = 1 (so N = 3), denote the columns of I 3 by e i , i = 1, 2, 3, and let and clearly (β −1 , e 2 ) is an eigenpair of S, 0 is a double algebraic, simple geometric eigenvalue, e 3 is the corresponding eigenvector and e 1 is the generalised eigenvector or principal eigenvector of grade 2. This behaviour is generic in N × N problems with the block structure of (2) as was shown by Malkus [9], who considered the Weierstrass-Kronecker canonical form of (2), and Ericsson [5], who considered the Jordan form for the shift-invert transformation of a variety of generalised eigenvalue problems. (Incidently, both authors restrict attention to problems with symmetric A but several of their results, at least to do with Jordan structure, extend to the case when A is nonsymmetric).…”

“…(Incidently, both authors restrict attention to problems with symmetric A but several of their results, at least to do with Jordan structure, extend to the case when A is nonsymmetric). To summarise the important results on S, where A and B have the block structure of (2), we have the following theorem, which can be readily deduced from Theorem 2.7 in Ericsson [5] : Theorem 1. S defined by (2) and (3) has n − m nonzero eigenvalues, a zero eigenvalue of algebraic multiplicity 2m and geometric multiplicity m. The order of the Jordan blocks corresponding to the defective eigenvalue 0 is two.…”

“…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.…”

“…Here M is the mass matrix of the velocity elements and K is nonsymmetric because of the linearisation of the convection term in the NavierStokes equations. As is well known, see Malkus [9], Ericsson [5] and Cliffe, Garratt, and Spence [3], (2) can have infinite eigenvalues, corresponding to eigenvectors of the form (0 T , p T ) T . These have no physical relevance and in applications one would only be concerned about the calculation of a small number of 'stability determining' finite eigenvalues.…”