8. Generalized Non-Hermitian Eigenvalue Problems
Gérard L. G. Sleijpen, Henk A. van der Vorst, Axel Ruhe, Zhaojun Bai, Thomas Ericsson, T. Kowalski, Bo Kågström, Ren-Cang Li Ren-Cang Li · Society for Industrial and Applied Mathematics eBooks · 2000
8.1 Introduction This chapter is devoted to the numerical solution of the (right) generalized non-Hermitian eigenvalue problem (GNHEP) Ax=λBx, 8.1 where A and B are general n by n matrices. Occasionally, one may also seek the solution of the left GNHEP y* A=λ y* B. 8.2 The algebraic and analytic theory of the generalized eigenvalue problem is considerably more complicated than the corresponding theory of the standard eigenvalue problem. This has been discussed in §2.6. This chapter covers a variety of numerical techniques for treating different sorts of the generalized eigenvalue problems. In §8.2, a brief sketch of what is called the QZ algorithm for the generalized eigenvalue problem is presented. This is currently the most powerful method for dense problems and it is an analog of the QR algorithm. However, the QZ method is only suitable for small- to moderate-sized problems because of the requirements of floating point operations and memory locations. A common approach for a large scale generalized eigenvalue problem is to reduce the problem (8.1) or (8.2) to a standard eigenvalue problem and then apply an iterative method, as described in Chapter 7. We refer to this technique as reduction to standard form. This reduction to standard form requires, for each iteration, the solution of a linear system with A or B or a combination of A and B. This will be discussed in more detail in §8.3. The Jacobi—Davidson method, presented in §8.4, avoids the transformation of to a standard eigenproblem. It does not need the exact solution of a linear system, only that a preconditioned iteration be available for a system with the matrix A − σ B, which offers possibilities to improve overall efficiency. The rational Krylov algorithm, introduced in §8.5, is a further development of shifted and inverted Arnoldi.