Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem

Abstract

Abstract.A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A 1 is presented. From the eigenpairs of some real matrix A 0 , the eigenpairs of A(t) ≡ (1 − t)A 0 + tA 1 are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A 1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix:• bifurcation, • ill conditioning due to nonorthogonal eigenvectors, • jumping of eigenpaths. These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented.

An adaptive homotopy approach for non-selfadjoint eigenvalue problems

“…Homotopy methods in the context of nonsymmetric matrix eigenvalue problems are discussed in [33][34][35]37]. In [36] an extension to the eigenvalue problem for selfadjoint partial differential operators is presented.…”

“…For simple and well-separated eigenvalues that do not bifurcate during the homotopy process, as considered here, it is known [26] that every eigenvalue λ (t) of the generalized eigenvalue problems (3.2) and (3.3) is an analytic function in t. Choosing appropriate homotopy stepsizes, the eigenvalues can therefore be continued on an analytic path towards the eigenvalues of (A + C , B ) [34,37]. The evolution of an eigenpair as a function of t is called an eigenpath and is denoted by (λ (t), u (t)) and (λ (t), u (t)), respectively.…”

“…To design an adaptive algorithm for the convection-diffusion problem a homotopy method is used. Homotopy methods are well established for nonsymmetric matrix eigenvalue problems [33][34][35]37]. Here, the homotopy approach is used not only on the matrix level but on the level of the differential operator as well.…”

Critical Eigenvalues Tracing for Power System Analysis via Continuation of Invariant Subspaces and Projected Arnoldi Method

“…Then , where is the m-dimension identity matrix, could be a choice of the additional conditions. However, the usual normalization may not be differentiable [21]. Thus, it is generally preferred to consider a linearized constraint (3) where is a fixed matrix with rank .…”

Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems

“…The numerical continuation of one eigenvalue can be considered a classical topic in numerical analysis; see, e.g., [23,26]. In contrast, the numerical continuation of several eigenvalues has not been investigated to a large extent for nonlinear eigenvalue problems, with the exception of the work [4,9] on polynomial eigenvalue problems.…”