The Infinite Bi-Lanczos Method for Nonlinear Eigenvalue Problems

Abstract

We propose a two-sided Lanczos method for the nonlinear eigenvalue problem (NEP). This two-sided approach provides approximations to both the right and left eigenvectors of the eigenvalues of interest. The method implicitly works with matrices and vectors with infinite size, but because particular (starting) vectors are used, all computations can be carried out efficiently with finite matrices and vectors. We specifically introduce a new way to represent infinite vectors that span the subspace corresponding to the conjugate transpose operation for approximating the left eigenvectors. Furthermore, we show that also in this infinite-dimensional interpretation the short recurrences inherent to the Lanczos procedure offer an efficient algorithm regarding both the computational cost and the storage.Key words. Nonlinear eigenvalue problem, two-sided Lanczos method infinite bi-Lanczos method, infinite two-sided Lanczos method.AMS subject classifications. 65F15, 65H17, 65F50 1. Introduction. Let M : C → C n×n be a matrix depending on a parameter with elements that are analytic in ρD, where ρ > 0 a constant, D is the open unit disk andD its closure. We present a new method for the nonlinear eigenvalue problem:We are interested in both the left and the right eigenvectors of the problem. The simultaneous approximation of both left and right eigenvectors is useful, e.g., in the estimation of the eigenvalue condition number and the vectors can be used as initial values for locally convergent two-sided iterative methods, e.g., those described in [21]. The NEP (1.1) has received considerable attention in the numerical linear algebra community, and there are several competitive numerical methods. There are for instance, so-called single vector methods such as Newton type methods [21,22,8], which often can be improved with subspace acceleration, see [25], and Jacobi-Davidson methods [6]. These have been extended in a block sense [14]. There are methods specialized for symmetric problems that have an (easily computed) Rayleigh functional [23]. There is also a recent class of methods which can be interpreted as either dynamically extending an approximation or carrying out an infinite-dimensional algorithm, see for instance [13,3,10] and references therein. For recent developments see the summary papers [20,17,26] and the benchmark collection [5].We propose a new method that is based on the two-sided Lanczos method for non-Hermitian problems. An intuitive derivation of the main idea of this paper is the following. Suppose (λ, x) is a solution to (1.1a). By adding trivial identities we have * Version

3 Case studies of model order reduction for acoustics and vibrations

“…In order to ensure that the resulting ROM equations of motion comply with this Lagrangian structure, a direct substitution in the Lagrangian can be performed. For a constant reduced-order basis, one can substitute for the reduced degrees of freedom q: 22) which results in the Lagrangian expressed as a function of the reduced-order degrees of freedom q:…”

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Compact Two-Sided Krylov Methods for Nonlinear Eigenvalue Problems

“…The boundary conditions are v x (0, t) = v x (π/2, t) = 0. After discretization with central differences with n = 5000, as in [12], the associated second order delay eigenvalue problem becomes, where C 0 , C 1 ∈ C 5000×5000 . We use a Taylor polynomial around λ = 0 for approximating the exponential as in [18]:…”

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Broyden's Method for Nonlinear Eigenproblems

“…where the matrices are the same as those in [13]. The simulations of this section are intended to illustrate method properties, and we do not claim that this method is the best method for this type of problem.…”