We investigate algorithms for solving large sparse symmetric matrix eigenvalue problems resulting from finite element discretizations of steady state electromagnetic fields in accelerating cavities. The methods have been applied to the new design of the accelerating cavity for the PSI 590 MeV ring cyclotron. The solutions of this kind of eigenvalue problems can be polluted by so-called spurious...

A structure preserving Sort-Jacobi algorithm for computing eigenvalues or singular values is presented. The proposed method applies to an arbitrary semisimple Lie algebra on its (−1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra an...

We consider the quadratic eigenvalue problem λ2Ax+ λBx+Cx = 0. Suppose that u is an approximation to an eigenvector x (for instance, obtained by a subspace method) and that we want to determine an approximation to the corresponding eigenvalue λ. The usual approach is to impose the Galerkin condition r(θ, u) = (θ2A+ θB +C)u ⊥ u, from which it follows that θ must be one of the two solutions to th...

In this paper we introduce a method for designing efficient Jacobi-like algorithms for eigenvalue decomposition of a real normal matrix. The algorithms use only real arithmetic and achieve ultimate quadratic convergence. A theoretical analysis is conducted and some experimental results are presented.

Results of lattice analysis indicate that the static potential in SU(3) gauge theory is proportional to eigenvalue of quadratic Casimir operator for the corresponding representation with a good accuracy. We discuss the pattern of deviation from this Casimir scaling in gluodynamics in terms of correlators of path-ordered gauge-invariant operators defined on the worldsheet of the confining string.

We present an iterative method, based on a block generalization of the Rayleigh Quotient Iteration method, to search for the p lowest eigenpairs of the generalized matrix eigenvalue problem Au = Bu. We prove its local quadratic convergence when B A is symmetric. The benefits of this method are the well-conditioned linear systems produced and the ability to treat multiple or nearly degenerate ei...

This paper studies existence and uniqueness results and interlacing properties of nonlinear modifications of small rank of symmetric eigenvalue problems. Approximation properties of the Rayleigh functional are used to design numerical methods the local convergence of which is quadratic or even cubic. Numerical examples demonstrate their efficiency.

In [2] Conca et al. stated two inclusion theorems for quadratic eigenvalue problems the proof of which are not complete. In this note we demonstrate by simple examples that the assertions as they stand are false. Taking advantage of an appropriate enumeration for eigenvalues of nonlinear eigenproblems we adjust the results.

We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic P (λ) ≡ λA1 + λA0 + A1, with A0, A1 ∈ Cn×n and A0 = A0. The perturbation of palindromic eigenvalues and eigenvectors, in terms of general matrix polynomials, palindromic linearizations, (semi-Schur) anti-triangular canonical forms, differentiation and Sun’s implicit function approach, are discussed.

The multivariate eigenvalue problem (MEP) which originally arises from the canonical correlation analysis is an important generalization of the classical eigenvalue problem. Recently, the MEP also finds applications in many other areas and continues to receive interest. However, the existing algorithms for the MEP are the generalization of the power iteration for the classical eigenvalue proble...