In this paper, we present a normwise perturbation theory for the regular generalized eigenproblem Ax = Bx, when is a simple and nite eigenvalue, which departs from the classical analysis with the chordal norm 9]. A backward error and a condition number are derived for a choice of exible measure to represent independent perturbations in the matrices A and B. The concept of optimal backward error...

We present a parallel implementation of the Davidson method for the numerical solution of large-scale, sparse, generalized eigenvalue problems. The implementation is done in the context of SLEPc, the Scalable Library for Eigenvalue Problem Computations. In this work, we focus on the Hermitian version of the method, with several optimizations. We compare the developed solver with other available...

In this paper, we propose the palindromic doubling algorithm (PDA) for the palindromic generalized eigenvalue problem (PGEP) A∗x = λAx. We establish a complete convergence theory of the PDA for PGEPs without unimodular eigenvalues, or with unimodular eigenvalues of partial multiplicities two (one or two for eigenvalue 1). Some important applications from the vibration analysis and the optimal c...

We investigate eigensolvers for computing a few of the smallest eigenvalues of a generalized eigenvalue problem resulting from the finite element discretization of the time independent Maxwell equation. Various multilevel preconditioners are employed to improve the convergence and memory consumption of the JacobiDavidson algorithm and of the locally optimal block preconditioned conjugate gradie...

We consider a problem of simultaneous reduction of a sequence of matrices by means of orthogonal transformations. We show that such reduction can be performed by a series of the deflation steps. At each deflation step a simultaneous eigenvalue problem, which is a direct generalization of the generalized eigenvalue problem, is solved. A fast variant of Gauss-Newton algorithm for its solution was...

Gradient iterations for the Rayleigh quotient are simple and robust solvers to determine a few of the smallest eigenvalues together with the associated eigenvectors of (generalized) matrix eigenvalue problems for symmetric matrices. Sharp convergence estimates for the Ritz values and Ritz vectors are derived for various steepest descent/ascent gradient iterations. The analysis shows that poores...

This paper presents a new method for designing two channel biorthogonal IIR lter banks, which satisfy both the perfect reconstruction and causal stable conditions. The proposed method is based on the formulation of a generalized eigenvalue problem by using Remez multiple exchange algorithm. Therefore, the lter coe cients can be computed by solving the eigenvalue problem, and the optimal solutio...

The paper is about a generalization of a classical eigenvalue-decomposition method originally developed for errors–in-variables linear system identification to handle an important class of nonlinear problems. A number of examples are presented to call the attention to the most critical part of the procedure turning the identification problem to a generalized eigenvalue-eigenvector calculation p...

The blind separation of multiple co-channel binary digital signals using an antenna array involves finding a factorization of a data matrix X into X AS, where all entries of S are 1 or −1. It is shown that this problem can be solved exactly and non-iteratively, via a certain generalized eigenvalue decomposition. As indicated by simulations, the algorithm is robust in the presence of noise. An i...

For an eigenvalue λ0 of a Hermitian matrix A, the formula Thompson and McEnteggert gives explicit expression adjugate λ0I−A, Adj(λ0I−A), in terms eigenvectors A for all its eigenvalues. In this paper Thompson-McEnteggert's is generalized to include any with entries arbitrary field. addition, nonsingular elementary divisors Adj(A) provided those A. Finally, generalization eigenvalue-eigenvector ...