Pseudoperturbation method for computation of E. Schmidt eigenvalues

In two problems with approximately given n -multiple generalized E. Schmidt eigenvalue with relevant Jordan chains pseudoperturbation method is applied for their sharpening. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Accurate computation of higher Sturm-Liouville eigenvalues

Summary The computation of eigenvalues of regular Sturm-Liouville problems is considered. It is shown that a simple step-dependent linear multistep method can be used to reduce the error of the orderk ⁴ h ² of the centered finite difference estimate of thek-th eigenvalue with uniform step lengthh, to an error of orderkh ². By an appropriate minimization of the local error term of the method one can obtain even more accurate results. A comparison of the simple correction techniques of Paine, de Hoog and Anderssen and of Andrew and Paine is given. Numerical examples demonstrate the usefulness of this correction even for low values ofk.Research Director of the National Fund for Scientific Research (N.F.W.O. Belgium)     

Fliess series approach to the computation of the eigenvalues of fourth-order Sturm-Liouville problems

This paper is an extension to our work on the computation of the eigenvalues of regular fourth-order Sturm-Liouville problems using Fliess series. The purpose here is twofold. First, we consider general self-adjoint separated boundary conditions. Second, we modify the algorithm presented in an earlier paper to ease considerably the computation of the iterated integrals involved.     

On the fast Lanczos method for computation of eigenvalues of Hankel matrices using multiprecision arithmetics

The use of the fast Fourier transform (FFT) accelerates Lanczos tridiagonalisation method for Hankel and Toeplitz matrices by reducing the complexity of matrix–vector multiplication. In multiprecision arithmetics, the FFT has overheads that make it less competitive compared with alternative methods when the accuracy is over 10000 decimal places. We studied two alternative Hankel matrix–vector multiplication methods based on multiprecision number decomposition and recursive Karatsuba‐like multiplication, respectively. The first method was uncompetitive because of huge precision losses, while the second turned out to be five to 14 times faster than FFT in the ranges of matrix sizes up to n = 8192 and working precision of b = 32768 bits we were interested in. We successfully applied our approach to eigenvalues calculations to studies of spectra of matrices that arise in research on Riemann zeta function. The recursive matrix–vector multiplication significantly outperformed both the FFT and the traditional multiplication in these studies. Copyright © 2016 John Wiley & Sons, Ltd.     

A difference-analytical method for the computation of eigenvalues of the fourth-order equations with separated boundary conditions

We propose a method for the computation of eigenvalues with odd multiplicities for spectral boundary-value problems of the 4th order with separated boundary conditions. In this method approximate eigenvalues appear as zeros of a certain function f(λ) which admits an explicit representation.     

Reliable computation of eigenvalues of the magnetostatic integral operator

Some variational problems in magnetostatics can be reformulated as eigenvalue problems for vector surface integral operators in appropriate function spaces, e.g., the magnetostatic integral operator is of considerable interest in the theory of permanent magnetization of compact bodies. In the case that the underlying surface is either a sphere, a spheroid, or a triaxial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lamé functions and surface ellipsoidal harmonics. Since there is an apparent lack in literature we provide an new effective scheme for the reliable computation of these functions and of the corresponding eigenvalues of the magnetostatic operator.     

Error estimates for the computation of eigenvalues of self-adjoint operators

A convergence theorem is given for the numerical computation of eigenvalues of self-adjoint operators. The computed eigenvectors are used to construct trial solutions which can be used to obtain easily computable and accurate error estimates.     

Discontinuous Galerkin computation of the Maxwell eigenvalues on simplicial meshes

This paper is concerned with the discontinuous Galerkin approximation of the Maxwell eigenproblem. After reviewing the theory developed in [A. Buffa, I. Perugia, Discontinuous Galerkin approximation of the Maxwell eigenproblem, Technical Report 24-PV, IMATI-CNR, Pavia, Italy, 2005 〈$〈$http://www.imati.cnr.it/∼$\sim$annalisa/PS/maxwell.pdf〉$〉$], we present a set of numerical experiments which both validate the theory, and provide further insight regarding the practical performance of discontinuous Galerkin methods, particularly in the case when non-conforming meshes, characterized by the presence of hanging nodes, are employed.     

On the computation of all eigenvalues for the eigenvalue complementarity problem

In this paper, a parametric algorithm is introduced for computing all eigenvalues for two Eigenvalue Complementarity Problems discussed in the literature. The algorithm searches a finite number of nested intervals \([\bar{l}, \bar{u}]\) in such a way that, in each iteration, either an eigenvalue is computed in \([\bar{l}, \bar{u}]\) or a certificate of nonexistence of an eigenvalue in \([\bar{l}, \bar{u}]\) is provided. A hybrid method that combines an enumerative method [1] and a semi-smooth algorithm [2] is discussed for dealing with the Eigenvalue Complementarity Problem over an interval \([\bar{l}, \bar{u}]\) . Computational experience is presented to illustrate the efficacy and efficiency of the proposed techniques.     

The computation of negative eigenvalues of singular Sturm-Liouville problems

In this work we shall develop a new interpolation method forthe computation of eigenvalues of singular Sturm–Liouvilleproblems. Basic properties of the Jost solutions are used todetermine the growth of the boundary function and an appropriateinterpolation basis. This leads to a good approximation of thenegative eigenvalues.     

On computation of real eigenvalues of matrices via the Adomian decomposition

The problem of matrix eigenvalues is encountered in various fields of engineering endeavor. In this paper, a new approach based on the Adomian decomposition method and the Faddeev-Leverrier’s algorithm is presented for finding real eigenvalues of any desired real matrices. The method features accuracy and simplicity. In contrast to many previous techniques which merely afford one specific eigenvalue of a matrix, the method has the potential to provide all real eigenvalues. Also, the method does not require any initial guesses in its starting point unlike most of iterative techniques. For the sake of illustration, several numerical examples are included.     

The extrapolated first order method for solving systems with complex eigenvalues

An extrapolated form of the basic first order stationary iterative method for solving linear systems when the associated iteration matrix possesses complex eigenvalues, is investigated. Sufficient (and necessary) conditions are given such that convergence is assured. An analytic determination of good (and sometimes optimum) values of the involved real parameter is presented in terms of certain bounds on the eigenvalues of the iteration matrix. The usefulness of the developed theory is shown through a simple application to the conventional Jacobi method.     

A quadratically convergent Jacobi-like method for real matrices with complex eigenvalues

A real Jacobi-like algorithm for diagonalizing arbitrary real matrices with complex eigenvalues is described and its applicability discussed. Numerical results are given and compared with those of the well-known real and complex algorithm of Eberlein.     

Iterative computation of derivatives of repeated eigenvalues and the corresponding eigenvectors

Recently the authors proposed a simultaneous iteration algorithm for the computation of the partial derivatives of repeated eigenvalues and the corresponding eigenvectors of matrices depending on several real variables. This paper analyses the properties of that algorithm and extends it in several ways. The previous requirement that the repeated eigenvalue be dominant is relaxed, and the new generalized algorithm given here allows the simultaneous treatment of simple and repeated eigenvalues. Methods for accelerating convergence are examined. Numerical results support our theoretical analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

Graeffe's method for eigenvalues

Let an entire functionF(z) of finite genus have infinitely many zeros which are all positive, and take real values for realz. Then it is shown how to give two-sided bounds for all the zeros ofF in terms of the coefficients of the power series ofF, in fact in terms of the coefficients obtained byGraeffe's algorithm applied toF. A simple numerical illustration is given for a Bessel function.     

On complex eigenvalues ofM andP matrices

Summary Inequalities are obtained for the complex eigenvalues of anM matrix or aP matrix which depend only on the order of the matrix.This research was supported in part by the National Science Foundation under Grant GP-20555.     

Estimates on the eigenvalues of complex nonlocal Sturm-Liouville problems

The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems. The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.