A method based on Rayleigh quotient gradient flow for extreme and interior eigenvalue problems

Recently, a continuous method has been proposed by Golub and Liao as an alternative way to solve the minimum and interior eigenvalue problems. According to their numerical results, their method seems promising. This article is an extension along this line. In this article, firstly, we convert an eigenvalue problem to an equivalent constrained optimization problem. Secondly, using the Karush-Kuhn-Tucker conditions of this equivalent optimization problem, we obtain a variant of the Rayleigh quotient gradient flow, which is formulated by a system of differential-algebraic equations. Thirdly, based on the Rayleigh quotient gradient flow, we give a practical numerical method for the minimum and interior eigenvalue problems. Finally, we also give some numerical experiments of our method, the Golub and Liao method, and EIGS (a Matlab implementation for computing eigenvalues using restarted Arnoldi’s method) for some typical eigenvalue problems. Our numerical experiments indicate that our method seems promising for most test problems.     

Estimation of the effect of numerical integration in finite element eigenvalue approximation

Summary Finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration. The effect of this numerical integration on the eigenvalue and eigenfunction error is estimated. Specifically, for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well-known estimates for approximations without quadrature, provided the quadrature rules have appropriate degrees of precision.The work of this author was partially supported by the National Science Foundation under Grant DMS-84-10324     

A Newton iteration process for inverse eigenvalue problems

Summary We suppose an inverse eigenvalue problem which includes the classical additive and multiplicative inverse eigenvalue problems as special cases. For the numerical solution of this problem we propose a Newton iteration process and compare it with a known method. Finally we apply it to a numerical example.     

Asymptotische Entwicklungen für Eigenwerte und Eigenvektoren bei der Approximation parameternichtlinearer Eigenwertaufgaben

Summary In order to apply extrapolation processes to the numerical solution of eigenvalue problems depending nonlinear on a parameter the existence of asymptotic expansions for eigenvalues and eigenvectors is studied. At the end of the paper some numerical examples are given.

     

On linearizations of the quadratic two-parameter eigenvalue problem

We present several transformations that can be used to solve the quadratic two-parameter eigenvalue problem (QMEP), by formulating an associated linear multiparameter eigenvalue problem. Two of these transformations are generalizations of the well-known linearization of the quadratic eigenvalue problem and linearize the QMEP as a singular two-parameter eigenvalue problem. The third replaces all nonlinear terms by new variables and adds new equations for their relations. The QMEP is thus transformed into a nonsingular five-parameter eigenvalue problem. The advantage of these transformations is that they enable one to solve the QMEP using existing numerical methods for multiparameter eigenvalue problems. We also consider several special cases of the QMEP, where some matrix coefficients are zero     

A finite element lumped mass scheme for solving eigenvalue problems of circular arches

Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.     

Structured eigenvalue condition numbers and linearizations for matrix polynomials

This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations. For all structures under consideration, we show that this cannot happen if the matrix polynomial is well scaled: there is always a structured linearization for which the structured eigenvalue condition number does not differ much. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.     

On the modification of an eigenvalue problem that preserves an eigenspace

Eigenvalue problems arise in many application areas ranging from computational fluid dynamics to information retrieval. In these fields we are often interested in only a few eigenvalues and corresponding eigenvectors of a sparse matrix. In this paper, we comment on the modifications of the eigenvalue problem that can simplify the computation of those eigenpairs. These transformations allow us to avoid difficulties associated with non-Hermitian eigenvalue problems, such as the lack of reliable non-Hermitian eigenvalue solvers, by mapping them into generalized Hermitian eigenvalue problems. Also, they allow us to expose and explore parallelism. They require knowledge of a selected eigenvalue and preserve its eigenspace. The positive definiteness of the Hermitian part is inherited by the matrices in the generalized Hermitian eigenvalue problem. The position of the selected eigenspace in the ordering of the eigenvalues is also preserved under certain conditions. The effect of using approximate eigenvalues in the transformation is analyzed and numerical experiments are presented.     

Numerical methods for finding multiple eigenvalues of matrices depending on parameters

Summary. Let be a square matrix dependent on parameters and , of which we choose as the eigenvalue parameter. Many computational problems are equivalent to finding a point such that has a multiple eigenvalue at . An incomplete decomposition of a matrix dependent on several parameters is proposed. Based on the developed theory two new algorithms are presented for computing multiple eigenvalues of with geometric multiplicity . A third algorithm is designed for the computation of multiple eigenvalues with geometric multiplicity but which also appears to have local quadratic convergence to semi-simple eigenvalues. Convergence analyses of these methods are given. Several numerical examples are presented which illustrate the behaviour and applications of our methods. Received December 19, 1994 / Revised version received January 18, 1996     

On the quadratic two-parameter eigenvalue problem and its linearization

We introduce the quadratic two-parameter eigenvalue problem and linearize it as a singular two-parameter eigenvalue problem. This, together with an example from model updating, shows the need for numerical methods for singular two-parameter eigenvalue problems and for a better understanding of such problems.There are various numerical methods for two-parameter eigenvalue problems, but only few for nonsingular ones. We present a method that can be applied to singular two-parameter eigenvalue problems including the linearization of the quadratic two-parameter eigenvalue problem. It is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils.     

Inverse unitary eigenproblems and related orthogonal functions

Summary. This paper explores the relationship between certain inverse unitary eigenvalue problems and orthogonal functions. In particular, the inverse eigenvalue problems for unitary Hessenberg matrices and for Schur parameter pencils are considered. The Szeg? recursion is known to be identical to the Arnoldi process and can be seen as an algorithm for solving an inverse unitary Hessenberg eigenvalue problem. Reformulation of this inverse unitary Hessenberg eigenvalue problem yields an inverse eigenvalue problem for Schur parameter pencils. It is shown that solving this inverse eigenvalue problem is equivalent to computing Laurent polynomials orthogonal on the unit circle. Efficient and reliable algorithms for solving the inverse unitary eigenvalue problems are given which require only O() arithmetic operations as compared with O() operations needed for algorithms that ignore the structure of the problem. Received April 3, 1995 / Revised version received August 29, 1996     

The combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems

Summary. The paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains are not subdomains of the original domain and when at the same time numerical integration is used for computing the involved bilinear forms. The considerations are restricted to piecewise linear approximations. The optimum rate of convergence for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used. In the case of a simple exact eigenvalue the optimum rate of convergence for approximate eigenfunctions in the -norm is proved while in the -norm an almost optimum rate of convergence (i.e. near to is achieved. In both cases a quadrature formula of first degree of precision is used. Quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to the convergence without the rate of convergence is proved. In the case of a multiple exact eigenvalue the approximate eigenfunctions are compard (in contrast to standard considerations) with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter . Received September 18, 1993 / Revised version received September 26, 1994     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

Numerical solution of a secular equation

Summary. A method is proposed for the solution of a secular equation, arising in modified symmetric eigenvalue problems and in several other areas. This equation has singularities which make the application of standard root-finding methods difficult. In order to solve the equation, a class of transformations of variables is considered, which transform the equation into one for which Newton's method converges from any point in a certain given interval. In addition, the form of the transformed equation suggests a convergence accelerating modification of Newton's method. The same ideas are applied to the secant method and numerical results are presented. Received July 1, 1994     

NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

In this article, we describe on a state of the art of validated numerical computations for solutions of differential equations. A brief overview of the main techniques for self-validating numerics for initial and boundary value problems in ordinary and partial differential equations including eigenvalue problems will be presented. A fairly detailed introductions are given for the author's own method related to second-order elliptic boundary value problems. Many references which seem to be useful for readers are supplied at the end of the article.     

Finite element analysis of the vibration problem of a plate coupled with a fluid

Summary. We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas elements and a non conforming coupling is used on the interface. In order to prove that the method is locking free we consider a family of problems, one for each thickness , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when . We prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t. Finally we present numerical results confirming the good performance of the method. Received February 4, 1998 / Revised version received May 26, 1999 / Published online June 21, 2000     

Continuous methods for extreme and interior eigenvalue problems

In this paper, continuous methods are introduced to compute both the extreme and interior eigenvalues and their corresponding eigenvectors for real symmetric matrices. The main idea is to convert the extreme and interior eigenvalue problems into some optimization problems. Then a continuous method which includes both a merit function and an ordinary differential equation (ODE) is introduced for each resulting optimization problem. The convergence of each ODE solution is proved for any starting point. The limit of each ODE solution for any starting point is fully studied. Both the extreme and the interior eigenvalues and their corresponding eigenvectors can be easily obtained under a very mild condition. Promising numerical results are also presented.     

Preconditioned iterative methods for a class of nonlinear eigenvalue problems

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to nonlinear eigenvalue problems with very large sparse ill-conditioned matrices monotonically depending on the spectral parameter. To compute the smallest eigenvalue of large-scale matrix nonlinear eigenvalue problems, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors, and inner products of vectors. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.     

Exact bounds for the solution branches of nonlinear eigenvalue problems

Summary A method is constructed which yields a strip containing the full solution sets of nonlinear eigenvalue problems of the formu=Tu.The strip can be narrowed iteratively, and the method applies for both stable and unstable branches. Its high degree of accuracy is demonstrated by numerical examples. In particular, a lower bound is given for the critical value at which criticality is lost in the thermal ignition problem for the unit ball.     

Implementation of a modified Marder–Weitzner method for solving nonlinear eigenvalue problems

Our goal is to propose four versions of modified Marder–Weitzner methods and to present the implementation of the new-type methods with incremental unknowns for solving nonlinear eigenvalue problems. By combining with compact schemes and modified Marder–Weitzner methods, six schemes well suited for the calculation of unstable solutions are obtained. We illustrate the efficiency of the new algorithms by using numerical computations and by comparing them with existing methods for some two-dimensional problems.