A block Newton method for nonlinear eigenvalue problems

We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viability.     

A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems

The inverse-free preconditioned Krylov subspace method of Golub and Ye [G.H. Golub, Q. Ye, An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems, SIAM J. Sci. Comp. 24 (2002) 312-334] is an efficient algorithm for computing a few extreme eigenvalues of the symmetric generalized eigenvalue problem. In this paper, we first present an analysis of the preconditioning strategy based on incomplete factorizations. We then extend the method by developing a block generalization for computing multiple or severely clustered eigenvalues and develop a robust black-box implementation. Numerical examples are given to illustrate the analysis and the efficiency of the block algorithm.     

A collocation method for eigenvalue problems

Approximations to the eigenvalues ofmth-order linear eigenvalue problems are determined by using a collocation method with piecewise-polynomial functions of degreem+d possessingm continuous derivatives as basis functions. It is shown that for a general class of problems the error in these approximations is at leastO([h(II)] ^d+1) whereh(II) is the maximal subinterval length. The question of stability of the discretized problems is also considered. It is not assumed that the eigenvalue problems are in any sense self-adjoint.     

A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems

For generalized eigenvalue problems, we consider computing all eigenvalues located in a certain region and their corresponding eigenvectors. Recently, contour integral spectral projection methods have been proposed for solving such problems. In this study, from the analysis of the relationship between the contour integral spectral projection and the Krylov subspace, we conclude that the Rayleigh–Ritz-type of the contour integral spectral projection method is mathematically equivalent to the Arnoldi method with the projected vectors obtained from the contour integration. By this Arnoldi-based interpretation, we then propose a block Arnoldi-type contour integral spectral projection method for solving the eigenvalue problem.     

A Ulm-like method for inverse eigenvalue problems

We propose a Ulm-like method for solving inverse eigenvalue problems, which avoids solving approximate Jacobian equations comparing with other known methods. A convergence analysis of this method is provided and the R-quadratic convergence property is proved under the assumption of the distinction of given eigenvalues. Numerical experiments as well as the comparison with the inexact Newton-like method are given in the last section.     

A method for solving stochastic eigenvalue problems

We present a novel approach for calculating stochastic eigenvalues of differential and integral equations as well as for random matrices. Five examples based on very different types of problem have been analysed and detailed numerical results obtained. It would seem that the method has considerable promise. The essence of the method is to replace the stochastic eigenvalue problem λ(ξ)?(ξ)=A(ξ)?(ξ), where ξ is a set of random variables, by the introduction of an auxiliary equation in which . This changes the problem from an eigenvalue one to an initial value problem in the new pseudo-time variable t. The new linear time-dependent equation may then be solved by a polynomial chaos expansion (PCE) and the stochastic eigenvalue and its moments recovered by a limiting process. This technique has the advantage of avoiding the non-linear terms in the conventional method of stochastic eigenvalue calculation by PCE, but it does introduce an additional, ‘pseudo-time’, independent variable t. The paper illustrates the viability of this approach by application to several examples based on realistic problems.     

Inverse eigenvalue problems for linear complementarity systems

Traditionally an inverse eigenvalue problem is about reconstructing a matrix from a given spectral data. In this work we study the set of real matrices A of order n such that the linear complementarity system

     

A new superconvergent collocation method for eigenvalue problems

Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree , we show that the proposed method exhibits an error of the order of for eigenvalue approximation and of the order of for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order can be obtained. This improves upon the order for eigenvalue approximation in the collocation/iterated collocation method and the orders and for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.

     

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.     

Kron's method for symmetric eigenvalue problems

Kron's method has been used successfully by engineers for over 30 years to find eigenvalues of large symmetric matrices. These matrices arise from domain decomposition of nonoverlapping domains of self-adjoint partial differential operators. This paper discusses some theoretical aspects of Kron's method. Specifically, the poles and zeroes of eigenvalues of the Kron matrix and their relationships with the eigenvalues of the original matrix are given.     

Arnoldi-Riccati method for large eigenvalue problems

This paper describes a method for computing the dominant/right-most eigenvalues of large matrices. The method consists of refining the approximate eigenelements of a large matrix obtained by classical methods such as Arnoldi. The refinement process leads to a Riccati equation to be solved approximately. Numerical evidence of the improvements achieved by using the proposed approach is reported.Part of this work was done while visiting CERFACS (France) in Sept–Oct 1995 under contract HCM # ERBCHRXCT930420.     

A projection based multiscale optimization method for eigenvalue problems

We present a projection based multiscale optimization method for eigenvalue problems. In multiscale optimization, optimization steps using approximations at a coarse scale alternate with corrections by occasional calculations at a finer scale. We study an example in the context of electronic structure optimization. Theoretical analysis and numerical experiments provide estimates of the expected efficiency and guidelines for parameter selection.     

A new method for solving Pareto eigenvalue complementarity problems

In this paper, we introduce a new method, called the Lattice Projection Method (LPM), for solving eigenvalue complementarity problems. The original problem is reformulated to find the roots of a nonsmooth function. A semismooth Newton type method is then applied to approximate the eigenvalues and eigenvectors of the complementarity problems. The LPM is compared to SNM_min and SNM_FB, two methods widely discussed in the literature for solving nonlinear complementarity problems, by using the performance profiles as a comparing tool (Dolan, Moré in Math. Program. 91:201–213, 2002). The performance measures, used to analyze the three solvers on a set of matrices mostly taken from the Matrix Market (Boisvert et al. in The quality of numerical software: assessment and enhancement, pp. 125–137, 1997), are computing time, number of iterations, number of failures and maximum number of solutions found by each solver. The numerical experiments highlight the efficiency of the LPM and show that it is a promising method for solving eigenvalue complementarity problems. Finally, Pareto bi-eigenvalue complementarity problems were solved numerically as an application to confirm the efficiency of our method.     

A mixed finite element method for fourth order eigenvalue problems

A mixed finite element method for approximating eigenpairs of IV order elliptic eigenvalue problems with Dirichlet boundary conditions has been given. The method can be applied to the vibration analysis of anisotropic/orthotropic/isotropic/biharmonic plates. Computer implementation procedures for this mixed method are given along with the results of numerical experiments.     

A linear eigenvalue algorithm for the nonlinear eigenvalue problem

The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted ${\mathcal {B}}$ . We consider the Arnoldi method for the operator ${\mathcal {B}}$ and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator ${\mathcal {B}}$ in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples.     

The projection Kantorovich method for eigenvalue problems

A modified projection method for eigenvalues and eigenvectors of a compact operator T on a Banach space is defined and analyzed. The method is derived from the Kantorovich regularization for second-kind equations involving the operator T. It is shown that when T is a positive self-adjoint operator on a Hilbert space and the projections are orthogonal, the modified method always gives eigenvalue approximations which are at least as accurate as those obtained from the projection method. For self-adjoint operators, the required computation is essentially the same for both methods. Numerical computations for two integral operators are presented. One has T positive self-adjoint, while in the other T is not self-adjoint. In both cases the eigenvalue approximations from the modified method are more accurate than those from the projection method.     

A sign-based linear method for horizontal linear complementarity problems

Numerical Algorithms - For the horizontal linear complementarity problem, we establish a linear method based on the sign patterns of the solution of the equivalent modulus equation under the...     

A mixed method of subspace iteration for Dirichlet eigenvalue problems

A full multigrid scheme was used in computing some eigenvalues of the Laplace eigenvalue problem with the Dirichlet boundary condition. We get a system of algebraic equations with an aid of finite difference method and apply subspace iteration method to the system to compute first some eigenvalues. The result shows that this is very effective in calculating some eigenvalues of this problem.     

Deflated block Krylov subspace methods for large scale eigenvalue problems

We discuss a class of deflated block Krylov subspace methods for solving large scale matrix eigenvalue problems. The efficiency of an Arnoldi-type method is examined in computing partial or closely clustered eigenvalues of large matrices. As an improvement, we also propose a refined variant of the Arnoldi-type method. Comparisons show that the refined variant can further improve the Arnoldi-type method and both methods exhibit very regular convergence behavior.