Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1981"

Computational methods of linear algebra

Abstract: The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

On multiple positive solutions of nonlinear elliptic eigenvalue problems

Abstract: (1981). On multiple positive solutions of nonlinear elliptic eigenvalue problems. Communications in Partial Differential Equations: Vol. 6, No. 8, pp. 951-961.

Balancing the Generalized Eigenvalue Problem

Abstract: An algorithm is presented for balancing the A and B matrices prior to computing the eigensystem of the generalized eigenvalue problem $Ax = \lambda Bx$. The three-step algorithm is specifically designed to precede the $QZ$-type algorithms, but improved performance is expected from most eigensystem solvers. Permutations and two-sided diagonal transformations are applied to A and B to produce matrices with certain desirable properties. Test cases are presented to illustrate the improved accuracy of the computed eigenvalues.

Corner transfer matrices

Abstract: For two-dimensional lattice models with interactions only between nearest (and diagonally nearest) neighbour spins, a well-known concept is the row-to-row transfer matrix. Less well-known is the “corner” transfer matrix (CTM). This has some very useful properties. If it is normalized so that its largest eigenvalue is unity, and the eigenvalues are arranged in numerically decreasing order, then each eigenvalue tends to a limit as the lattice becomes large. For those models which have been solved exactly (notably the Ising, eight-vertex and hard hexagon models), this limiting eigenvalue distribution is very simple, being basically that of a direct product of two-by-two matrices. From it the order parameter can easily be obtained. For all models one can write down formally exact matrix relations for the CTM, but the matrices are of infinite size. If one uses a representation in which the CTM is diagonal, and then truncates these relations to finite size, then one obtains a quite accurate approximation. The larger the size the greater the accuracy. I.G. Enting and I have thereby obtained comparatively long series expansions for the Ising model in a field, and for the hard squares model.

Entropy numbers of diagonal operators with an application to eigenvalue problems

Abstract: 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.

A Newton iteration process for inverse eigenvalue problems

Abstract: 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.

Solving Sparse Symmetric Generalized Eigenvalue Problems without Factorization

Abstract: In this paper we discuss an iterative technique for finding the algebraically smallest (or largest) eigenvalue of the generalized eigenvalue problem $A - \lambda M$, where A and M are real, symmetric, and M is positive definite. We assume that A and M are such that it is undesirable to factor the matrix $A - \sigma M$ for any value of $\sigma $. We prove that the algorithm is globally convergent, and that convergence is asymptotically quadratic. Finally, we discuss the modifications required in the algorithm to make it computationally feasible.

Collocation approximation to eigenvalues of an ordinary differential equation: numerical illustrations

Abstract: We display the numerical results associated with the collocation of three eigenvalue problems using from one to four Gauss points per partition interval in order to document the sharpness of the error bounds we have previously obtained. The ordinary differential operators involved are real with constant coefficients; two of the problems have an eigenvalue whose ascent exceeds one. We propose an explanation for the observed manner in which a set of simple approximate eigenvalues can approach a single multiple eigenvalue.

On generalised and quadratic eigenvalue problems

Abstract: The generalised eigenvalue problem Tx = λVx and the quadratic eigenvalue problem Ay = ∊By + ∊2Cy are considered for self-adjoint linear operators on Hilbert spaces. Conditions are given for the reduction of these problems to the classical case Su = vu where S is compact and self-adjoint

Deferred correction for the integral equation eigenvalue problem

Abstract: This paper considers the improvement of approximate eigenvalues and eigenfunctions of integral equations using the method of deferred correction. A convergence theorem is proved and a numerical example illustrating the theory is given.

Variational Methods for One and Several Parameter Non-Linear Eigenvalue Problems

Abstract: Wn(\\)xn = 0 ^ xn (1.1) Wn(\\) = Tn+ Vn{\\) n = 1,2, . . . , * , where X £ R while Tn and Vn(\\) are self-adjoint linear operators on a Hilbert space i?\"w. If X = (Xi, . . . , Xk) € R fc and x = (xi, . . . , xk) Ç 0 L i #n satisfy (1.1) then we call X an eigenvalue, x an eigenvector and (X, x) an eigenpair. While our main thrust is towards the general case of several parameters Xw, the method ultimately involves reduction to a sequence of one parameter problems. Our chief contributions are (i) to generalise the conditions under which this reduction is possible, and (ii) to develop methods for the one parameter problem particularly suited to the multiparameter application. For example, we give rather general results on the magnitude and direction of the movement of non-linear eigenvalues under perturbation. Until Section 8, we restrict ourselves to the case where the Tn have compact resolvents and the Vn{\\) are bounded, for each n and X. This includes, for example, second order linear ordinary differential equations (de) on intervals [an, bn] with self-adjoint boundary conditions. The case where the Vn(\\) are linear in X has received attention from many authors, starting explicitly for k = 2 with Klein's investigation of Lamé's equation [11]. Not long afterwards, Bôcher [6] showed, under certain restrictions on the de, that for any non-negative integer multi-index i = (ii, . . . , ik) there exists a unique eigenpair (X, x) so that xn has in zeros in [ani bn]. There are also various results on monotonie and continuous dependence of both the eigenvalues and the zeros (or, more generally, of the focal points) of the eigenvector. Such results go back to Sturm for the case k = 1 and may be found for the general multiparameter case in [4]. Non-linear one parameter problems have an extensive literature; see for example [17] and the references there. Their generalisation to several

Irregular multipoint eigenvalue problems

Abstract: We prove asymptotic estimates for the Green's function of irregular multipoint eigenvalue problems, these estimates are fundamental for the expansion of functions into a series of eigenfunctions of irregular eigenvalue problems.

On error bounds in strong approximations for eigenvalue problems

Abstract: Abstract Some corrections of error bounds obtained by Chatelin and Lemordant for the first three terms of the asymptotic case of a strong approximation are given. The error bounds for the approximations of order 2 in the Galerkin method are compared with the Rayleigh quotients constructed with the eigenvectors in the Sloan method. A numerical experiment is also carried out.

Transformation of a class of nonselfadjoint eigenvalue problems

Abstract: A class of non-self-adjoint boundary value problems possessing countably many real eigenvalues can be made self-adjoint by means of a nonsingular transformation. A set of criteria for such problems to be self-adjoint is derived.

Solution of the inverse eigenvalue problem of a vibrating continuum with the method of intermediate operators

Abstract: The problem of searching into the unknown “shape” of a continuum to have prescribed eigenvalues will be identified with an eigenvalue problem belonging to an intermediate problem in the sense of Weinstein, Bazley and Fox. Those eigenvalues of this latter are prescribed, which are defined by a linear algebraic eigenvalue problem. A special base operator enables the prescribed m eigenvalues to be the first m ones of the structure to be designed. Unknown parameters of the shape are looked for in an appropriate family of functions. A nonlinear algebraic system of equations is obtained for the unknown coefficients. Method is illustrated with a straight rod performing plane flexural vibration. A numerical example is given.

Eigenvalue problem for an infinite tridiagonal matrix

Abstract: A method is developed for the calculation of the eigenvectors of an infinite tridiagonal matrix. Possible application of this method to study the problem of localization in a disordered linear chain is also discussed.

Convergence and errors for spectral components of the projective-finite element approximation to variational eigenvalue problems

Abstract: We analyze the Galerkin-finite element approximation to the variational eigenvalue problem a(u,v) = λb(u,v) on a product of reflexive Banach spaces B1 × B2 We obtain eigenvalue convergence results for infinitely generated spectra in the non-Hermitian case and eigensolution convergence and errors in the Hermitian case.