Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1980"
TL;DR: It is shown that for each shift several eigenvectors will converge after very few steps of the Lanczos algorithm, and the most effective combination of shifts and Lanczos runs is determined for different sizes and sparsity properties of the matrices.
Abstract: A new algorithm is developed which computes a specified number of eigenvalues in any part of the spectrum of a generalized symmetric matrix eigenvalue problem. It uses a linear system routine (factorization and solution) as a tool for applying the Lanczos algorithm to a shifted and inverted problem. The algorithm determines a sequence of shifts and checks that all eigenvalues get computed in the intervals between them. It is shown that for each shift several eigenvectors will converge after very few steps of the Lanczos algorithm, and the most effective combination of shifts and Lanczos runs is determined for different sizes and sparsity properties of the matrices. For large problems the operation counts are about five times smaller than for traditional subspace iteration methods. Tests on a numerical example, arising from a finite element computation of a nuclear power piping system, are reported, and it is shown how the performance predicted bears out in a practical situation.
338 citations
TL;DR: In this paper, the authors considered linear and nonlinear eigenvalue problems with an indefinite weight function and showed that linear eigenvalues can be solved with an arbitrary weight function.
Abstract: (1980). On some linear and nonlinear eigenvalue problems with an indefinite weight function. Communications in Partial Differential Equations: Vol. 5, No. 10, pp. 999-1030.
269 citations
TL;DR: The existence of positive solutions of elliptic eigenvalue problems of the form (0.1) has been investigated extensively in recent years; see as discussed by the authors for a survey article.
130 citations
TL;DR: In this article, the authors extended the analysis of regular problems containing the eigenvalue parameter in the boundary conditions given by Walter (1973) and myself (1977) to singular problems which involve the eigvalue parameter linearly in a regular or a limit-circle boundary condition at the left endpoint.
Abstract: In this paper I extend the analysis of regular problems containing the eigenvalue parameter in the boundary conditions given by Walter (1973) and myself (1977) to singular problems which involve the eigenvalue parameter linearly in a regular or a limit-circle boundary condition at the left endpoint. The formulation of the limit-circle boundary conditions follows that given in another paper by the present author in 1977, and has the advantage that a λ-dependent boundary condition at a regular endpoint becomes a special case of a λ-dependent boundary condition at a limit-circle endpoint. The simplicity of the spectrum is also built into the formulation given, and the spectral function is shown to have bounded total variation over (−∞, ∞) which is known in terms of the parameters of the λ-dependent boundary condition independently of the limit-circle/limit-point classification at the right endpoint. The theory is applied to the constant coefficient equation in [0, ∞) and the Bessel equation of order zero in (0, ∞), explicit formulae for the spectral function being obtained in each case. Finally, the question is posed as to whether the classical Weyl theory for problems not involving λ in the boundary conditions can also be formulated so as to involve spectral functions having bounded total variation.
102 citations
TL;DR: An algorithm is described which, given an approximate simple eigenvalue and a corresponding approximate eigenvector, provides rigorous error bounds for improved versions of them, which may indeed correspond to non-linear elementary divisors.
Abstract: An algorithm is described which, given an approximate simple eigenvalue and a corresponding approximate eigenvector, provides rigorous error bounds for improved versions of them. No information is required on the rest of the eigenvalues, which may indeed correspond to non-linear elementary divisors. A second algorithm is described which gives more accurate improved versions than the first but provides only error estimates rather than rigorous bounds. Both algorithms extend immediately to the generalized eigenvalue problem.
43 citations
36 citations
TL;DR: In this paper, the authors characterize the J-selfadjoint operators generated by a symmetric ordinary differential expression on an open real interval ι and prove eigenfunction expansion results using Hilbertspace-techniques.
Abstract: In provided with a J-innerproduct we characterize the J-selfadjoint operators generated by a symmetric ordinary differential expression on an open real interval ι. For a subclass of these operators we prove eigenfunction expansion results using Hilbertspace-techniques.
34 citations
32 citations
23 citations
TL;DR: In this paper, a numerical method for the fast solution of discretized elliptic eigenvalue problems was described and the results of this technique applied to the biharmonic (plate) equation were reported.
Abstract: A recently published paper describes a numerical method for the fast solution of discretized elliptic eigenvalue problems. Here we report the results of this technique applied to the biharmonic (plate) equation. 46 eigenvalues (24 single and 11 double ones) are computed and the corresponding eigenfunctions are plotted.
18 citations
TL;DR: The Leverrier algorithm as modified by Faddeev gives the characteristic equation of a matrix A, its inverse, and the eigenvector corresponding to a simple eigenvalue λ of A.
TL;DR: In this paper, a method is constructed which yields a strip containing the full solution sets of nonlinear eigenvalue problems of the formu=?Tu.672======¯¯¯¯¯¯
Abstract: 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.
TL;DR: In this paper, a class of two-person nonzero sum games where the strategy choices are constrained in some form for each player is analyzed, and the equivalent nonlinear programs which must be solved for the Cournot-Nash equilibrium are shown.
TL;DR: In this article, it was shown that the squares of the eigenfunctions may be linearly dependent on nontrivial subintervals of [0, 1] where ϕein C is positive.
Abstract: We consider the eigenvalue problem u″ + λϕ = 0, u(0)=u(1) = 0, where ϕein C[0, 1] is positive. It is well known that the eigenfunctions corresponding to distinct eigenvalues are linearly independent. It is shown in this paper that the squares of the eigenfunctions may be linearly dependent on nontrivial subintervals of [0,1]. This result has relevance in the variational analysis of eigenvalue problems.
TL;DR: In this article, the eigenvalue problem for the differential equation describing scalar waves in a clad inhomogeneous planar waveguide is solved using an asymptotic formulation derived in the preceding paper.
Abstract: For pt.I see ibid. vol.18, p.3057 (1980). The eigenvalue problem for the differential equation describing scalar waves in a clad inhomogeneous planar waveguide is solved using an asymptotic formulation derived in the preceding paper. The results of this earlier paper are summarised, and a method proposed for the extraction of the eigenvalue from the implicit eigenvalue equation. Although in general these calculations are conveniently carried out by numerical methods, approximate closed-form expressions can be obtained for the eigenvalue and these are given for all the asymptotic regimes of interest.
TL;DR: A new method is proposed which transforms the eigenvalue with the smallest real part (poorest attenuation component) to the Eigenvalue of largest absolute value of a new matrix, the S-method.
Abstract: Conventional methods for dynamic stability analysis such as the eigenvalue method (QR method), frequency response method, direct numerical integration method, etc., are not applicable to very large-scale systems because of limitations of memory capacity, computing time, and computation accuracy. To analyze a larger power system, it is necessary to take advantage of sparsity and regularity of matrices and to derive eigenvalues more efficiently (that is, with smaller memory capacity and shorter computing time). In this paper, we propose a new method which transforms the eigenvalue with the smallest real part (poorest attenuation component) to the eigenvalue of largest absolute value of a new matrix. We call the new matrix, Stability matrix (S-matrix) and call the proposed method, the S-method. The theoretical background of the S-method is presented and the effectiveness of the S-method is shown.
TL;DR: In this article, a systematic iteration scheme is presented to numerically solve the eigenvalue problem described by a coupled set of second-order differential equations under the boundary conditions that the solution is localized near the origin.
Abstract: A systematic iteration scheme is presented to numerically solve the eigenvalue problem described by a coupled set of second-order differential equations under the boundary conditions that the solution is localized near the origin. The method consists of the application of Newton's iteration scheme to the shooting method. A useful analytical expression for the correction to the trial eigenvalue is obtained. Application is made for two examples, simple harmonic oscillator and electro-magnetic drift-wave in a finite-β plasma in a sheared magnetic field, to demonstrate a remarkable stability, efficiency and accuracy of the method. In the latter example, reduction of the CPU time by factor 20–30 is obtained as compared with the standard simplex method.
01 Jan 1980
TL;DR: In this paper, a survey of various methods for effective estimation of the eigenvalues of a single discontinuous coefficient in a composite material mechanics problem is provided, where the main idea is to transform the one-dimensional Sturm-Liouville problems of concern to Liouville normal form.
Abstract: Eigenvalue problems with discontinuous coefficients occur naturally in many areas of composite material mechanics In previous work, based on mixed variational schemes, an approximation technique of Rayleigh-Ritz type applied to a modified “new quotient” has been developed by Nemat-Nasser and coworkers and applied in estimating eigenvalues and eigenfunctions for such problems in a wide variety of contexts Alternative approaches, resulting from modification of classical Sturm-Liouville theory, have been established recently by the present authors The central idea is to transform the one-dimensional Sturm-Liouville problems of concern to Liouville normal form This leads to a problem with a single discontinuous coefficient which moreover occurs in an undifferentiated term Eigenvalue estimates based on the transformed problem are established This paper provides a survey of these various methods for effective estimation of the eigenvalues of such problems Related issues arising in the area of eigenvalue optimization are briefly discussed
TL;DR: A modification of the Danilewski method is presented, permitting the solution of the eigenvalue problem for a constant sparse matrix of large order to be reduced to the solution for a polynomial matrix of lower order.
Abstract: A modification of the Danilewski method is presented, permitting the solution of the eigenvalue problem for a constant sparse matrix of large order to be reduced to the solution of the same problem for a polynomial matrix of lower order. Certain solution algorithms are proposed for a partial eigenvalue problem for the polynomial matrix. Questions of the realization of the algorithms on a model PRORAB computer are examined.
TL;DR: The influence of non-conservatism on the dynamics of rotating systems modelled by 3-D elasticity is considered in this article, where the fields are treated as small dynamic excursions about a potentially large initial state.
TL;DR: In this article, a normalized process is used for solving the eigenvalue problem, permitting the determination of linearly independent eigenvectors corresponding to the zero eigen value of matrixD(λ) and to the zeros of matrixA0 and A2.
Abstract: The solution of the eigenvalue problem is examined for the polynomial matrixD(λ)=Aoλs+A1λs−1+...+As when the matricesA0 andA2 (or one of them) are singular. A normalized process is used for solving the problem, permitting the determination of linearly independent eigenvectors corresponding to the zero eigenvalue of matrixD(λ) and to the zero eigenvalue of matrixA0. The computation of the other eigenvalues ofD(λ) is reduced to the same problem for a constant matrix of lower dimension. An ALGOL program and test examples are presented.
TL;DR: In this paper, the smoothness of the N-dimensional joint eigenvalue distribution for the volume element in the space of N × N orthogonal matrices is checked by re-deriving the exact results for N = 2.
Abstract: Exact distributions are given for the two-dimensional case when the mean of the off-diagonal element is non-zero. The joint eigenvalue distribution for theN dimensional case, derived using the volume element in the space ofN ×N orthogonal matrices, is checked by rederiving the exact results forN=2. The smooth nature of theN-dimensional joint distribution supports the claim of the method of moments that the single eigenvalue distribution is a smooth function of the ratio of mean-to-mean square deviation.
TL;DR: In this paper, a series of finite element algorithms for the calculation of the post-buckling behaviour of structures are proposed, which represent a finite element implementation of a variational principle for nonlinear eigenvalue problems.
Abstract: A series of finite element algorithms for the calculation of the post-buckling behaviour of structures are proposed. The algorithms represent a finite element implementation of a variational principle for nonlinear eigenvalue problems. Incremental amplitude solution schemes are defined. Numerical results are presented for the post-buckling behaviour of specific structures with linear and nonlinear pre-buckling states.
01 Jan 1980
TL;DR: The Lanczos algorithm was originally used to tridiagonalize symmetric matrices, but it was soon replaced by more effective methods based on explicit orthogonal similarity transformations, and was later revived as an effective scheme for solving sparse symmetric eigenvalue problems.
Abstract: The Lanczos algorithm was originally used to tridiagonalize symmetric matrices, but it was soon replaced by more effective methods based on explicit orthogonal similarity transformations. The algorithm was later revived as an effective scheme for solving sparse symmetric eigenvalue problems. The historical development and the current state of the Lanczos algorithm are surveyed.
TL;DR: In this article, the critical slab problem has been analyzed as an example of such an inverse eigenvalue problem of a Fredholm integral equation using the theory of perturbation of a class of positive, analytic operatorvalued functions in Banach space.
Abstract: Defines a general class of problem that has been termed the inverse eigenvalue problem. Basically similar problems have already been studied as isolated and specific examples in the analysis of time eigenvalues appearing in neutron transport theory. In this work, however, the authors present a general unified method for their treatment using functional analytic methods. Specifically, the critical slab problem has been analysed as an example of such an inverse eigenvalue problem of a Fredholm integral equation using the theory of perturbation of a class of positive, analytic operator-valued functions in Banach space. Numerical calculations of the critical thickness are given. These results are encouraging, considering the simplicity of the method, which does not involve an explicit solution of the Fredholm equation.
TL;DR: The bisection strategy reported by Rick and Evans for the calculation of eigenvalues of a real symme.
TL;DR: In this article, a simple, rapidly convergent procedure is described for solving a third-order symmetric eigenvalue problem Au = λ Bu typically arising in vibration analysis. But this procedure is not applicable to the problem we consider in this paper.
Abstract: A simple, rapidly convergent procedure is described for solving a third-order symmetric eigenvalue problem Au = λ Bu typically arising in vibration analysis. The eigenvalue problem is represented in terms of its variational dual, the Rayleigh quotient, and the eigenosolution is obtained through a topographical search for points of quotient stationarity. The associated computer routine is compact and can easily be incorporated within the calling program. Degenerate eigensolutions cause no difficulty. An example FORTRAN routine is given.
TL;DR: In this paper, lower bounds for the lowest eigenvalue a0 of the boundary value problem were derived for p = 1,2 and ∞ by solving a variational principle and the quality of the approximations thus obtained is discussed for Mathieu's equation and an application in magnetohydrodynamics.
Abstract: For Hill's equation
the lowest eigenvalue a0 of the boundary value problem y(x + 1) = y(x) is considered. Introducing Lp norms of the function f(x), lower bounds for a0 which depend only on this norm are derived for p = 1,2 and ∞ by solving a variational principle. For these lower bounds analytical expressions are obtained. The quality of the approximations thus obtained is discussed for Mathieu's equation and an application in magnetohydrodynamics is considered.