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Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1976"


Journal ArticleDOI
TL;DR: In this article, a Gaussian probability density function with the same mean and variance is used to calculate the eigenvalue spectrum of a large symmetric square matrix, each of whose upper triangular elements is described by the Gaussian distribution.
Abstract: A new and straightforward method is presented for calculating the eigenvalue spectrum of a large symmetric square matrix each of whose upper triangular elements is described by a Gaussian probability density function with the same mean and variance. Using the n to 0 method, the authors derive the semicircular eigenvalue spectrum when the mean of each element is zero and show that there is a critical finite mean value above which a single eigenvalue splits off from the semicircular continuum of eigenvalues.

187 citations


Journal ArticleDOI
TL;DR: In this article, a FORTRAN IV algorithm is presented for determining sets of dominant eigenvalues and corresponding eigenvectors of symmetric matrices, which is also extended to the solution of the natural vibration of a structure for which symmetric stiffness and mass matrices are available.
Abstract: A FORTRAN IV algorithm is presented for determining sets of dominant eigenvalues and corresponding eigenvectors of symmetric matrices. It is also extended to the solution of the equations of natural vibration of a structure for which symmetric stiffness and mass matrices are available. The matrices are stored and processed in variable bandwidth form, thus enabling advantage to be gained from sparseness in the equations. Some of the procedures may also be used to solve symmetric positive definite equations such as those arising from the static analysis of structures loaded within the elastic range.

162 citations


Journal ArticleDOI
TL;DR: In this paper, a method for computing a nested sequence of orthonormal bases for the dominant invariant subspaces of a non-Hermitian matrix is described, which is particularly suited to large sparse eigenvalue problems, since it requires only that one can form the product of the matrix in question with a vector.
Abstract: This paper describes a simultaneous iteration technique for computing a nested sequence of orthonormal bases for the dominant invariant subspaces of a non-Hermitian matrix. The method is particularly suited to large sparse eigenvalue problems, since it requires only that one be able to form the product of the matrix in question with a vector. A convergence theory for the method is developed and practical details are discussed.

150 citations


Journal ArticleDOI
TL;DR: In this article, it was proved that a real symmetric tridiagonal matrix with positive codiagonal elements is uniquely determined by its eigenvalues and the eigen values of the largest leading principal submatrix, and can be constructed from these data.

149 citations


Journal ArticleDOI
TL;DR: In this paper, the Hadamard-Fischer inequality was shown to hold for the case of BE W(n) and A E 7(1I) if A E W( n) and 0, A.
Abstract: 1) Spec A[Jl.l n IR =1= t/>, for t/> c Jl. S (n), 2) I(A[J-L]) « I(A[v]), if t/> c v S Jl. S (n), where I(A[Jl.]) = min(Spec A[Jl.l n IR). For A, BE W(n), define A «, B by I(A[J-L]) « I(B[J-L]), for t/> c Jl. S ( n ). By definition, A E7(1I) if A E W(n) and 0 «, A. For 0 « , A «t B (where A, BE W(n» it is shown that 3) 0 « det A « det B-det(B-I(A)I) « det B. For A E 7(11 ) , A «, A[Jl.l El1 A{J1.), and hence we obtain the Hadamard-Fischer inequality 4) 0 « det A « det A[Jl.l det A{J1.)

50 citations


Journal ArticleDOI
TL;DR: In this article, the eigenvalue problem for real symmetric matrices is considered, where A and B are symmetric real matrices, and the expression (x * Ax + Bx) is bounded away from zero.
Abstract: The eigenvalue problem $Ax = \lambda Bx$ is considered where A and B are real symmetric matrices. Perturbation bounds are obtained in case the expression $(x^ * Ax)^2 + (x^ * Bx)^2 $ is bounded away from zero. Numerical methods for the solution of the problem are discussed.

47 citations


Journal ArticleDOI
TL;DR: In this paper, an elementary proof is given that the statistical mechanical transfer matrix, when symmetric, has a maximum eigenvalue which is non-degenerate and larger than the absolute value of any other eigen value.
Abstract: An elementary proof is given that the statistical mechanical transfer matrix, when symmetric, has a maximum eigenvalue which is non-degenerate and larger than the absolute value of any other eigenvalue. Moreover, the corresponding eigenvector can be chosen so that all its entire entries are strictly positive.

41 citations




Book ChapterDOI
01 Jan 1976
TL;DR: The literature on algorithms for solving the generalized eigenvalue problem Ax = λBx, where A and B are real symmetric matrices, B is positive definite, and A and A are large and sparse is surveyed.
Abstract: This paper surveys the literature on algorithms for solving the generalized eigenvalue problem Ax = λBx, where A and B are real symmetric matrices, B is positive definite, and A and B are large and sparse.

33 citations


Journal ArticleDOI
TL;DR: Bathe's algorithm of subspace iteration for the solution of the eigenvalue problem with symmetric matrices is improved by incorporating an acceleration technique using Chebyshev polynomials as mentioned in this paper.
Abstract: Bathe's algorithm of subspace iteration for the solution of the eigenvalue problem with symmetric matrices is improved by incorporating an acceleration technique using Chebyshev polynomials. This method of acceleration is particularly effective for this kind of iteration. The rate of convergence of the iteration scheme presented is considerably improved when compared with the original one, and satisfactory rates of convergence can be obtained for a wider range of eigenvalues.

Journal ArticleDOI
TL;DR: In this article, the eigenvalue problem arising in the free vibration and stability analysis of gyroscopic systems is associated with a λ-matrix in which λ as well as its square appears.

Journal ArticleDOI
Raymond H. Plaut1
TL;DR: In this paper, Meirovitch et al. derived an equivalent formulation for the symmetric and skew-symmetric matrices, which leads to a new condition for flutter instability and a technique for determining approximations to frequencies of vibration.
Abstract: Eigenvalue problems associated with discrete gyroscopic systems have the property that the characteristic equation is a function of the square of the eigenvalue. It may be advantageous, therefore, to replace the given system by an equivalent one that only involves this squared parameter. Meirovitch has recently derived such an equivalent problem for a certain class of gyroscopic systems. The original problem, involving symmetric and skew-symmetric matrices, can be replaced by one involving only symmetric matrices. These results are extended to a more general class of systems in the present paper. In addition, a particular form of gyroscopic system is considered that occurs in problems such as rotating shafts and fluid-conveying pipes. For these systems an equivalent formulation is obtained which leads to a new condition for flutter instability and a technique for determining approximations to frequencies of vibration. An example is presented to illustrate these features.

Journal ArticleDOI
Tohru Morita1
TL;DR: In this paper, it was shown that the thermodynamic properties and the distribution functions of the Ising systems on the Cayley tree are generally obtained in terms of the solution of a recurrence formula, when the interaction is of finite range.
Abstract: It is shown that the thermodynamic properties and the distribution functions of the Ising systems on the Cayley tree are generally obtained in terms of the solution of a recurrence formula, when the interaction is of finite range. For the Bethe lattice where the boundary effects are ignored, the properties are given in terms of the solution of a nonlinear eigenvalue problem. It is further shown that a certain approximation in the cluster variation method is exact for this system and the equations determining the parameters occurring in this method are equivalent to the obtained nonlinear eigenvalue problem.

Journal ArticleDOI
TL;DR: In this article, a non-perturbative method for eigenvalue problems is presented, which involves the use of an appropriately scaled set of basis functions for the determination of each eigen value.
Abstract: A method for eigenvalue problems is presented. As an example, we have obtained very accurate eigenvalues and eigenfunctions of the quartic anharmonic oscillator. The method is non-perturbative and involves the use of an appropriately scaled set of basis functions for the determination of each eigenvalue. The claimed accuracy for all eigenvalues is 15 significant figures. The method does not deteriorate for higher eigenvalues.

Journal ArticleDOI
Leonard Meirovitch1
TL;DR: An entirely new method for the calculation of the natural frequencies and natural modes of a spinning axisymmetric spacecraft with flexible appendages is presented and a computational algorithm is developed that solves the "reduced eigenvalue problem" more efficiently than existing algorithms.
Abstract: This paper presents an entirely new method for the calculation of the natural frequencies and natural modes of a spinning axisymmetric spacecraft with flexible appendages. The approach is based on a method for the solution of the eigenvalue problem for general linear gyroscopic systems developed recently by this author. When the structure is axisymmetric, the matrices defining the eigenvalue problem are of a special type. The present paper takes advantage of the special nature of the system to reduce the order of the eigenvalue problem by a factor of two. Moreover, it develops a computational algorithm that solves the "reduced eigenvalue problem" more efficiently than existing algorithms. The method should find its application to a large variety of spinning axisymmetric structures.


Journal ArticleDOI
TL;DR: In this paper, an accurate reduction scheme for structural eigenvalue problems is deduced from a variational theorem in which the displacement, velocity and/or momentum fields are taken to be independent.
Abstract: An accurate reduction scheme for structural eigenvalue problems is deduced from a variational theorem in which the displacement, velocity and/or momentum fields are taken to be independent.

Journal ArticleDOI
TL;DR: In this paper, a numerical technique is presented for locating the eigenvalues of two point linear differential eigenvalue problems, namely, the Orr-Sommerfeld equation of the plane Poiseuille flow.

Journal ArticleDOI
TL;DR: In this paper, two globally convergent Jacobi-like normdecreasing methods for diagonalizing the so-called J-symmetric matrices are presented, depending on whether the real or the imaginary parts of the eigenvalues are better separated.
Abstract: Two globally convergent Jacobi-like normdecreasing methods for diagonalising the so-calledJ-symmetric matrices are presented. The properties ofJ-symmetric matrices and their connection with various generalized symmetric eigenvalue problems are briefly discussed. The choice between the two methods depends on whether the real or the imaginary parts of the eigenvalues are better separated.

Journal ArticleDOI
TL;DR: A modification of the central-difference method is given in this article which greatly improves the convergence when applied to a certain class of singular eigenvalue problems, including the Klein-Gordon equation.
Abstract: A modification of the central-difference method is given which greatly improves the convergence when applied to a certain class of singular eigenvalue problems, including the Klein-Gordon equation. The singularity given special treatment is at the finite end.


01 Aug 1976
TL;DR: The development of the tridiagonal reduction method and its implementation in NASTRAN are described for real eigenvalue analysis as typified by structural vibration and buckling problems.
Abstract: The development of the tridiagonal reduction method and its implementation in NASTRAN are described for real eigenvalue analysis as typified by structural vibration and buckling problems. This method is an automatic matrix reduction scheme whereby the eigensolutions in the neighborhood of a specified point in the eigenspectrum can be accurately extracted from a tridiagonal eigenvalue problem whose order is much lower than that of the full problem. The process is effected without orbitrary lumping of masses or other physical quantities at selected node points and thus avoids one of the basic weaknesses of other techniques.

Journal ArticleDOI
TL;DR: In this paper, the authors considered quadratic eigenvalue problems of the form (2I-?A-B)x=0 with compact, positive definite operators A, B in a complex, separable Hilbert space.
Abstract: We consider eigenvalue problems of the form (?2I-?A-B)x=0 with compact, positive definite operatorsA, B in a complex, separable Hilbert space. Computational methods for upper bounds for the positive eigenvalues of this quadratic eigenvalue problem are derived, using a construction of Aronszajn, applied to linear eigenvalue problems by him and Bazley and Fox. We prove their convergence and give an example.

Journal ArticleDOI
TL;DR: In this paper, the monotonicity of the first eigenvalue λ 1 (D) of (1) as a functional of the domainD is studied and the monotoneness of the eigenvalues is investigated.
Abstract: This paper is concerned with the monotonicity of the first eigenvalue λ1 (D) of (1) as a functional of the domainD.