Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1992"

Numerical Methods for Large Eigenvalue Problems

Abstract: Preface to the Classics Edition Preface 1. Background in matrix theory and linear algebra 2. Sparse matrices 3. Perturbation theory and error analysis 4. The tools of spectral approximation 5. Subspace iteration 6. Krylov subspace methods 7. Filtering and restarting techniques 8. Preconditioning techniques 9. Non-standard eigenvalue problems 10. Origins of matrix eigenvalue problems References Index.

Lower eigenvalue estimates for Dirac operators

Abstract: We derive upper eigenvalue estimates for generalized Dirac operators on closed Riemannian manifolds. In the case of the classical Dirac operator the estimates on the first eigenvalues are sharp for spheres of constant curvature.

A chart of numerical methods for structured eigenvalue problems

Abstract: We consider eigenvalue problems for real and complex matrices with two of the following algebraic properties: symmetric, Hermitian, skew symmetric, skew Hermitian, symplectic, conjugate symplectic, J-symmetric, J-Hermitian, J-skew symmetric, J-skew Hermitian. In the complex case we found numerically stable algorithms that preserve and exploit both structures in 24 out of the 44 nontrivial cases with such a twofold structure. Of the remaining 20, we found algorithms that preserve part of the structure of 9 pairs. In the real case we found algorithms for all pairs studied. The algorithms are constructed from a small set of numerical tools, including orthogonal reduction to Hessenberg form, simultaneous diagonalization of commuting normal matrices, Francis’ QR algorithm, the Quaternion QR-algorithm and structure revealing, symplectic, unitary similarity transformations.

The extreme eigenvalues and stability of real symmetric interval matrices

Abstract: A novel Khoritonov-like algorithm for computing the minimal and maximal eigenvalues of n*n dimensional symmetric interval matrices is presented. It is proved that the maximal eigenvalue of a given set of interval matrices coincides with the maximal eigenvalue of a special set of 2/sup n-1/ symmetric vertex matrices, whereas its minimal eigenvalue coincides with the minimal of another special set of 2/sup n-1/ symmetric vertex matrices. As immediate corollaries of this algorithm, weak necessary and sufficient conditions for testing the Hurwitz and Schur stability of symmetric interval matrices, where one has to test the stability of 2/sup n-1/ and 2/sup n/ symmetric vertex matrices, respectively, are obtained. >

Numerical methods for large eigenvalue problems : theory and algorithms

Abstract: A high velocity liquid jet cutting nozzle in which a nozzle member is supported in communicative connection with the outlet end of a high pressure liquid pipe by means of a mounting which is seated in a cup-shaped holder which is connected to the outlet end of the pipe.

Neural networks for computing eigenvalues and eigenvectors

Abstract: Two new algorithms and associated neuron-like network architectures are proposed for solving the eigenvalue problem in real-time. The first approach is based on the solution of a set of nonlinear algebraic equations by employing optimization techniques. The second approach employs a multilayer neural network with linear artificial neurons and it exploits the continuous-time error back-propagation learning algorithm. The second approach enables us to find all the eigenvalues and the associated eigenvectors simultaneously by training the network to match some desired patterns, while the first approach is suitable to find during one run only one particular eigenvalue (e.g. an extreme eigenvalue) and the corresponding eigenvector in realtime. In order to find all eigenpairs the optimization process must be repeated in this case many times for different initial conditions. The performance and convergence behaviour of the proposed neural network architectures are investigated by extensive computer simulations.

Divide and conquer algorithms for the bandsymmetric eigenvalue problem

Abstract: Divide and conquer algorithms are formulated for the solution of the eigenvalue problem for symmetric band matrices. The new algorithms are compared to the traditional solutionspaths offered by Eispack , tridiagonalization of the band matrix followed by the tridiagonal QR algorithm.

Practical improvement of the divide-and-conquer eigenvalue algorithms

Abstract: We present a practical modification of the recent divide-and-conquer algorithms of [3] for approximating the eigenvalues of a real symmetric tridiagonal matrix. In this modified version, we avoid the numerical stability problems of the algorithms of [3] but preserve their insensitivity to clustering the eigenvalues and the possibility to give a priori upper bounds on their computational cost for any input matrix. We confirm the theoretical effectiveness of our algorithms by numerical experiments.

Eigenvalue and Eigenvector Approximate Analysis for Repeated Eigenvalue Problems

Abstract: The method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problems in terms of a single positive-valued parameter. The resulting equations yield first order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem.

The extreme eigenvalues and stability of Hermitian interval matrices

Abstract: The author presents a Kharitonov-like algorithm to find the minimal and maximal eigenvalues, i.e. the root clustering interval, of a set of (n*n)-dimensional Hermitian interval matrices. It is proven that the maximal eigenvalue of a given set of Hermitian interval matrices coincides with the maximal eigenvalue of a special set of Hermitian vertex matrices while its minimal eigenvalue coincides with the minimal eigenvalue of another such special set of the same size. As immediate corollaries of this algorithm, necessary and sufficient conditions for testing Hurwitz and Schur stability of Hermitian interval matrices wherein one has to test stability of certain Hermitian vertex matrices are obtained. >

On condition numbers of a nondefective multiple eigenvalue

Abstract: This paper concerns with measures of the sensitivity of a nondefective multiple eigenvalue of a matrix. Different condition numbers are introduced starting from directional derivatives of the multiple eigenvalue. Properties of the condition numbers defined by Stewart and Zhang [4] are studied; especially, the Wilkinson's theorem on matrices with a very ill-conditioned eigenproblem is extended.

The eigenvalue spreads of a hermitian matrix and its principal submatrices

Abstract: Inequalities are proved connecting the eigenvalue spread of a Hermitian matix to the eigenvalue spreads of its collection of principal submatrices. An application is made to the numerical range of general matrices.

On minimizing the largest eigenvalue of a symmetric matrix

Abstract: The problem of minimizing the largest eigenvalue over an affine family of symmetric matrices is considered. This problem has a variety of applications, such as the stability analysis of dynamic systems or the computation of structured singular values. Given in >or=0, an optimality condition is given which ensures that the largest eigenvalue is within in error bound of the solution. A novel line search rule is proposed and shown to have good descent property. When the multiplicity of the largest eigenvalue at solution is known, a novel algorithm for the optimization problem under consideration is derived. Numerical experiments show that the algorithm has good convergence behavior. >

A parallel approach to the modified Numerov-like eigenvalue determination for Sturm-Liouville problems.

Abstract: A modified Numerov-like eigenvalue algorithm, previously introduced, is parallelized. An inplementation of this algorithm on a Helios based parallel processing transputer system is discussed. Time savings with respect to a sequential approach are commented.

An inverse eigenvalue problem: computing B -stable Runge-Kutta methods having real poles

Abstract: The implementation of implicit Runge-Kutta methods requires the solution of large sets of nonlinear equations. It is known that on serial machines these costs can be reduced if the stability function of ans-stage method has only ans-fold real pole. Here these so-called singly-implicit Runge-Kutta methods (SIRKs) are constructed utilizing a recent result on eigenvalue assignment by state feedback and a new tridiagonalization, which preserves the entries required by theW-transformation. These two algorithms in conjunction with an unconstrained minimization allow the numerical treatment of a difficult inverse eigenvalue problem. In particular we compute an 8-stage SIRK which is of order 8 andB-stable. This solves a problem posed by Hairer and Wanner a decade ago. Furthermore, we finds-stageB-stable SIRKs (s=6,8) of orders, which are evenL-stable.

Bit-level systolic algorithms for real symmetric and Hermitian eigenvalue problems

Abstract: Arithmetic algorithms are presented that speed up the parallel Jacobi method for the eigen-decomposition of real symmetric and complex Hermitian matrices. The 2×2 submatrices to which the Jacobi rotations are applied form a Clifford algebra, hence they can be decomposed into a sum of even and odd part. This decomposition enables the application of the rotations from a single side instead of both, thus removing some sequentiality from the original Jacobi method. Moreover, with the help of implicit CORDIC algorithms, the rotations are evaluated and applied in a fully concurrent fashion on triangular arrays of specialized processors. The CORDIC algorithms employed in the complex case are genuine 3- and 4-dimensional generalizations of the 2-dimensional algorithm used in the real case. Because these algorithms are implicit, variants are obtained with minor modifications that perform rotations whose resolution is poor at first and slowly increases to become high in the last steps of the Jacobi method. Such variants further reduce the total computation time.

A large, sparse, and indefinite generalized eigenvalue problem from fluid mechanics

Abstract: A numerical method for calculating the minimum positive eigenvalue of a sparse, indefinite, Hermitian algebraic problem has been developed. The method is based on inverse iteration and is a generalization of a procedure previously employed for the simpler problem of finding the smallest eigenvalue of a positive-definite matrix. Motivation was provided by a three-dimensional research problem from hydrodynamic stability. Stability limits obtained from the application of the method to a previously studied problem are compared to independently determined results.

On the inverse eigenvalue problem for matrices (atomic corrections)

Abstract: The inverse eigenvalue problem for matrices is studied with the objective of obtaining an efficient method for correcting energy levels in atomic systems, though the results are applicable to any eigenvalue problem. The approach is a development of earlier work by S. Friedland (1977). The diagonal elements of a real symmetric matrix with given off-diagonal elements are adjusted to yield a given spectrum. The authors discuss cases where there are real solutions and no real solutions, with particular emphasis on the latter. Problems of slow convergence arise. They demonstrate the cause of this slow convergence, give a geometrical interpretation of the problem and show how it can be avoided. Also the matrices encountered arise from complex calculations and are subject to error. They develop an error analysis that permits us, among other things, to judge whether corrections in any particular case are justified in view of anticipated errors in the given computed off-diagonal matrix elements. Finally, the method is demonstrated in an application to certain sets of levels in 12 times ionized (neon-like) titanium.

Symmetric solutions and eigenvalue problems of Toeplitz systems

Abstract: Algorithms and properties of symmetric solutions of a Toeplitz system are studied. It is shown that the numbers of positive and negative eigenvalues associated with symmetric (antisymmetric) eigenvectors are the same as the numbers of positive and negative predictor errors of symmetric (antisymmetric) filters. Based on the odd symmetric solutions and the property that all roots of symmetric and antisymmetric filters are on the unite circle, a method for Pisarenko's decomposition is introduced. Compared with some other methods, it reduces the number of iterations and the computational cost in each iteration considerably. >

A two-dimensional bisection method for solving two-parameter eigenvalue problems

Abstract: It has long been known that separation of variables can be applied to the Helmholtz equation in 11 three-dimensional coordinate systems. As a result, a multiparameter eigenvalue problem is formed. In this paper, the well-known bisection method for ordinary eigenvalue problems is generalized to a special class of discrete two-parameter eigenvalue problems. The range of the real roots of the problem is discussed and some numerical results are given.

Nonlinear equations at resonance and generalized eigenvalue problems

Abstract: IN A 1967 PAPER, Loud [25] obtained sharp nonresonance conditions for the second order differential equation -xn = g(x) + h(t). (1) More precisely, assuming g to be an odd function of class C’ and h to be a continuous T-periodic function, h being even and odd-harmonic, he proved the existence (and uniqueness) of a T-periodic solution to (l), whenever the range of the derivative of g does not interact with the set of eigenvalues ((21~i/T)~, i = 0, 1 , . . .) of the differential operator, i.e. for some n E N,

Lanczos algorithm for acoustic boundary element eigenvalue problems

Abstract: Acoustic eigenvalue analysis by the boundary element method results in a generalized eigensystem of the form [K{x}=λ[M]{x}, where [K] and [M] are fully populated unsymmetric matrices. With a brief outline of the boundary element procedure, a Lanczos‐two‐sided recursion algorithm is presented to solve the unsymmetric eigenproblem. Details on the implementation of the scheme as coded in the ansysR general purpose computer program are described. Numerical examples on two‐dimensional (2‐D) acoustic eigenvalue analysis are included to demonstrate the effectiveness of the Lanczos algorithm in extracting the first few eigenfrequencies of large problems. Issues relating to the effectiveness of the boundary element procedure when compared to the acoustic finite element eigenvalue formulation are discussed.

Algorithm to solve the partial eigenvalue problem for large profile matrices

Abstract: An algorithm is proposed for computing some of the eigenvalues and the corresponding eigenvectors of the generalized eigenvalue problem for symmetric positive-definite matrices stored in profile form on disk

Universal eigenvalue equations

Abstract: ing d = f · b. Conversely, if this homomorphic condition holds everywhere, then apply b to both its sides, observe that Cχb = iA and by abstracting c get h = Cχa with a = h · b. Q.D.E. 5.8 Definitions. Keep the notation of 5.4 and let B be the set of bases of α with index X. Consider a family k: B → F(FXA)C, for some C, and a b ∈ B. If for all b′ ∈ B and M :X → A (43) kbM = kb′(M ◦ b′) , then we say that the function kb: FXA → C, as well as family k, are (absolutely) invariant. (In fact, if kb is invariant, then any kb′ is, as it follows from 5.1 through easy passages.) See [38] for a (lone) definition of universal invariance in F. Klein’s synthetic form. We call an invariant function η: FXA → D general if for any invariant family k as above there is a function f : B → FDC, such that kb′ = fb′ · η for all b′ ∈ B. 5.9 Corollary. If χ is an analytic representative, then η = Cχ is a general invariant function. Proof. By 5.6 η = rb−1: FXA → H αα. Hence, η is an invariant function if (43) holds for the k such that ka = ra−1 for all a ∈ B, i.e. if rb′(ηM) = M ◦ b′ for all M :X → A and b′ ∈ B. This is a trivial identity because of our notation of r, η and ◦ in 1.2, 5.9 and 5.4. Also, η is general, since it is one to one. (We can take fa = ka · η−1 for all a ∈ B.) Q.D.E. 5.10 Example. By the preceding corollary any function or predicate of the endomorphism associated to a family M is invariant. Hence, in based algebras all present theory about universal eigenvalue equations concerns invariants only. E.g. by 5.7(A) independence is an invariant predicate. This differs from Universal Algebra, where noninvariant notions are accepted, as we are going to show. In fact, we disprove the invariance of the “C/Ci–independences” as in [15]. These are weaker notions of independence, sometimes [11] and [17] considered akin of independence itself. In a vector space, they express the lack of certain linear dependencies among the elements (columns) of a family M and are equivalent to the independence of M . Here, we recall a couple of them that correspond to conditions (C3) and (10) of [15]. (However, it is easy to see that the next conterexample works even for conditions (C1) and (C2) ibidem.) Given M and α as in 5.8, consider the following two conditions: ⋂ (C↑V ) = C( ⋂ V ) , for all V ⊆ PM (44) and Mx ∈ Cv implies Mx = vx , for all v ⊆ M and x ∈ X , (45)