Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1996"

The real and the symmetric nonnegative inverse eigenvalue problems are different

Abstract: We show that there exist real numbers Al, A2 ... A, that occur as the eigenvalues of an entry-wise nonnegative n-by-n matrix but do not occur as the eigenvalues of a symmetric nonnegative n-by-n matrix. This solves a problem posed by Boyle and Handelman, Hershkowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue problem by appending 0's to given spectral data is refined.

Conjugate gradient methods for solving the smallest eigenpair of large symmetric eigenvalue problems

Abstract: SUMMARY In this paper, a detailed description of CG for evaluating eigenvalue problems by minimizing the Rayleigh quotient is presented from both theoretical and computational viewpoints. Three variants of CG together with their asymptotic behaviours and restarted schemes are discussed. In addition, it is shown that with a generally selected preconditioning matrix the actual performance of the PCG scheme may not be superior to an accelerated inverse power method. Finally, some test problems in the finite element simulation of 2-D and 3-D large scale structural models with up to 20200 unknowns are performed to examine and demonstrate the performances.

A Matlab implementation of the Implicitly Restarted Arnoldi Method for solving large-scale eigenvalue problems

Abstract: This thesis describes a Matlab implementation of the Implicitly Restarted Arnoldi Method for computing a few selected eigenvalues of large structured matrices. Shiftand-invert methods allow the calculation of the eigenvalues nearest any point in the complex plane, and polynomial acceleration techniques aid in computing eigenvalues of operators which are de ned by mles instead of Matlab matrices. These new Matlab functions will be incorporated into the upcoming version 5 of Matlab and will greatly extend Matlab's capability to deal with many real-world eigenvalue problems that were intractable in version 4. The thesis begins with a discussion of the Implicitly Restarted Arnoldi Method. The bulk of the thesis is a user's manual for the Matlab functions which implement this algorithm. The user's guide not only describes the functions' syntax and structure but also discusses some of the di culties that were overcome during their development. The thesis concludes with several examples of the functions applied to realistic test problems which illustrate the versatility and power of this new tool.

Design of IIR digital filters based on eigenvalue problem

Abstract: This paper presents a new method for designing IIR digital filters with optimum magnitude response in the Chebyshev sense and different order numerator and denominator. The proposed procedure is based on the formulation of a generalized eigenvalue problem by using Remez exchange algorithm. Since there exist more than one eigenvalue in the general eigenvalue problem, we introduce a very simple selection rule for the eigenvalue to be sought for where the rational interpolation is performed if and only if the positive minimum eigenvalue is chosen. Therefore, the solution of the rational interpolation problem can be obtained by computing only one eigenvector corresponding to the positive minimum eigenvalue, and the optimal filter coefficients are easily obtained through a few iterations. The design algorithm proposed in this paper not only retains the speed inherent in the Remez exchange algorithm but also simplifies the interpolation step because it has been reduced to the computation of the positive minimum eigenvalue. Some properties of the filters such as lowpass filters, bandpass filters, and so on are discussed, and several design examples are presented to demonstrate the effectiveness of this method.

On conjugate gradient-like methods for eigen-like problems

Abstract: Numerical analysts, physicists, and signal processing engineers have proposed algorithms that might be called conjugate gradient for problems associated with the computation of eigenvalues. There are many variations, mostly one eigenvalue at a time though sometimes block algorithms are proposed. Is there a correct “conjugate gradient” algorithm for the eigenvalue problem? How are the algorithms related amongst themselves and with other related algorithms such as Lanczos, the Newton method, and the Rayleigh quotient?

Parallel Eigenvalue Algorithms for Large-Scale Control-Optimization Problems

Abstract: In a recent article, the authors presented the formulation and outline of parallel algorithms for the integrated structural/control optimization problem. The solutions of the Riccati equation, open-loop system of equations, and closed-loop system of equations encountered in this problem require repeated solution of the complex eigenvalue problem of a general unsymmetric matrix. This is the bottleneck for simultaneous optimization of structural and control systems and its application to design of large adaptive/smart structures. In this paper, robust parallel-vector algorithms are presented for the solution of the eigenvalue problem of a general unsymmetric real matrix employing the architecture of shared memory supercomputers such as Cray YMP 8/8128. Judicious combination of vectorization, microtasking, and macrotasking is explored in order to achieve maximum efficiency. The algorithms are applied to large matrices including one resulting from a 21-story space truss structure. It is shown that the speedup due to both vectorization and parallel processing increases with the size of the problem, thus making the algorithms particularly attractive for integrated structural/control optimization of large adaptive structures.

A theorem on the asymptotic eigenvalue distribution of Toeplitz-block-Toeplitz matrices

Abstract: We extend Szego's eigenvalue distribution theorem to block Toeplitz matrices. A sequence of such matrices arises from the autocorrelation of a 2-D discrete random process. We show that the eigenvalues of the matrix sequence are asymptotically distributed like the samples of the random process' 2-D power spectrum.

A quasi-Newton adaptive algorithm for generalized symmetric eigenvalue problem

Abstract: We first recast the generalized symmetric eigenvalue problem, where the underlying matrix pencil consists of symmetric positive definite matrices, into an unconstrained minimization problem by constructing an appropriate cost function. We then extend it to the case of multiple eigen-vectors using an inflation technique. Based on this asymptotic formulation, we derive a quasi-Newton-based adaptive algorithm for estimating the required generalized eigen-vectors in the data case. The resulting algorithm is modular and parallel, and it is globally convergent with probability one. We also analyze the effect of inexact inflation on the convergence of this algorithm and that of inexact knowledge of one of the matrices (in the pencil) on the resulting eigenstructure. Simulation results demonstrate that the performance of this algorithm is almost identical to that of the rank-one updating algorithm of Karasalo (1986). Further, the performance of the proposed algorithm has been found to remain stable even over 1 million updates without suffering from any error accumulation problems.

Estimates for the first eigenvalue in some Sturm-Liouville problems

Abstract: Contents Introduction §1. On estimates of the first eigenvalue in a Sturm-Liouville problem §2. On other estimates for the first eigenvalue §3. On a more general Sturm-Liouville problem §4. On estimates for all the eigenvalues §5. On an estimate for the first eigenvalue of a Sturm-Liouville problem for a higher-order operator §6. On the problem of Lagrange §7. Appendix. Technical lemmas Bibliography

Sign-patterns which require a positive eigenvalue

Abstract: We investigate matrices which have a positive eigenvalue by virtue of their sign‐pattern and regardless of the magnitudes of the entries. When all the o‐diagonal entries are nonzero, we show that an n◊n

Eigenvalue bounds for transformations of solvable potentials

Abstract: We study smooth transformations of potentials for which exact bound-state solutions of Schrodinger's equation are known. Eigenvalue approximation formulae are obtained which provide lower or upper energy bounds according to whether the transformation function g is convex or concave. Detailed results are presented for perturbed Coulomb potentials of the form and

Numerical algorithms for solutions of large eigenvalue problems in piezoelectric resonators

Abstract: Two algorithms for eigenvalue problems in piezoelectric finite element analyses are introduced. The first algorithm involves the use of Lanczos method with a new matrix storage scheme, while the second algorithm uses a Rayleigh quotient iteration scheme. In both solution methods, schemes are implemented to reduce storage requirements and solution time. Both solution methods also seek to preserve the sparsity structure of the stiffness matrix to realize major savings in memory. In the Lanczos method with the new storage scheme, the bandwidth of the stiffness matrix is optimized by mixing the electrical degree of freedom with the mechanical degrees of freedom. The unique structural pattern of the consistent mass matrix is exploited to reduce storage requirements. These major reductions in memory requirements for both the stiffness and mass matrices also provided large savings in computational time. In the Rayleigh quotient iteration method, an algorithm for generating good initial eigenpairs is employed to improve its overall convergence rate, and its convergence stability in the regions of closely spaced eigenvalues and repeated eigenvalues. The initial eigenvectors are obtained by interpolation from a coarse mesh. In order for this multi-mesh iterative method to be effective, an eigenvector of interest in the fine mesh must resemble an eigenvector in the coarse mesh. Hence, the method is effective for finding the set of eigenpairs in the low-frequency range, while the Lanczos method with a mixed electromechanical matrix can be used for any frequency range. Results of example problems are presented to show the savings in solution time and storage requirements of the proposed algorithms when compared with the existing algorithms in the literature.

Arnoldi-Riccati method for large eigenvalue problems

Abstract: 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.

Jacobi-Davidson methods for generalized MHD-eigenvalue problems

Abstract: A Jacobi-Davidson algorithm for computing selected eigenvalues and associated eigenvectors of the generalized eigenvalue problem Ax = Bx is presented. In this paper the emphasis is put on the case where one of the matrices, say the B-matrix, is Hermitian positive denite. The method is an inner-outer iterative scheme, in which the inner iteration process consists of solving linear systems to some accuracy. The factorization of either matrix is avoided. Numerical experiments are presented for problems arising in magnetohydrodynamics (MHD).

Bounds for eigenvalues of symmetric block jacobi scaled matrices

Abstract: The paper presents upper bounds for the largest eigenvalue of a block Jacobi scaled symmetric positive-definite matrix which depend only on such parameters as the block semibandwidth of a matrix and its block size. From these bounds we also derive upper bounds for the smallest eigenvalue of a symmetric matrix with identity diagonal blocks. Bibliography: 4 titles.

Chebyshev acceleration techniques for large complex non hermitian eigenvalue problems

Abstract: We propose an extension of the Arnoldi-Chebyshev algorithm to the large complex non Hermitian case. We demonstrate the algorithm on two applied problems.