Eigenvalue Value Problems Research Articles

Asymptotic methods for solving boundary value eigenvalue problems

The aim of the study is an approximate construction with a given accuracy of solutions of boundary value problems for eigenvalues under various types of boundary conditions. It is shown that the problem of finding approximate large eigenvalues of boundary value problems is reduced to the analysis and solution of singularly perturbed differential equations with variable coefficients. Methods used: asymptotic diagram method developed to construct the asymptotic behavior of solutions of singularly perturbed differential equations and systems; methods of numerical integration of boundary value problems. The main results obtained are: the asymptotics of the required accuracy are constructed in the analytical form for the eigenvalues and eigenfunctions of the boundary value problems under various boundary conditions; analysis of the computational capabilities of the practical use of the constructed asymptotics in comparison with the results of numerical integration.

General solution of the coupled-wave equations of acousto-optics.

The basic problem of the diffraction of an optical plane wave by an acoustic plane wave in an anisotropic homogeneous medium is considered. The acousto-optical interaction is considered indifferently of the isotropic or of the birefringent type. Coupled-wave equations are obtained rigorously and cast into an eigenvalue value problem. A general solution is obtained for the diffraction efficiency of diffracted orders, for any interaction length and diffraction regime. The theory includes the Bragg regime, the Raman-Nath regime, and all intermediate situations in the same formulation. The method of solution is both exact and computationally efficient. It is similar in character to the rigorous coupled-wave analysis of Moharam and Gaylord but differs by the choice of basis functions adapted to propagating rather than static gratings. Examples are given for acousto-optical interaction in paratellurite, TeO2.

High Performance Algorithms for Structured Matrix Problems

Edited by: Peter Arbenz, Marcin Paprzycki, Ahmed Sameh and Vivek Sarin Nova Science Publishers, Commack, NY, 1998, 197 pp. ISBN 1-56072-594-X, $79.95 The book is a well-organized comprehensive collection of algorithms and ideas in the area of high performance solutions for structured linear systems, eigenvalue and singular value problems. The content of this text is impressive, not only summarizing the state of the art in the area, but also discussing many detailed implementation issues with supporting examples on numerous parallel and distributed architectures from vector computers, shared and distributed memory multiprocessors to cluster of workstations. Hence, this book is an excellent reference for the working professionals in scientific and engineering computing with application areas. The book's targets can be mainly classified into four categories, namely linear system solver, eigenvalue problems, matrices with special structure and parallel computation. In the first part, linear system solver, direct solution techniques have been discussed mainly in term of performance and stability for large and sparse linear systems with a certain structure inherent in the problems. The effectiveness of different approaches is demonstrated by extensive experiments carried out on different architectures from vector computer, shared and distributed memory parallel computers to workstations. In the second part, eigenvalue problems, several approaches are highlighted for computing eigenvalues and singular values of structured matrices. They are presented by, either reducing banded matrices to bidiagonal and tridiagonal form by numerically stable and efficient orthogonal transformations, or improving bisection and divide-and-conquer algorithms, on parallel architectures. The theoretical considerations are verified by experimental results on vector and distributed memory computers. In the third part, matrices with special structures, some special structured matrices such as Hankel, Toeplitz systems from many applications, for example, signal processing, are treated. The authors propose several fast algorithms for the solution of such linear systems by considering the special properties of the structure and the numerical stability constraints. In the last part, parallel computation, the issues related to the question how to efficiently implement the existing algorithms for structured matrices are covered; ranging from load balance to optimized implementation. In summary, this book addresses a variety of algorithm design issues by either comprehensive survey or in-depth discussions for structured matrices. The only small disappointment is that the discussions on iterative methods for parallel and distributed architectures should be touched as well. Overall, this text would be a very valuable reference to professionals or to students working on structured matrix related problems. Laurence Tianruo Yang St. Francis Xavier University