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Wednesday, April 22, 2020 | History

3 edition of Solving large sparse eigenvalue problems on supercomputers found in the catalog.

Solving large sparse eigenvalue problems on supercomputers

B. Philippe

Solving large sparse eigenvalue problems on supercomputers

  • 8 Want to read
  • 6 Currently reading

Published by Research Institute for Advanced Computer Science, NASA Ames Research Center in [Moffett Field, Calif.?] .
Written in English

    Subjects:
  • Eigenvalues.

  • Edition Notes

    StatementBernard Philippe, Youcef Saad.
    SeriesRIACS technical report -- 88.38., NASA CR -- 185421., RIACS technical report -- TR 88-38., NASA contractor report -- NASA CR-185421.
    ContributionsSaad, Y., Research Institute for Advanced Computer Science (U.S.)
    The Physical Object
    FormatMicroform
    Pagination1 v.
    ID Numbers
    Open LibraryOL15274069M

    In computational mathematics, a matrix-free method is an algorithm for solving a linear system of equations or an eigenvalue problem that does not store the coefficient matrix explicitly, but accesses the matrix by evaluating matrix-vector products. Such methods can be preferable when the matrix is so big that storing and manipulating it would cost a lot of memory and computer Hardware: CPU cache, TLB, Cache-oblivious . A significantly revised and improved introduction to a critical aspect of scientific computation Matrix computations lie at the heart of most scientific computational tasks. For any scientist or engineer doing large-scale simulations, an understanding of the topic is essential. Fundamentals of Matrix Computations, Second Edition explains matrix computations and the accompanying . Eigenvalues of Large, Sparse Matrices, I Eigenvalues of Large, Sparse Matrices, II Sensitivity of Eigenvalues and Eigenvectors Methods for the Symmetric Eigenvalue Problem The Generalized Eigenvalue Problem 7 Iterative Methods for Linear Systems A Model Problem The Classical Iterative. @article{osti_, title = {The two-phase method for finding a great number of eigenpairs of the symmetric or weakly non-symmetric large eigenvalue problems}, author = {Dul, F.A. and Arczewski, K.}, abstractNote = {Although it has been stated that [open quotes]an attempt to solve (very large problems) by subspace iterations seems futile[close quotes], we will show that the .


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Solving large sparse eigenvalue problems on supercomputers by B. Philippe Download PDF EPUB FB2

1 Introduction. The numerical solution of large sparse eigenvalue problems arises in numerous important scientific applications that can be termed supercomputing applications. The advances in supercomputing technology allow today to tackle very large eigenvalue problems that were not feasible a few years by: 5.

CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): An important problem in scientific computing consists in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix.

The most popular methods to solve these problems are based on projection techniques on appropriate subspaces. The main attraction.

Get this from a library. Solving large sparse eigenvalue problems on supercomputers. [Bernard Philippe; Youcef Saad; Research Institute for Advanced Computer Science (U.S.)].

This book presents a unified treatment of recently developed techniques and current understanding about solving systems of linear equations and large scale eigenvalue problems on high-performance computers.

It provides a rapid introduction to the world of vector and parallel processing for these linear algebra by: The topics presented in the book, including novel numerical algorithms, high-performance implementation techniques, software developments and sample applications, will contribute to various fields that involve solving large-scale eigenvalue problems.

Solving large sparse eigenvalue problems on supercomputers. problem in scientific computing Solving large sparse eigenvalue problems on supercomputers book in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix. The most popular methods to solve these problems are based on projection techniques on appropriate subspaces.

The main attraction of these Author: Bernard Philippe and Youcef Saad. () New Methods for Calculations of the Lowest Eigenvalues of the Real Symmetric Generalized Eigenvalue Problem. Journal of Computational Physics() Jacobi—Davidson algorithm and its application to modeling RF/Microwave detection by: Solving large sparse eigenvalue problems RAT_KRYLOV, eigenvalues; The FEAST algorithm for Hermitian eigenproblems FEAST, RKFUN, eigenvalues; Solving nonlinear eigenvalue problems util_nleigs; Moving the poles of a rational Krylov space RAT_KRYLOV, poles; Structure of rational Krylov projections RAT_KRYLOV, semiseparable; Fitting an artificial.

Saad Y. () Projection methods for solving large sparse eigenvalue problems. In: Kågström B., Ruhe A. (eds) Matrix Pencils. Lecture Notes in Mathematics, vol Cited by: He was head of the Department of Computer Science and Engineering from to He received the Doctorat d'Etat from the University of Grenoble (France) in His current research interests include numerical linear algebra, sparse matrix computations, iterative methods, parallel computing, and numerical methods for eigenvalue problems/5(9).

Solving large sparse eigenvalue problems Mario Berljafa Stefan Guttel 5 References 6 1 Introduction The rst use of rational Krylov methods was for the solution of large sparse eigenvalue problems Ax = Bx, where Aand Bare N Nmatrices and (;x) are the wanted eigen- and we can equivalently solve the standard eigenvalue problem H m yK mz.

3. Block Methods for Solving Sparse Linear Systems 4. A Recursive Analysis of Dissection Strategies 5. Applications of an Element Model for Gaussian Elimination 6. An Optimization Problem Arising from Tearing Methods II. Eigenvalue Problems 1.

A Bibliographical Tour of the Large, Sparse Generalized Eigenvalue Problem Edition: 1. This revised edition discusses numerical methods for computing eigenvalues and eigenvectors of large sparse matrices.

It provides an in-depth view of the numerical methods that are applicable for solving matrix eigenvalue problems that arise in various engineering and scientific applications. Each chapter was updated by shortening or deleting. Under the support of this award, the filter algorithm for solving large sparse eigenvalue problems was developed at Stanford to address the computational difficulties in the previous methods with the goal to enable accelerator simulations on then the world largest unclassified supercomputer at NERSC for this class of : Gene Golub, Kwok Ko.

by the second class of problems. Several books dealing with numerical methods for solving eigenvalue prob-lems involving symmetric (or Hermitian) matrices have been written and there are a few software packages both public and commercial available. The book by Parlett [] is an excellent treatise of the problem.

Despite a rather strongFile Size: 2MB. Numerical Methods for Large Eigenvalue Problems This book was originally published by Manchester University Press (Oxford rd, Manchester, UK) in -- (ISBN 0 1) and in the US under Halstead Press (John Wiley, ISBN 0 7).

It is currently out of print. 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. the past two decades for solving large sparse symmetric eigenvalue problems see, e.g., [5, 6, 7, 20, 26, 47, 35].

In most cases, these packages deal with the situation where a relatively small number of eigenvalues are sought on either end of the spectrum.

Computing eigenvalues located well inside the spectrum is often supported by provid. Solving Sparse Linear Least Squares Problems On Some Supercomputers By Using A Sequence Of Large Dense Blocks Article (PDF Available) in BIT 37(3) February with 50 Reads How we measure 'reads'.

The power of ARPACK is that it can compute only a specified subset of eigenvalue/eigenvector pairs. This is accomplished through the keyword following values of which are available.

which = 'LM': Eigenvalues with largest magnitude (eigs, eigsh), that is, largest eigenvalues in the euclidean norm of complex numbers.

which = 'SM': Eigenvalues with. The power of ARPACK is that it can compute only a specified subset of eigenvalue/eigenvector pairs.

This is accomplished through the keyword following values of which are available. which = 'LM': Eigenvalues with largest magnitude (eigs, eigsh), that is, largest eigenvalues in the euclidean norm of complex numbers.; which = 'SM': Eigenvalues with.

B = I (identity matrix), equation (1) is a standard eigenvalue problem. We are interested in problems where A and B are very large and sparse symmetric positive definite (SPD) matrices. Furthermore, they are supposed to have no general sparsity pattern.

In this case, the n eigenvalues of the system will be positive real and denoted byFile Size: KB. Solution of large, dense symmetric generalized eigenvalue problems using secondary storage Article (PDF Available) in ACM Transactions on Mathematical Software 14(3) September with.

Since the eigenvalues of A −1 are the reciprocals of the eigenvalues of A, the smallest eigenvalue of A is the largest eigenvalue of A −1, and we can compute it by applying the power method to A −1. This is not done by using the iteration x k = A – 1 x k – 1 but by solving the system A x k = x k – 1 for each k.

Stathopoulos A and Fischer C Reducing synchronization on the parallel Davidson method for the large sparse, eigenvalue problem Proceedings of the ACM/IEEE conference on Supercomputing, () Barnard S, Pothen A and Simon H A spectral algorithm for envelope reduction of sparse matrices Proceedings of the ACM/IEEE conference on.

The package is designed to compute a few eigenvalues and corresponding eigenvectors of large sparse or structured matrices, using the Implicitly Restarted Arnoldi Method (IRAM) or, in the case of symmetric matrices, the corresponding variant of.

I am aware of the paper "Some Modified Matrix Eigenvalue Problems" by Gene Golub, but do not see a way to efficiently incorporate the generalized inverse in the sparse case.

I am also aware of this question, but do not want to rely on a sophisticated solver for generalized eigenvalue problems; I'm really looking for something "not so different. Large-Scale Sparse Singular Value Computations.

Krylov subspace methods on supercomputers. SIAM J. Sci. Statist. Comput. 10(6): On solving the large sparse generalized eigenvalue problem. PhD thesis. University of Illinois at Urbana-Champaign, Urbana, by: Written for researchers in applied mathematics and scientific computing, this book discusses numerical methods for computing eigenvalues and eigenvectors of large sparse matrices.

It provides an in-depth view of the numerical methods that are applicable for solving matrix eigenvalue problems that arise in various engineering and scientific.

underlying potential. For fundamental problems of interest, the matrix dimension often exceeds and the number of nonzero matrix elements may saturate available storage on present-day leadership class facilities. We survey recent results and advances in solving this large sparse matrix eigenvalue problem.

Get this from a library. Numerical methods for large eigenvalue problems. [Y Saad; Society for Industrial and Applied Mathematics.] -- This revised edition discusses numerical methods for computing eigenvalues and eigenvectors of large sparse matrices.

It provides an in-depth view of the numerical methods that are applicable for. This book presents a unified treatment of recently developed techniques and current understanding about solving systems of linear equations and large scale eigenvalue problems on high-performance computers.

It provides a rapid introduction to the world of vector and parallel processing for these linear algebra applications. Topics include major elements of advanced.

Solving large sparse linear systems is a fundamental prob-lem in high performance scienti c and engineering comput-ing. Application developers face the challenge of predicting which solver will converge fastest or at all for a given linear system.

The time to solve di erent linear systems depends on var-ious factors. The main challenge is to. * A new chapter on iterative methods, including the powerful preconditioned conjugate-gradient method for solving symmetric, positive definite systems * An introduction to new methods for solving large, sparse eigenvalue problems including the popular implicitly-restarted Arnoldi and Jacobi-Davidson methods/5(3).

Iterative Methods for Solving Large-scale Eigenvalue Problems Chao Yang Lawrence Berkeley National Laboratory, Berkeley, CA I Large-scale eigenvalue problem Ax = x or Ax = Bx I A, B large, sparse, or structured I y Ax and y Bx can be computed e ciently I Set of nonlinear equations in x, since = xT Iterative Methods for Solving Large.

Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigenvalue problem, and the singular value decomposition. We consider dense, band and sparse matrices.

This book presents a unified treatment of recently developed techniques and current understanding about solving systems of linear equations and large scale eigenvalue problems on high-performance computers.

and solution of large sparse eigenvalue problems. Parallel two level block ILU Preconditioning techniques for solving large sparse. Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation (−) =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.

When k = 1, the vector is called simply an eigenvector, and the. To get good and useful answers, I suggest you make your question more specific. Try to avoid the word 'best' as it is hard to define what is the best.

How about, 'how to solve large sparse indefinite linear systems?' $\endgroup$ – Jan Jul 17 '13 at ARPACK: An implementation of the Implicitly Re-started Arnoldi Iteration that computes some of the eigenvalues and eigenvectors of a large sparse matrix.

Available from [email protected] under the directory scalapack - Lehoucq, Sorensen, et al. - (Show Context). The numerical solution of the eigenproblem for the eigenvalues and eigenvectors is generally the most time-consuming part of computing synthesis mechanisms on a supercomputer.

A team using Argonne Leadership Computing Facility’s supercomputer, Mira, has successfully tested an approach to harnessing Mira's massively parallel architecture to solve this part of the [email protected]{osti_, title = {Algorithms for sparse matrix eigenvalue problems.

[DBLKLN, block Lanczos algorithm with local reorthogonalization strategy]}, author = {Lewis, J.G.}, abstractNote = {Eigenvalue problems for an n by n matrix A where n is large and A is sparse are considered.

A is assumed to be unstructured: it cannot be reordered to have narrow .Book Description: Tremendous progress has been made in the scientific and engineering disciplines regarding the use of iterative methods for linear systems. This second edition gives an in-depth, up-to-date view of practical algorithms for solving large-scale linear systems of equations, including a wide range of the best methods available today/5(9).