Implicitly Restarted Arnoldi/lanczos Methods for Large Scale Eigenvalue Calculations
نویسنده
چکیده
This report provides an introductory overview of the numerical solution of large scale algebraic eigenvalue problems. The main focus is on a class of methods called Krylov subspace projection methods. The Lanczos method is the premier member of this class and the Arnoldi method is a generalization to the nonsymmetric case. A recently developed and very promising variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in detail. This method is highlighted because of its suitability as a basis for software development. It may be viewed as a truncated form of the implicitly shifted QR-algorithm that is appropriate for very large problems. Based on this technique, a public domain software package called ARPACK has been developed in Fortran 77 for nding a few eigenvalues and eigenvectors of large scale symmetric, nonsymmetric, standard or generalized problems. This package has performed well on workstations, parallel-vector supercomputers, distributed memory parallel systems and clusters of workstations. The important features of this package are presented along with a discussion some applications and performance indicators. AMS classi cation: Primary 65F15; Secondary 65G05
منابع مشابه
Efficient Computation of the Maximum Eigenvalue of Large Symmetric Matrices
Though the implicitly restarted Arnoldi/Lanczos method in ARPACK is a reliable method for computing a few eigenvalues of large-scale matrices, it can be inefficient because it only checks for convergence at restarts. Significant savings in runtime can be obtained by checking convergence at each Lanczos iteration. We describe a new convergence test for the maximum eigenvalue that is numerically ...
متن کاملThick-Restart Lanczos Method for Symmetric Eigenvalue Problems
For real symmetric eigenvalue problems, there are a number of algorithms that are mathematically equivalent, for example, the Lanczos algorithm, the Arnoldi method and the unpreconditioned Davidson method. The Lanczos algorithm is often preferred because it uses signiicantly fewer arithmetic operations per iteration. To limit the maximum memory usage, these algorithms are often restarted. In re...
متن کاملAccelerating the LSTRS Algorithm
In a recent paper [Rojas, Santos, Sorensen: ACM ToMS 34 (2008), Article 11] an efficient method for solving the Large-Scale Trust-Region Subproblem was suggested which is based on recasting it in terms of a parameter dependent eigenvalue problem and adjusting the parameter iteratively. The essential work at each iteration is the solution of an eigenvalue problem for the smallest eigenvalue of t...
متن کاملImplicitly Restarted Arnoldi Methods and Eigenvalues of the Discretized Navier Stokes Equations
Implicitly restarted Arnoldi methods and eigenvalues of the discretized Navier Stokes equations. Abstract We are concerned with nding a few eigenvalues of the large sparse nonsymmetric generalized eigenvalue problem Ax = Bx that arises in stability studies of incompressible uid ow. The matrices have a block structure that is typical of mixed nite-element discretizations for such problems. We ex...
متن کامل