Large-Scale Computation of Pseudospectra Using ARPACK and Eigs

The Arnoldi Eigenvalue Iteration with Exact Shifts Can Fail

The restarted Arnoldi algorithm, implemented in the ARPACK software library and MATLAB’s eigs command, is among the most common means of computing select eigenvalues and eigenvectors of a large, sparse matrix. To assist convergence, a starting vector is repeatedly refined via the application of automatically-constructed polynomial filters whose roots are known as ‘exact shifts’. Though Sorensen...

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 ...

A reduced basis approach to large-scale pseudospectra computation

For studying spectral properties of a non-normal matrix A ∈ Cn×n, information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε-pseudospectra, that is the level-sets of the resolvent norm function g(z) = ‖(zI−A)‖2. The computation of ε-pseudospectra requires determining the smallest singular values σmin(zI − A) on for all z on a ...

ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods

Crpc Research Into Linear Algebra Software for High Performance Computers

In this paper we look at a number of approaches being investigated in the Center for Research on Parallel Computation (CRPC) to develop linear algebra software for high-performance computers. These approaches are exempliied by the LAPACK, templates, and ARPACK projects. LAPACK is a software library for performing dense and banded linear algebra computations, and was designed to run eeciently on...