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Manuscript Title: Software for Computing Eigenvalue Bounds for Iterative Subspace Matrix Methods.
Authors: Ron Shepard, Michael Minkoff, Yunkai Zhou
Program title: SUBROUTINE BOUNDS_OPT
Catalogue identifier: ADVE_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 170(2005)109
Programming language: Standard FORTRAN 90.
Computer: any computer that supports a Fortran 90 compiler.
Operating system: any computer that supports a Fortran 90 compiler.
RAM: 5m+5 working-precision and 2m+7 integer for m Ritz values
Word size: The floating point working precision is parameterized with the symbolic constant WP.
Keywords: Bounds, Eigenvalue, Subspace, Ritz, Hermitian, Generalized, Gap, Spread.
PACS: 02.10, 02.60, 02.70, 89.80.
Classification: 4.8.

Nature of problem:
The computational solution of eigenvalue problems using iterative subspace methods has widespread applications in the physical sciences and engineering as well as other areas of mathematical modeling (economics, social sciences, etc.). The accuracy of the solution of such problems and the utility of those errors is a fundamental problem that is of importance in order to provide the modeler with information of the reliability of the computational results. Such applications include using these bounds to terminate the iterative procedure at specified accuracy limits.

Solution method:
The Ritz values and their residual norms are computed and used as input for the procedure. While knowledge of the exact eigenvalues is not required, we require that the Ritz values are isolated from the exact eigenvalues outside of the Ritz spectrum and that there are no skipped eigenvalues within the Ritz spectrum. Using a multipass refinement approach, upper and lower bounds are computed for each Ritz value.

Running time:
While typical applications would deal with m<20, for m=100,000, the running time is 0.12 s. on an Apple PowerBook.