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Abstract: A procedure is presented for the computation of bounds to eigenvalues of the generalized hermitian eigenvalue problem and to the standard hermitian eigenvalue problem. This procedure is applicable to iterative subspace eigenvalue methods and to both outer and inner eigenvalues. The Ritz values and their corresponding residual norms, all of which are computable quantities, are needed by the procedure. Knowledge of the exact eigenvalues is not needed by the procedure, but it must be known that the computed Ritz values are isolated from exact eigenvalues outside of the Ritz spectrum and that there are no skipped eigenvalues within the Ritz spectrum range. A multipass refinement procedure is described to compute the bounds for each Ritz value. This procedure requires effort where m is the subspace dimension for each pass. [Copyright &y& Elsevier]

Abstract: This paper describes software for computing eigenvalue bounds to the standard and generalized hermitian eigenvalue problem as described in [Y. Zhou, R. Shepard, M. Minkoff, Computing eigenvalue bounds for iterative subspace matrix methods, Comput. Phys. Comm. 167 (2005) 90–102]. The software discussed in this manuscript applies to any subspace method, including Lanczos, Davidson, SPAM, Generalized Davidson Inverse Iteration, Jacobi–Davidson, and the Generalized Jacobi–Davidson methods, and it is applicable to either outer or inner eigenvalues. This software can be applied during the subspace iterations in order to truncate the iterative process and to avoid unnecessary effort when converging specific eigenvalues to a required target accuracy, and it can be applied to the final set of Ritz values to assess the accuracy of the converged results. Program summary: Title of program: SUBROUTINE BOUNDS_OPT Catalogue identifier: ADVE Program obtainable from: CPC Program Library, Queen''s University of Belfast, N. Ireland Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADVE Computers: any computer that supports a Fortran 90 compiler Operating systems: any computer that supports a Fortran 90 compiler Programming language: Standard Fortran 90 High speed storage required: working-precision and integer for m Ritz values No. of bits in a word: The floating point working precision is parameterized with the symbolic constant WP No. of lines in distributed program, including test data, etc.: 2452 No. of bytes in distributed program, including test data, etc.: 281 543 Distribution format: tar.gz Nature of physical 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. Method of solution: 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. Typical running time: While typical applications would deal with , for , the running time is 0.12 s on an Apple PowerBook. [Copyright &y& Elsevier]