From: [email-masked]
Date: Fri, 28 Feb 86 00:38:39 cst

INDEX

LANCZOS CODES WITH NO REORTHOGONALIZATION

The Argonne Library contains Lanczos codes with no reorthogonalization
for computing eigenvalues and eigenvectors of (1) real symmetric
matrices and (2) of Hermitian matrices,
and (3) for computing singular values and corresponding
singular vectors of real, rectangular matrices.  (8/84 codes)
Similar codes for computing eigenvalues and eigenvectors of
(4) factored inverses of real symmetric matrices, (5) of real symmetric
generalized problems, and (6) of diagonalizable, complex symmetric
matrices are available from the authors.   Block Lanczos codes
for computing a few extreme eigenvalues and corresponding
eigenvectors of real symmetric matrices are also available from
the authors.  The Block codes use limited reorthogonalization of
the Lanczos vectors with respect to converged Ritz vectors.
The authors are currently developing Lanczos codes for
diagonalizable nonsymmetric matrices which should be available
by the end of 1986.

             Jane Cullum and Ralph Willoughby
             IBM Research
             Yorktown Heights, N.Y. 10598
             Phone:  914-945-2227

Please contact the authors if you encounter any problems.
Detailed comments and documentation for these and for the other
Lanczos codes are available in the 2-volume book.
Anyone contemplating using these codes should read the detailed
documentation first.

       LANCZOS ALGORITHMS FOR LARGE SYMMETRIC EIGENVALUE COMPUTATIONS
       Jane K. Cullum and Ralph A. Willoughby
       Volume 1. Theory: ($30)  Volume 2. Programs ($50)
       Volumes 3 and 4 in Progress in Scientific Computing
       Series, Edited by S. Abarbanel, R. Glowinski, G. Golub,
       P. Henrici, H. O. Kreiss. Published by Birkhauser,
       Basel, Switzerland, January 1985.

Observed user problems with these codes have been (1) Failure to
define large enough working arrays; (2) Subtle bugs in the user-
supplied subroutines which define the matrix and perform the
matrix-vector multiplies or solves (see USPEC, CMATV, AMATV, BSOLV,
and LSOLV below), and (3) The BISEC subroutine assumes that no more
than KMAX/2 eigenvalues are to be computed in any subinterval supplied
to it.  These Lanczos codes assume that all of the required
computations are done in at least IBM double precision arithmetic.
The presence of bugs in the user-supplied subroutines should be
readily identifiable by the subsequent observed nonconvergence
of 'all' of the eigenvalues approximations.

LANCZOS CODES FOR REAL SYMMETRIC MATRICES (No reorthogonalization)

DOCUMENTATION:  The file LEVALHED contains the required documentation
for the Real Symmetric, the Hermitian, the Factored Inverse,
and the Generalized Real Symmetric Lanczos Codes with no
reorthogonalization.

LEVAL:  This is the main program for computing eigenvalues of a real
        symmetric matrix using a Lanczos recursion with no
        reorthogonalization. Eigenvalue/eigenvector computations proceed
        in 2 stages.  Eigenvalues must be computed first.  Then a
        second program LEVEC can be used to compute eigenvectors
        corresponding to each of the eigenvalues in a
        user-specified subset of the eigenvalues computed by LEVAL.
        LEVAL calls subroutines from the set of subroutines LEMULT
        and from the set of subroutines LESUB.  The USPEC and CMATV
        subroutines in the LEMULT file provided are samples
        of matrix-specification and matrix-vector multiply subroutines.
        Users may use these sample subroutines.  However, in general
        maximum efficiency will be achieved only when the user supplies
        similar subroutines which are specialized for the particular
        matrix or family of matrices being used.  The main Lanczos
        programs do not 'see' the user's matrix.  All information about
        the matrix is passed to the program as vectors of the form A*x.
        These subroutines must be efficient, consistent, and accurate.

LEVEC:  This is the Main Program for computing eigenvectors of a real
        symmetric matrix using the Lanczos recursion used in LEVAL.
        This program accepts eigenvalue
        approximations computed using corresponding LEVAL program.
        For each such eigenvalue supplied a corresponding
        eigenvector will be computed.

LEMULT: This file contains the subroutine LANCZS which computes
        the Lanczos matrices used in both LEVAL and LEVEC.  It
        also contains sample matrix specification subroutines
        USPEC and sample matrix-vector multiply subroutines
        CMATV for three sets of matrices, (1) Real, sparse,
        and symmetric matrices, (2) Diagonal matrices and
        (3) Poisson Matrices.  The user should devise
        analogous USPEC and CMATV subroutines for the particular
        matrices being used. Three additional subroutines,
        EXEVG, EXERR and EXVEC, are applicable only for the
        Poisson test matrices and compute respectively, the
        true eigenvalues of Poisson test matrices,
        the corresponding
        errors in the eigenvalue approximations computed
        by LEVAL, and the corresponding
        errors in the eigenvector approximations
        computed by LEVEC.

LESUB:  Contains the other subroutines used by the main
        programs for the Lanczos programs with no
        reorthogonalization for Real Symmetric matrices, for Hermitian
        matrices, for Factored Inverses of Real Symmetric
        Matrices, and for Generalized Real Symmetric Problems.
        In particular these subroutines are called by
        the two main programs LEVAL and LEVEC.

        BISEC:    Computes eigenvalues of real symmetric Lanczos
                  matrices by bisection
        INVERR:   Uses inverse iteration on real symmetric Lanczos
                  matrix to compute error estimates for computed
                  eigenvalue approximations.
        TNORM:    Computes scaling factors used in setting tolerances.
        LUMP:     'Combines' 'good' Lanczos eigenvalues.
        ISOEV:    Determines isolated 'good' Lanczos eigenvalues.
        PRTEST:   Checks for 'hidden' eigenvalues once convergence is
                  achieved.
        STURMI:   Determines initial sizes of Lanczos matrices used
                  in eigenvector computations.
        INVERM:   Computes Lanczos eigenvectors in LEVEC.
        LBISEC:   Recomputes eigenvalue for eigenvector computations.
        CINPRD:   Computes Hermitian inner products.
        LPERM:    Computes permutation of vector, used in factored
                  versions and generalized real symmetric codes.
        LECOMPAC: Optional program used to determine sparse format.

LEVALIO: Listing of input and output files which must be
         defined for LEVAL and sample input file for LEVAL.

LEVECIO: Listing of input and output files which must be
         defined for LEVEC and sample input file for LEVEC.

LANCZOS CODES FOR HERMITIAN MATRICES (No reorthogonalization)

HLEVAL: This is the main program for computing eigenvalues of a
        Hermitian matrix using a Lanczos recursion with no
        reorthogonalization. Eigenvalue/eigenvector computations proceed
        in 2 stages.  Eigenvalues must be computed first.  Then a
        second program HLEVEC can be used to compute eigenvectors
        corresponding to each of the eigenvalues in a
        user-specified subset of the eigenvalues computed by HLEVAL.
        HLEVAL calls subroutines from the set of subroutines HLEMULT
        and from the set of subroutines LESUB.  The USPEC and CMATV
        subroutines in the HLEMULT file provided are samples
        of matrix-specification and matrix-vector multiply subroutines.
        Users may use these sample subroutines.  However, in general,
        maximum efficiency will be achieved only when the user supplies
        similar subroutines which are specialized for the particular
        matrix or family of matrices being used.  The main Lanczos
        programs do not 'see' the user's matrix.  All information about
        the matrix is passed to the program as vectors of the form A*x.
        CMATV computations must be efficient, consistent and accurate.

HLEVEC: This is the Main Program for computing eigenvectors of a
        Hermitian matrix using a Lanczos recursion with no
        reorthogonalization.  Program accepts eigenvalue
        approximations computed using corresponding HLEVAL program.
        For each such eigenvalue supplied a corresponding
        eigenvector will be computed.

HLEMULT:This file contains the subroutine LANCZS which computes
        the Lanczos matrices used in both HLEVAL and HLEVEC.  It
        also contains sample matrix specification subroutines
        USPEC and sample matrix-vector multiply subroutines
        CMATV for four sets of matrices, (1) sparse and
        Hermitian matrices, (2) and (3) Hermitian 'Poisson' matrices
        cases I and II, and (4) Hermitian tridiagonal matrices.
        The user should devise analogous subroutines specific
        to the particular matrices being used. Two
        additional subroutines, EXEVG and HEXVEC, are applicable
        only for the 'Poisson' test matrices, case I, and
        compute respectively, the true eigenvalues of
        these test matrices, and the corresponding
        errors in the eigenvalue approximations computed
        by HLEVAL, and in the corresponding
        eigenvector approximations computed by HLEVEC.

HLEVALIO: Listing of input and output files which must be
          defined for HLEVAL and sample input file for HLEVAL.

HLEVECIO: Listing of input and output files which must be
          defined for HLEVEC and sample input file for HLEVEC.

LANCZOS CODES FOR REAL, RECTANGULAR MATRICES (No reorthogonalization)

DOCUMENTATION:  The file LSVALHED contains the required documentation
for the real, rectangular singular value and vector Lanczos computaions.

LSVAL:  This is the main program for computing singular values of a
        real, rectangular matrix using a Lanczos recursion with no
        reorthogonalization. Singular value/vector computations proceed
        in 2 stages.  Singular values must be computed first.  Then a
        second program LSVEC can be used to compute singluar vectors
        corresponding to each of the singular values in a
        user-specified subset of the singular values computed by LSVAL.
        LSVAL calls subroutines from the set of subroutines LSMULT
        and from the set of subroutines LSSUB.  The USPEC, STRAN and
        SVMAT subroutines in the LSMULT file provided are samples,
        of matrix-specification and matrix-vector multiply subroutines.
        LSVAL and LSVEC use both A*x and A-transpose*y subroutines.
        Users may use these sample subroutines.  However, in general
        maximum efficiency will be achieved only when the user supplies
        similar subroutines which are specialized for the particular
        matrix or family of matrices being used.  The main Lanczos
        programs do not 'see' the user's matrix.  All information about
        the matrix is passed to the program as vectors of the form A*x
        and A-transpose*y and these computations must be efficient,
        consistent and accurate.

LSVEC:  This is the main program for computing singular vectors of a
        real rectangular matrix using a Lanczos recursion with no
        reorthogonalization.  This program accepts singular value
        approximations computed using corresponding LSVAL program.
        For each such singular value supplied a corresponding
        singular vector will be computed.

LSMULT: This file contains the subroutine LANCZS which computes
        the Lanczos matrices used in both LSVAL and LSVEC.  It
        also contains sample matrix specification subroutines
        USPEC and sample matrix-vector multiply subroutines
        STRAN and SVMAT for two sets of matrices, (1) Real, sparse,
        and rectangular matrices and (2) Diagonal matrices.
        The user should devise analogous subroutines specific
        to the particular matrix being used.

LSSUB:  Contains the other subroutines used by the main
        programs for the singular value/vector Lanczos
        codes with no reorthogonalization. These subroutines
        are called by the two main programs LSVAL and LSVEC.

        BISEC:    Computes eigenvalues of real symmetric Lanczos
                  matrices by bisection
        INVERR:   Uses inverse iteration on real symmetric Lanczos
                  matrix to compute error estimates for computed
                  eigenvalue approximations.
        TNORM:    Computes scaling factors used in setting tolerances.
        LUMP:     'Combines' 'good' Lanczos eigenvalues.
        ISOEV:    Determines isolated 'good' Lanczos eigenvalues.
        PRTEST:   Checks for 'hidden' singular values once convergence
                  is achieved.
        STURMI:   Determines initial sizes of Lanczos matrices used
                  in singular vector computations.
        INVERM:   Computes Lanczos eigenvectors in LEVEC.
        LBISEC:   Recomputes singular value for singular vector
                  computations.
        LSCOMPAC: Optional program used to determine sparse format.

LSVALIO: Listing of input and output files which must be
         defined for LSVAL and sample input file for LSVAL.

LSVECIO: Listing of input and output files which must be
         defined for LSVEC and sample input file for LSVEC.