PDSYGVX(l)

NAME
PDSYGVX - compute all the eigenvalues, and optionally, the eigenvectors of a real generalized SY-definite eigenproblem

SYNOPSIS

SUBROUTINE PDSYGVX(
    IBTYPE, JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO )

CHARACTER
    JOBZ, RANGE, UPLO

INTEGER
    IA, IB, IBTYPE, IL, INFO, IU, IZ, JA, JB, JZ, LIWORK, LWORK, M, N, NZ

DOUBLE
    PRECISION ABSTOL, ORFAC, VL, VU

INTEGER
    DESCA( * ), DESCB( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * )

DOUBLE
    PRECISION A( * ), B( * ), GAP( * ), W( * ), WORK( * ), Z( * )

INTEGER
    BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_

PARAMETER
    ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 )

DOUBLE
    PRECISION ONE

PARAMETER
    ( ONE = 1.0D+0 )

DOUBLE
    PRECISION FIVE, ZERO

PARAMETER
    ( FIVE = 5.0D+0, ZERO = 0.0D+0 )

INTEGER
    IERRNPD

PARAMETER
    ( IERRNPD = 16 )

LOGICAL
    ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ

CHARACTER
    TRANS

INTEGER
    IACOL, IAROW, IBCOL, IBROW, ICOFFA, ICOFFB, ICTXT, IROFFA, IROFFB, LIWMIN, LWMIN, MQ0, MYCOL, MYROW, NB, NEIG, NN, NP0, NPCOL, NPROW

DOUBLE
    PRECISION EPS, SCALE

INTEGER
    IDUM1( 5 ), IDUM2( 5 )

LOGICAL
    LSAME

INTEGER
    ICEIL, INDXG2P, NUMROC

DOUBLE
    PRECISION PDLAMCH

EXTERNAL
    LSAME, ICEIL, INDXG2P, NUMROC, PDLAMCH

EXTERNAL
    BLACS_GRIDINFO, CHK1MAT, DGEBR2D, DGEBS2D, DSCAL, PCHK1MAT, PCHK2MAT, PDPOTRF, PDSYEVX, PDSYGST, PDTRMM, PDTRSM, PXERBLA

INTRINSIC
    ABS, DBLE, ICHAR, MAX, MIN, MOD

PURPOSE
PDSYGVX computes all the eigenvalues, and optionally, the eigenvectors of a real generalized SY-definite eigenproblem, of the form sub( A )*x=(lambda)*sub( B )*x, sub( A )*sub( B )x=(lambda)*x, or sub( B )*sub( A )*x=(lambda)*x. Here sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ) is assumed to be SY, and sub( B ) denoting B( IB:IB+N-1, JB:JB+N-1 ) is assumed to be symmetric positive definite.

NOTES
Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location.

Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array".

NOTATION STORED IN EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
                        DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                        the BLACS process grid A is distribu-
                        ted over. The context itself is glo-
                        bal, but the handle (the integer
                        value) may vary.
M_A (global) DESCA( M_ ) The number of rows in the global
                        array A.
N_A (global) DESCA( N_ ) The number of columns in the global
                        array A.
MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
                        the rows of the array.
NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
                        the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                        row of the array A is distributed.
CSRC_A (global) DESCA( CSRC_ ) The process column over which the
                        first column of the array A is
                        distributed.
LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
                        array. LLD_A >= MAX(1,LOCr(M_A)).

Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column.
Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row.
The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC:
        LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
        LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
An upper bound for these quantities may be computed by:
        LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
        LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

ARGUMENTS

IBTYPE (global input) INTEGER
        Specifies the problem type to be solved:
        = 1: sub( A )*x = (lambda)*sub( B )*x
        = 2: sub( A )*sub( B )*x = (lambda)*x
        = 3: sub( B )*sub( A )*x = (lambda)*x

JOBZ (global input) CHARACTER*1
        = 'N': Compute eigenvalues only;
        = 'V': Compute eigenvalues and eigenvectors.

RANGE (global input) CHARACTER*1
        = 'A': all eigenvalues will be found.
        = 'V': all eigenvalues in the interval [VL,VU] will be found.
        = 'I': the IL-th through IU-th eigenvalues will be found.

UPLO (global input) CHARACTER*1
        = 'U': Upper triangles of sub( A ) and sub( B ) are stored;
        = 'L': Lower triangles of sub( A ) and sub( B ) are stored.

N (global input) INTEGER
        The order of the matrices sub( A ) and sub( B ). N >= 0.

A (local input/local output) DOUBLE PRECISION pointer into the
        local memory to an array of dimension (LLD_A, LOCc(JA+N-1)).
        On entry, this array contains the local pieces of the
        N-by-N symmetric distributed matrix sub( A ). If UPLO = 'U',
        the leading N-by-N upper triangular part of sub( A ) contains
        the upper triangular part of the matrix. If UPLO = 'L', the
        leading N-by-N lower triangular part of sub( A ) contains
        the lower triangular part of the matrix.

        On exit, if JOBZ = 'V', then if INFO = 0, sub( A ) contains
        the distributed matrix Z of eigenvectors. The eigenvectors
        are normalized as follows:
        if IBTYPE = 1 or 2, Z**T*sub( B )*Z = I;
        if IBTYPE = 3, Z**T*inv( sub( B ) )*Z = I.
        If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
        or the lower triangle (if UPLO='L') of sub( A ), including
        the diagonal, is destroyed.

IA (global input) INTEGER
        The row index in the global array A indicating the first
        row of sub( A ).

JA (global input) INTEGER
        The column index in the global array A indicating the
        first column of sub( A ).

DESCA (global and local input) INTEGER array of dimension DLEN_.
        The array descriptor for the distributed matrix A.
        If DESCA( CTXT_ ) is incorrect, PDSYGVX cannot guarantee
        correct error reporting.

B (local input/local output) DOUBLE PRECISION pointer into the
        local memory to an array of dimension (LLD_B, LOCc(JB+N-1)).
        On entry, this array contains the local pieces of the
        N-by-N symmetric distributed matrix sub( B ). If UPLO = 'U',
        the leading N-by-N upper triangular part of sub( B ) contains
        the upper triangular part of the matrix. If UPLO = 'L', the
        leading N-by-N lower triangular part of sub( B ) contains
        the lower triangular part of the matrix.

        On exit, if INFO <= N, the part of sub( B ) containing the
        matrix is overwritten by the triangular factor U or L from
        the Cholesky factorization sub( B ) = U**T*U or
        sub( B ) = L*L**T.

IB (global input) INTEGER
        The row index in the global array B indicating the first
        row of sub( B ).

JB (global input) INTEGER
        The column index in the global array B indicating the
        first column of sub( B ).

DESCB (global and local input) INTEGER array of dimension DLEN_.
        The array descriptor for the distributed matrix B.
        DESCB( CTXT_ ) must equal DESCA( CTXT_ )

VL (global input) DOUBLE PRECISION
        If RANGE='V', the lower bound of the interval to be searched
        for eigenvalues. Not referenced if RANGE = 'A' or 'I'.

VU (global input) DOUBLE PRECISION
        If RANGE='V', the upper bound of the interval to be searched
        for eigenvalues. Not referenced if RANGE = 'A' or 'I'.

IL (global input) INTEGER
        If RANGE='I', the index (from smallest to largest) of the
        smallest eigenvalue to be returned. IL >= 1.
        Not referenced if RANGE = 'A' or 'V'.

IU (global input) INTEGER
        If RANGE='I', the index (from smallest to largest) of the
        largest eigenvalue to be returned. min(IL,N) <= IU <= N.
        Not referenced if RANGE = 'A' or 'V'.

ABSTOL (global input) DOUBLE PRECISION
        If JOBZ='V', setting ABSTOL to PDLAMCH( CONTEXT, 'U') yields
        the most orthogonal eigenvectors.

        The absolute error tolerance for the eigenvalues.
        An approximate eigenvalue is accepted as converged
        when it is determined to lie in an interval [a,b]
        of width less than or equal to

                ABSTOL + EPS * max( |a|,|b| ) ,

        where EPS is the machine precision. If ABSTOL is less than
        or equal to zero, then EPS*norm(T) will be used in its place,
        where norm(T) is the 1-norm of the tridiagonal matrix
        obtained by reducing A to tridiagonal form.

        Eigenvalues will be computed most accurately when ABSTOL is
        set to twice the underflow threshold 2*PDLAMCH('S') not zero.
        If this routine returns with ((MOD(INFO,2).NE.0) .OR.
        (MOD(INFO/8,2).NE.0)), indicating that some eigenvalues or
        eigenvectors did not converge, try setting ABSTOL to
        2*PDLAMCH('S').

        See "Computing Small Singular Values of Bidiagonal Matrices
        with Guaranteed High Relative Accuracy," by Demmel and
        Kahan, LAPACK Working Note #3.

        See "On the correctness of Parallel Bisection in Floating
        Point" by Demmel, Dhillon and Ren, LAPACK Working Note #70

M (global output) INTEGER
        Total number of eigenvalues found. 0 <= M <= N.

NZ (global output) INTEGER
        Total number of eigenvectors computed. 0 <= NZ <= M.
        The number of columns of Z that are filled.
        If JOBZ .NE. 'V', NZ is not referenced.
        If JOBZ .EQ. 'V', NZ = M unless the user supplies
        insufficient space and PDSYGVX is not able to detect this
        before beginning computation. To get all the eigenvectors
        requested, the user must supply both sufficient
        space to hold the eigenvectors in Z (M .LE. DESCZ(N_))
        and sufficient workspace to compute them. (See LWORK below.)
        PDSYGVX is always able to detect insufficient space without
        computation unless RANGE .EQ. 'V'.

W (global output) DOUBLE PRECISION array, dimension (N)
        On normal exit, the first M entries contain the selected
        eigenvalues in ascending order.

ORFAC (global input) DOUBLE PRECISION
        Specifies which eigenvectors should be reorthogonalized.
        Eigenvectors that correspond to eigenvalues which are within
        tol=ORFAC*norm(A) of each other are to be reorthogonalized.
        However, if the workspace is insufficient (see LWORK),
        tol may be decreased until all eigenvectors to be
        reorthogonalized can be stored in one process.
        No reorthogonalization will be done if ORFAC equals zero.
        A default value of 10^-3 is used if ORFAC is negative.
        ORFAC should be identical on all processes.

Z (local output) DOUBLE PRECISION array,
        global dimension (N, N),
        local dimension ( LLD_Z, LOCc(JZ+N-1) )
        If JOBZ = 'V', then on normal exit the first M columns of Z
        contain the orthonormal eigenvectors of the matrix
        corresponding to the selected eigenvalues. If an eigenvector
        fails to converge, then that column of Z contains the latest
        approximation to the eigenvector, and the index of the
        eigenvector is returned in IFAIL.
        If JOBZ = 'N', then Z is not referenced.

IZ (global input) INTEGER
        The row index in the global array Z indicating the first
        row of sub( Z ).

JZ (global input) INTEGER
        The column index in the global array Z indicating the
        first column of sub( Z ).

DESCZ (global and local input) INTEGER array of dimension DLEN_.
        The array descriptor for the distributed matrix Z.
        DESCZ( CTXT_ ) must equal DESCA( CTXT_ )

WORK (local workspace/output) DOUBLE PRECISION array,
           dimension (LWORK)
        if JOBZ='N' WORK(1) = optimal amount of workspace
           required to compute eigenvalues efficiently
        if JOBZ='V' WORK(1) = optimal amount of workspace
           required to compute eigenvalues and eigenvectors
           efficiently with no guarantee on orthogonality.
           If RANGE='V', it is assumed that all eigenvectors
           may be required.

LWORK (local input) INTEGER
        See below for definitions of variables used to define LWORK.
        If no eigenvectors are requested (JOBZ = 'N') then
           LWORK >= 5 * N + MAX( 5 * NN, NB * ( NP0 + 1 ) )
        If eigenvectors are requested (JOBZ = 'V' ) then
           the amount of workspace required to guarantee that all
           eigenvectors are computed is:
           LWORK >= 5 * N + MAX( 5*NN, NP0 * MQ0 + 2 * NB * NB ) +
             ICEIL( NEIG, NPROW*NPCOL)*NN

           The computed eigenvectors may not be orthogonal if the
           minimal workspace is supplied and ORFAC is too small.
           If you want to guarantee orthogonality (at the cost
           of potentially poor performance) you should add
           the following to LWORK:
              (CLUSTERSIZE-1)*N
           where CLUSTERSIZE is the number of eigenvalues in the
           largest cluster, where a cluster is defined as a set of
           close eigenvalues: { W(K),...,W(K+CLUSTERSIZE-1) |
                        W(J+1) <= W(J) + ORFAC*2*norm(A) }
        Variable definitions:
           NEIG = number of eigenvectors requested
           NB = DESCA( MB_ ) = DESCA( NB_ ) = DESCZ( MB_ ) =
                DESCZ( NB_ )
           NN = MAX( N, NB, 2 )
           DESCA( RSRC_ ) = DESCA( NB_ ) = DESCZ( RSRC_ ) =
                        DESCZ( CSRC_ ) = 0
           NP0 = NUMROC( NN, NB, 0, 0, NPROW )
           MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, 0, 0, NPCOL )
           ICEIL( X, Y ) is a ScaLAPACK function returning
           ceiling(X/Y)

        When LWORK is too small:
           If LWORK is too small to guarantee orthogonality,
           PDSYGVX attempts to maintain orthogonality in
           the clusters with the smallest
           spacing between the eigenvalues.
           If LWORK is too small to compute all the eigenvectors
           requested, no computation is performed and INFO=-23
           is returned. Note that when RANGE='V', PDSYGVX does
           not know how many eigenvectors are requested until
           the eigenvalues are computed. Therefore, when RANGE='V'
           and as long as LWORK is large enough to allow PDSYGVX to
           compute the eigenvalues, PDSYGVX will compute the
           eigenvalues and as many eigenvectors as it can.

        Relationship between workspace, orthogonality & performance:
           Greater performance can be achieved if adequate workspace
           is provided. On the other hand, in some situations,
           performance can decrease as the workspace provided
           increases above the workspace amount shown below:

           For optimal performance, greater workspace may be
           needed, i.e.
              LWORK >= MAX( LWORK, 5 * N + NSYTRD_LWOPT,
                NSYGST_LWOPT )
              Where:
                LWORK, as defined previously, depends upon the number
                   of eigenvectors requested, and
                NSYTRD_LWOPT = N + 2*( ANB+1 )*( 4*NPS+2 ) +
                  ( NPS + 3 ) * NPS
                NSYGST_LWOPT = 2*NP0*NB + NQ0*NB + NB*NB

                ANB = PJLAENV( DESCA( CTXT_), 3, 'PDSYTTRD', 'L',
                   0, 0, 0, 0)
                SQNPC = INT( SQRT( DBLE( NPROW * NPCOL ) ) )
                NPS = MAX( NUMROC( N, 1, 0, 0, SQNPC ), 2*ANB )
                NB = DESCA( MB_ )
                NP0 = NUMROC( N, NB, 0, 0, NPROW )
                NQ0 = NUMROC( N, NB, 0, 0, NPCOL )

                NUMROC is a ScaLAPACK tool functions;
                PJLAENV is a ScaLAPACK envionmental inquiry function
                MYROW, MYCOL, NPROW and NPCOL can be determined by
                  calling the subroutine BLACS_GRIDINFO.

              For large N, no extra workspace is needed, however the
              biggest boost in performance comes for small N, so it
              is wise to provide the extra workspace (typically less
              than a Megabyte per process).

           If CLUSTERSIZE >= N/SQRT(NPROW*NPCOL), then providing
           enough space to compute all the eigenvectors
           orthogonally will cause serious degradation in
           performance. In the limit (i.e. CLUSTERSIZE = N-1)
           PDSTEIN will perform no better than DSTEIN on 1 processor.
           For CLUSTERSIZE = N/SQRT(NPROW*NPCOL) reorthogonalizing
           all eigenvectors will increase the total execution time
           by a factor of 2 or more.
           For CLUSTERSIZE > N/SQRT(NPROW*NPCOL) execution time will
           grow as the square of the cluster size, all other factors
           remaining equal and assuming enough workspace. Less
           workspace means less reorthogonalization but faster
           execution.

        If LWORK = -1, then LWORK is global input and a workspace
        query is assumed; the routine only calculates the size
        required for optimal performance on all work arrays.
        Each of these values is returned in the first entry of the
        corresponding work array, and no error message is issued by
        PXERBLA.

IWORK (local workspace) INTEGER array
        On return, IWORK(1) contains the amount of integer workspace
        required.

LIWORK (local input) INTEGER
        size of IWORK
        LIWORK >= 6 * NNP
        Where:
          NNP = MAX( N, NPROW*NPCOL + 1, 4 )

        If LIWORK = -1, then LIWORK is global input and a workspace
        query is assumed; the routine only calculates the minimum
        and optimal size for all work arrays. Each of these
        values is returned in the first entry of the corresponding
        work array, and no error message is issued by PXERBLA.

IFAIL (output) INTEGER array, dimension (N)
        IFAIL provides additional information when INFO .NE. 0
        If (MOD(INFO/16,2).NE.0) then IFAIL(1) indicates the order of
        the smallest minor which is not positive definite.
        If (MOD(INFO,2).NE.0) on exit, then IFAIL contains the
        indices of the eigenvectors that failed to converge.

        If neither of the above error conditions hold and JOBZ = 'V',
        then the first M elements of IFAIL are set to zero.

ICLUSTR (global output) integer array, dimension (2*NPROW*NPCOL)
        This array contains indices of eigenvectors corresponding to
        a cluster of eigenvalues that could not be reorthogonalized
        due to insufficient workspace (see LWORK, ORFAC and INFO).
        Eigenvectors corresponding to clusters of eigenvalues indexed
        ICLUSTR(2*I-1) to ICLUSTR(2*I), could not be
        reorthogonalized due to lack of workspace. Hence the
        eigenvectors corresponding to these clusters may not be
        orthogonal. ICLUSTR() is a zero terminated array.
        (ICLUSTR(2*K).NE.0 .AND. ICLUSTR(2*K+1).EQ.0) if and only if
        K is the number of clusters
        ICLUSTR is not referenced if JOBZ = 'N'

GAP (global output) DOUBLE PRECISION array,
           dimension (NPROW*NPCOL)
        This array contains the gap between eigenvalues whose
        eigenvectors could not be reorthogonalized. The output
        values in this array correspond to the clusters indicated
        by the array ICLUSTR. As a result, the dot product between
        eigenvectors correspoding to the I^th cluster may be as high
        as ( C * n ) / GAP(I) where C is a small constant.

INFO (global output) INTEGER
        = 0: successful exit
        < 0: If the i-th argument is an array and the j-entry had
              an illegal value, then INFO = -(i*100+j), if the i-th
              argument is a scalar and had an illegal value, then
              INFO = -i.
        > 0: if (MOD(INFO,2).NE.0), then one or more eigenvectors
                failed to converge. Their indices are stored
                in IFAIL. Send e-mail to scalapack@cs.utk.edu
              if (MOD(INFO/2,2).NE.0),then eigenvectors corresponding
                to one or more clusters of eigenvalues could not be
                reorthogonalized because of insufficient workspace.
                The indices of the clusters are stored in the array
                ICLUSTR.
              if (MOD(INFO/4,2).NE.0), then space limit prevented
                PDSYGVX from computing all of the eigenvectors
                between VL and VU. The number of eigenvectors
                computed is returned in NZ.
              if (MOD(INFO/8,2).NE.0), then PDSTEBZ failed to
                compute eigenvalues.
                Send e-mail to scalapack@cs.utk.edu
              if (MOD(INFO/16,2).NE.0), then B was not positive
                definite. IFAIL(1) indicates the order of
                the smallest minor which is not positive definite.