pdsyev.f

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      SUBROUTINE pdsyev( JOBZ, UPLO, N, A, IA, JA, DESCA, W,
     $                   Z, IZ, JZ, DESCZ, WORK, LWORK, INFO )
*
*  -- ScaLAPACK routine (version 1.7) --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
*     May 25, 2001
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            IA, INFO, IZ, JA, JZ, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            DESCA( * ), DESCZ( * )
      DOUBLE PRECISION   A( * ), W( * ), WORK( * ), Z( * )
*     ..
*
*  Purpose
*  =======
*
*  PDSYEV computes all eigenvalues and, optionally, eigenvectors
*  of a real symmetric matrix A by calling the recommended sequence
*  of ScaLAPACK routines.
*
*  In its present form, PDSYEV assumes a homogeneous system and makes
*  no checks for consistency of the eigenvalues or eigenvectors across
*  the different processes.  Because of this, it is possible that a
*  heterogeneous system may return incorrect results without any error
*  messages.
*
*  Notes
*  =====
*  A description vector is associated with each 2D block-cyclicly dis-
*  tributed matrix.  This vector stores the information required to
*  establish the mapping between a matrix entry and its corresponding
*  process and memory location.
*
*  In the following comments, the character _ should be read as
*  "of the distributed matrix".  Let A be a generic term for any 2D
*  block cyclicly distributed matrix.  Its description vector is DESCA:
*
*  NOTATION        STORED IN      EXPLANATION
*  --------------- -------------- --------------------------------------
*  DTYPE_A(global) DESCA( DTYPE_) The descriptor type.
*  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 distributed
*                                 matrix A.
*  N_A    (global) DESCA( N_ )    The number of columns in the distri-
*                                 buted matrix A.
*  MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
*                                 the rows of A.
*  NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
*                                 the columns of A.
*  RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
*                                 row of the matrix A is distributed.
*  CSRC_A (global) DESCA( CSRC_ ) The process column over which the
*                                 first column of A is distributed.
*  LLD_A  (local)  DESCA( LLD_ )  The leading dimension of the local
*                                 array storing the local blocks of the
*                                 distributed matrix A.
*                                 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 ).
*
*  Arguments
*  =========
*
*     NP = the number of rows local to a given process.
*     NQ = the number of columns local to a given process.
*
*  JOBZ    (global input) CHARACTER*1
*          Specifies whether or not to compute the eigenvectors:
*          = 'N':  Compute eigenvalues only.
*          = 'V':  Compute eigenvalues and eigenvectors.
*
*  UPLO    (global input) CHARACTER*1
*          Specifies whether the upper or lower triangular part of the
*          symmetric matrix A is stored:
*          = 'U':  Upper triangular
*          = 'L':  Lower triangular
*
*  N       (global input) INTEGER
*          The number of rows and columns of the matrix A.  N >= 0.
*
*  A       (local input/workspace) block cyclic DOUBLE PRECISION array,
*          global dimension (N, N), local dimension ( LLD_A,
*          LOCc(JA+N-1) )
*
*          On entry, the symmetric matrix A.  If UPLO = 'U', only the
*          upper triangular part of A is used to define the elements of
*          the symmetric matrix.  If UPLO = 'L', only the lower
*          triangular part of A is used to define the elements of the
*          symmetric matrix.
*
*          On exit, the lower triangle (if UPLO='L') or the upper
*          triangle (if UPLO='U') of A, including the diagonal, is
*          destroyed.
*
*  IA      (global input) INTEGER
*          A's global row index, which points to the beginning of the
*          submatrix which is to be operated on.
*
*  JA      (global input) INTEGER
*          A's global column index, which points to the beginning of
*          the submatrix which is to be operated on.
*
*  DESCA   (global and local input) INTEGER array of dimension DLEN_.
*          The array descriptor for the distributed matrix A.
*          If DESCA( CTXT_ ) is incorrect, PDSYEV cannot guarantee
*          correct error reporting.
*
*  W       (global output) DOUBLE PRECISION array, dimension (N)
*          If INFO=0, the eigenvalues in ascending order.
*
*  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 JOBZ = 'N', then Z is not referenced.
*
*  IZ      (global input) INTEGER
*          Z's global row index, which points to the beginning of the
*          submatrix which is to be operated on.
*
*  JZ      (global input) INTEGER
*          Z's global column index, which points to the beginning of
*          the submatrix which is to be operated on.
*
*  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)
*          Version 1.0:  on output, WORK(1) returns the workspace
*          needed to guarantee completion.
*          If the input parameters are incorrect, WORK(1) may also be
*          incorrect.
*
*          If JOBZ='N' WORK(1) = minimal=optimal amount of workspace
*          If JOBZ='V' WORK(1) = minimal workspace required to
*             generate all the eigenvectors.
*
*
*  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 + SIZESYTRD + 1
*          where
*             SIZESYTRD = The workspace requirement for PDSYTRD
*                         and is MAX( NB * ( NP +1 ), 3 * NB )
*          If eigenvectors are requested (JOBZ = 'V' ) then
*             the amount of workspace required to guarantee that all
*             eigenvectors are computed is:
*
*             QRMEM = 2*N-2
*             LWMIN = 5*N + N*LDC + MAX( SIZEMQRLEFT, QRMEM ) + 1
*
*          Variable definitions:
*             NB = DESCA( MB_ ) = DESCA( NB_ ) =
*                  DESCZ( MB_ ) = DESCZ( NB_ )
*             NN = MAX( N, NB, 2 )
*             DESCA( RSRC_ ) = DESCA( RSRC_ ) = DESCZ( RSRC_ ) =
*                              DESCZ( CSRC_ ) = 0
*             NP = NUMROC( NN, NB, 0, 0, NPROW )
*             NQ = NUMROC( MAX( N, NB, 2 ), NB, 0, 0, NPCOL )
*             NRC = NUMROC( N, NB, MYPROWC, 0, NPROCS)
*             LDC = MAX( 1, NRC )
*             SIZEMQRLEFT = The workspace requirement for PDORMTR
*                           when it's SIDE argument is 'L'.
*
*          With MYPROWC defined when a new context is created as:
*             CALL BLACS_GET( DESCA( CTXT_ ), 0, CONTEXTC )
*             CALL BLACS_GRIDINIT( CONTEXTC, 'R', NPROCS, 1 )
*             CALL BLACS_GRIDINFO( CONTEXTC, NPROWC, NPCOLC, MYPROWC,
*                                  MYPCOLC )
*
*          If LWORK = -1, the LWORK is global input and a workspace
*          query is assumed; the routine only calculates the minimum
*          size for the WORK array.  The required workspace is returned
*          as the first element of WORK and no error message is issued
*          by PXERBLA.
*
*  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 INFO = 1 through N, the i(th) eigenvalue did not
*                converge in DSTEQR2 after a total of 30*N iterations.
*                If INFO = N+1, then PDSYEV has detected heterogeneity
*                by finding that eigenvalues were not identical across
*                the process grid.  In this case, the accuracy of
*                the results from PDSYEV cannot be guaranteed.
*
*  Alignment requirements
*  ======================
*
*  The distributed submatrices A(IA:*, JA:*) and Z(IZ:IZ+M-1,JZ:JZ+N-1)
*  must verify some alignment properties, namely the following
*  expressions should be true:
*
*  ( MB_A.EQ.NB_A.EQ.MB_Z .AND. IROFFA.EQ.IROFFZ .AND. IROFFA.EQ.0 .AND.
*    IAROW.EQ.IZROW )
*  where
*  IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ).
*
*  =====================================================================
*
*  Version 1.4 limitations:
*     DESCA(MB_) = DESCA(NB_)
*     DESCA(M_) = DESCZ(M_)
*     DESCA(N_) = DESCZ(N_)
*     DESCA(MB_) = DESCZ(MB_)
*     DESCA(NB_) = DESCZ(NB_)
*     DESCA(RSRC_) = DESCZ(RSRC_)
*
*     .. Parameters ..
      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   FIVE, ONE, TEN, ZERO
      parameter( zero = 0.0d+0, one = 1.0d+0,
     $                     ten = 10.0d+0, five = 5.0d+0 )
      INTEGER            IERREIN, IERRCLS, IERRSPC, IERREBZ, ITHVAL
      parameter( ierrein = 1, ierrcls = 2, ierrspc = 4,
     $                   ierrebz = 8, ithval = 10 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, WANTZ
      INTEGER            CONTEXTC, CSRC_A, I, IACOL, IAROW, ICOFFA, 
     $                   iinfo, indd, indd2, inde, inde2, indtau, 
     $                   indwork, indwork2, iroffa, iroffz, iscale, 
     $                   izrow, j, k, ldc, llwork, lwmin, mb_a, mb_z, 
     $                   mycol, mypcolc, myprowc, myrow, nb, nb_a, nb_z,
     $                   np, npcol, npcolc, nprocs, nprow, nprowc, nq, 
     $                   nrc, qrmem, rsrc_a, rsrc_z, sizemqrleft, 
     $                   sizesytrd
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, 
     $                   smlnum
*     ..
*     .. Local Arrays ..
      INTEGER            DESCQR( 9 ), IDUM1( 3 ), IDUM2( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            INDXG2P, NUMROC, SL_GRIDRESHAPE
      DOUBLE PRECISION   PDLAMCH, PDLANSY
      EXTERNAL           lsame, numroc, pdlamch, pdlansy,
     $                   sl_gridreshape
*     ..
*     .. External Subroutines ..
      EXTERNAL           blacs_gridexit, blacs_gridinfo, chk1mat, dcopy,
     $                   descinit, dscal, dsteqr2, pchk1mat, pchk2mat,
     $                   pdelget, pdgemr2d, pdlascl, pdlaset, pdormtr,
     $                   pdsytrd, pxerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, ichar, max, min, mod, sqrt, int
*     ..
*     .. Executable Statements ..
*       This is just to keep ftnchek and toolpack/1 happy
      IF( block_cyclic_2d*csrc_*ctxt_*dlen_*dtype_*lld_*mb_*m_*nb_*n_*
     $    rsrc_.LT.0 )RETURN
*
*     Quick return
*
      IF( n.EQ.0 ) RETURN
*
*     Test the input arguments.
*
      CALL blacs_gridinfo( desca( ctxt_ ), nprow, npcol, myrow, mycol )
      info = 0
*
      wantz = lsame( jobz, 'V' )
      IF( nprow.EQ.-1 ) THEN
         info = -( 700+ctxt_ )
      ELSE IF( wantz ) THEN
         IF( desca( ctxt_ ).NE.descz( ctxt_ ) ) THEN
            info = -( 1200+ctxt_ )
         END IF
      END IF
      IF( info .EQ. 0 ) THEN
         CALL chk1mat( n, 3, n, 3, ia, ja, desca, 7, info )
         IF( wantz )
     $      CALL chk1mat( n, 3, n, 3, iz, jz, descz, 12, info )
*
         IF( info.EQ.0 ) THEN
*
*           Get machine constants.
*
            safmin = pdlamch( desca( ctxt_ ), 'Safe minimum' )
            eps = pdlamch( desca( ctxt_ ), 'Precision' )
            smlnum = safmin / eps
            bignum = one / smlnum
            rmin = sqrt( smlnum )
            rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
*
            nprocs = nprow*npcol
            nb_a = desca( nb_ )
            mb_a = desca( mb_ )
            nb = nb_a
            lower = lsame( uplo, 'L' )
*
            rsrc_a = desca( rsrc_ )
            csrc_a = desca( csrc_ )
            iroffa = mod( ia-1, mb_a )
            icoffa = mod( ja-1, nb_a )
            iarow = indxg2p( 1, nb_a, myrow, rsrc_a, nprow )
            iacol = indxg2p( 1, mb_a, mycol, csrc_a, npcol )
            np = numroc( n+iroffa, nb, myrow, iarow, nprow )
            nq = numroc( n+icoffa, nb, mycol, iacol, npcol )
 
            IF( wantz ) THEN
               nb_z = descz( nb_ )
               mb_z = descz( mb_ )
               rsrc_z = descz( rsrc_ )
               iroffz = mod( iz-1, mb_a )
               izrow = indxg2p( 1, nb_a, myrow, rsrc_z, nprow )
               sizemqrleft = max( ( nb_a*( nb_a-1 ) ) / 2, ( np+nq )*
     $                            nb_a ) + nb_a*nb_a
            ELSE
               sizemqrleft = 0
               iroffz = 0
               izrow = 0
            END IF
            sizesytrd = max( nb * ( np +1 ), 3 * nb )
*
*           Initialize the context of the single column distributed
*           matrix required by DSTEQR2. This specific distribution
*           allows each process to do 1/pth of the work updating matrix
*           Q during DSTEQR2 and achieve some parallelization to an
*           otherwise serial subroutine.
*
            ldc = 0
            IF( wantz ) THEN
               contextc = sl_gridreshape( desca( ctxt_ ), 0, 1, 1,
     $                                    nprocs, 1 )
               CALL blacs_gridinfo( contextc, nprowc, npcolc, myprowc,
     $                              mypcolc )
               nrc = numroc( n, nb_a, myprowc, 0, nprocs)
               ldc = max( 1, nrc )
               CALL descinit( descqr, n, n, nb, nb, 0, 0, contextc,
     $                        ldc, info )
            END IF
 
*
*           Set up pointers into the WORK array
*
            indtau = 1
            inde = indtau + n
            indd = inde + n
            indd2 = indd + n
            inde2 = indd2 + n
            indwork = inde2 + n
            indwork2 = indwork + n*ldc
            llwork = lwork - indwork + 1
*
*           Compute the total amount of space needed
*
            qrmem = 2*n-2
            IF( wantz ) THEN
               lwmin = 5*n + n*ldc + max( sizemqrleft, qrmem ) + 1
            ELSE
               lwmin = 5*n + sizesytrd + 1
            END IF
*
         END IF
         IF( info.EQ.0 ) THEN
            IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
               info = -1
            ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
               info = -2
            ELSE IF( lwork.LT.lwmin .AND. lwork.NE.-1 ) THEN
               info = -14
            ELSE IF( iroffa.NE.0 ) THEN
               info = -5
            ELSE IF( desca( mb_ ).NE.desca( nb_ ) ) THEN
               info = -( 700+nb_ )
            END IF
            IF( wantz ) THEN
               IF( iroffa.NE.iroffz ) THEN
                  info = -10
               ELSE IF( iarow.NE.izrow ) THEN
                  info = -10
               ELSE IF( desca( m_ ).NE.descz( m_ ) ) THEN
                  info = -( 1200+m_ )
               ELSE IF( desca( n_ ).NE.descz( n_ ) ) THEN
                  info = -( 1200+n_ )
               ELSE IF( desca( mb_ ).NE.descz( mb_ ) ) THEN
                  info = -( 1200+mb_ )
               ELSE IF( desca( nb_ ).NE.descz( nb_ ) ) THEN
                  info = -( 1200+nb_ )
               ELSE IF( desca( rsrc_ ).NE.descz( rsrc_ ) ) THEN
                  info = -( 1200+rsrc_ )
               ELSE IF( desca( ctxt_ ).NE.descz( ctxt_ ) ) THEN
                  info = -( 1200+ctxt_ )
               ENDIF
            END IF
         END IF
         IF( wantz ) THEN
            idum1( 1 ) = ichar( 'V' )
         ELSE
            idum1( 1 ) = ichar( 'N' )
         END IF
         idum2( 1 ) = 1
         IF( lower ) THEN
            idum1( 2 ) = ichar( 'L' )
         ELSE
            idum1( 2 ) = ichar( 'U' )
         END IF
         idum2( 2 ) = 2
         IF( lwork.EQ.-1 ) THEN
            idum1( 3 ) = -1
         ELSE
            idum1( 3 ) = 1
         END IF
         idum2( 3 ) = 3
         IF( lsame( jobz, 'V' ) ) THEN
            CALL pchk2mat( n, 3, n, 3, ia, ja, desca, 7, n, 3, n, 3,
     $                     iz, jz, descz, 12, 3, idum1, idum2, info )
         ELSE
            CALL pchk1mat( n, 3, n, 3, ia, ja, desca, 7, 3, idum1,
     $                     idum2, info )
         END IF
*
*     Write the required workspace for lwork queries.
*
         work( 1 ) = dble( lwmin )
      END IF
*
      IF( info.NE.0 ) THEN
         CALL pxerbla( desca( ctxt_ ), 'PDSYEV', -info )
         IF( wantz ) CALL blacs_gridexit( contextc )
         RETURN
      ELSE IF( lwork .EQ. -1 ) THEN
         IF( wantz ) CALL blacs_gridexit( contextc )
         RETURN
      END IF
*
*     Scale matrix to allowable range, if necessary.
*
      iscale = 0
*
      anrm = pdlansy( 'M', uplo, n, a, ia, ja, desca, work( indwork ) )
*
 
      IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
         iscale = 1
         sigma = rmin / anrm
      ELSE IF( anrm.GT.rmax ) THEN
         iscale = 1
         sigma = rmax / anrm
      END IF
*
      IF( iscale.EQ.1 ) THEN
         CALL pdlascl( uplo, one, sigma, n, n, a, ia, ja, desca, iinfo )
      END IF
*
*     Reduce symmetric matrix to tridiagonal form.
*
      CALL pdsytrd( uplo, n, a, ia, ja, desca, work( indd ),
     $              work( inde ), work( indtau ), work( indwork ),
     $              llwork, iinfo )
*
*     Copy the values of D, E to all processes.
*
      DO 10 i=1,n
         CALL pdelget( 'A', ' ', work(indd2+i-1), a,
     $                 i+ia-1, i+ja-1, desca )
 10   CONTINUE
      IF( lsame( uplo, 'U') ) THEN
          DO 20 i=1,n-1
             CALL pdelget( 'A', ' ', work(inde2+i-1), a, 
     $                     i+ia-1, i+ja, desca )
 20       CONTINUE
      ELSE
          DO 30 i=1,n-1
             CALL pdelget( 'A', ' ', work(inde2+i-1), a,
     $                     i+ia, i+ja-1, desca )
 30       CONTINUE
      ENDIF
*
      IF( wantz ) THEN
*
         CALL pdlaset( 'Full', n, n, zero, one, work( indwork ), 1, 1,
     $                 descqr )
*
*        DSTEQR2 is a modified version of LAPACK's DSTEQR.  The
*        modifications allow each process to perform partial updates
*        to matrix Q.
*
         CALL dsteqr2( 'I', n, work( indd2 ), work( inde2 ),
     $                 work( indwork ), ldc, nrc, work( indwork2 ), 
     $                 info )
*
         CALL pdgemr2d( n, n, work( indwork ), 1, 1, descqr, z, ia, ja,
     $                  descz, contextc )
*
         CALL pdormtr( 'L', uplo, 'N', n, n, a, ia, ja, desca,
     $                 work( indtau ), z, iz, jz, descz,
     $                 work( indwork ), llwork, iinfo )
*
      ELSE
*
         CALL dsteqr2( 'N', n, work( indd2 ), work( inde2 ),
     $                 work( indwork ), 1, 1, work( indwork2 ),
     $                 info )
      ENDIF
*
*     Copy eigenvalues from workspace to output array
*
      CALL dcopy( n, work( indd2 ), 1, w, 1 )
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( iscale .EQ. 1 ) THEN
         CALL dscal( n, one / sigma, w, 1 )
      END IF
*
*     Free up resources
*
      IF( wantz ) THEN
           CALL blacs_gridexit( contextc )
      END IF
*
*     Compare every ith eigenvalue, or all if there are only a few,
*     across the process grid to check for heterogeneity.
*
      IF( n.LE.ithval ) THEN
         j = n
         k = 1
      ELSE
         j = n/ithval
         k = ithval
      END IF
*
      DO 40 i = 1, j
         work( i+indtau ) = w( (i-1)*k+1 )
         work( i+inde ) = w( (i-1)*k+1 )
 40   CONTINUE
*
      CALL dgamn2d( desca( ctxt_ ), 'a', ' ', j, 1, work( 1+indtau ),
     $              j, 1, 1, -1, -1, 0 )
      CALL dgamx2d( desca( ctxt_ ), 'a', ' ', j, 1, work( 1+inde ),
     $              j, 1, 1, -1, -1, 0 )
*
      DO 50 i = 1, j
         IF( info.EQ.0 .AND. ( work( i+indtau )-work( i+inde )
     $        .NE. zero ) )THEN 
            info = n+1
         END IF
 50   CONTINUE
*
      RETURN
*
*     End of PDSYEV
*
      END