..
     .. Array Arguments ..
     ..
  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 max(3,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.
  Alignment requirements
  ======================
  The distributed submatrices A(IA:*, JA:*), C(IC:IC+M-1,JC:JC+N-1),
  and B( IB:IB+N-1, JB:JB+N-1 ) must verify some alignment properties,
  namely the following expressions should be true:
     DESCA(MB_) = DESCA(NB_)
     IA = IB = IZ
     JA = IB = JZ
     DESCA(M_) = DESCB(M_) =DESCZ(M_)
     DESCA(N_) = DESCB(N_)= DESCZ(N_)
     DESCA(MB_) = DESCB(MB_) = DESCZ(MB_)
     DESCA(NB_) = DESCB(NB_) = DESCZ(NB_)
     DESCA(RSRC_) = DESCB(RSRC_) = DESCZ(RSRC_)
     DESCA(CSRC_) = DESCB(CSRC_) = DESCZ(CSRC_)
     MOD( IA-1, DESCA( MB_ ) ) = 0
     MOD( JA-1, DESCA( NB_ ) ) = 0
     MOD( IB-1, DESCB( MB_ ) ) = 0
     MOD( JB-1, DESCB( NB_ ) ) = 0
  =====================================================================
     .. Parameters ..