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NAME

       PDSTEIN - compute the eigenvectors of a symmetric tridiagonal matrix in
       parallel, using inverse iteration

SYNOPSIS

       SUBROUTINE PDSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC,  Z,  IZ,  JZ,
                           DESCZ,  WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR,
                           GAP, INFO )

           INTEGER         INFO, IZ, JZ, LIWORK, LWORK, M, N

           DOUBLE          PRECISION ORFAC

           INTEGER         DESCZ( * ), IBLOCK( * ), ICLUSTR( * ), IFAIL( *  ),
                           ISPLIT( * ), IWORK( * )

           DOUBLE          PRECISION D( * ), E( * ), GAP( * ), W( * ), WORK( *
                           ), Z( * )

PURPOSE

       PDSTEIN computes the eigenvectors of a symmetric tridiagonal matrix  in
       parallel, using inverse iteration. The eigenvectors found correspond to
       user specified eigenvalues. PDSTEIN does not orthogonalize vectors that
       are   on  different  processes.  The  extent  of  orthogonalization  is
       controlled by the input parameter LWORK.  Eigenvectors that are  to  be
       orthogonalized are computed by the same process. PDSTEIN decides on the
       allocation of work among the processes and then calls DSTEIN2 (modified
       LAPACK  routine)  on each individual process. If insufficient workspace
       is allocated, the expected orthogonalization may not be done.

       Note : If the eigenvectors obtained are not orthogonal, increase
              LWORK and run the code again.

       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

       P = NPROW * NPCOL is the total number of processes

       N       (global input) INTEGER
               The order of the tridiagonal matrix T.  N >= 0.

       D       (global input) DOUBLE PRECISION array, dimension (N)
               The n diagonal elements of the tridiagonal matrix T.

       E       (global input) DOUBLE PRECISION array, dimension (N-1)
               The (n-1) off-diagonal elements of the tridiagonal matrix T.

       M       (global input) INTEGER
               The total number of eigenvectors to be found. 0 <= M <= N.

       W       (global input/global output) DOUBLE PRECISION array, dim (M)
               On input, the first M elements of W contain all the eigenvalues
               for which eigenvectors are  to  be  computed.  The  eigenvalues
               should  be grouped by split-off block and ordered from smallest
               to largest within the block (The output array  W  from  PDSTEBZ
               with   ORDER=’b’  is  expected  here).  This  array  should  be
               replicated on all processes.  On output, the first  M  elements
               contain the input eigenvalues in ascending order.

               Note  : To obtain orthogonal vectors, it is best if eigenvalues
               are computed to highest accuracy ( this can be done by  setting
               ABSTOL  to  the underflow threshold = DLAMCH(’U’) --- ABSTOL is
               an input parameter to PDSTEBZ )

       IBLOCK  (global input) INTEGER array, dimension (N)
               The  submatrix  indices  associated  with   the   corresponding
               eigenvalues  in  W  -- 1 for eigenvalues belonging to the first
               submatrix from the top, 2 for those  belonging  to  the  second
               submatrix,  etc.  (The  output  array  IBLOCK  from  PDSTEBZ is
               expected here).

       ISPLIT  (global input) INTEGER array, dimension (N)
               The splitting points, at which T breaks  up  into  submatrices.
               The  first  submatrix  consists of rows/columns 1 to ISPLIT(1),
               the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
               and  the  NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1
               through ISPLIT(NSPLIT)=N (The output array ISPLIT from  PDSTEBZ
               is expected here.)

       ORFAC   (global input) DOUBLE PRECISION
               ORFAC  specifies  which  eigenvectors should be orthogonalized.
               Eigenvectors that correspond to eigenvalues  which  are  within
               ORFAC*||T||  of  each other are to be orthogonalized.  However,
               if the workspace is insufficient (see  LWORK),  this  tolerance
               may  be  decreased  until all eigenvectors to be orthogonalized
               can be stored in one process.   No  orthogonalization  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,
               dimension  (DESCZ(DLEN_), N/npcol + NB) Z contains the computed
               eigenvectors associated with  the  specified  eigenvalues.  Any
               vector  which  fails  to converge is set to its current iterate
               after MAXITS iterations ( See  DSTEIN2  ).   On  output,  Z  is
               distributed across the P processes in block cyclic format.

       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.

       WORK    (local workspace/global output) DOUBLE PRECISION array,
               dimension  (  LWORK ) On output, WORK(1) gives a lower bound on
               the workspace (  LWORK  )  that  guarantees  the  user  desired
               orthogonalization (see ORFAC).  Note that this may overestimate
               the minimum workspace needed.

       LWORK   (local input) integer
               LWORK  controls the extent of orthogonalization  which  can  be
               done. The number of eigenvectors for which storage is allocated
               on each process is NVEC  =  floor((  LWORK-  max(5*N,NP00*MQ00)
               )/N).   Eigenvectors  corresponding  to  eigenvalue clusters of
               size NVEC - ceil(M/P) + 1 are guaranteed to be orthogonal ( the
               orthogonality  is similar to that obtained from DSTEIN2).  Note
               :  LWORK   must  be  no  smaller  than:  max(5*N,NP00*MQ00)   +
               ceil(M/P)*N,  and  should  have  the  same  input  value on all
               processes.  It is the minimum value of LWORK input on different
               processes that is significant.

               If LWORK = -1, then LWORK 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.

       IWORK   (local workspace/global output) INTEGER array,
               dimension  (  3*N+P+1 ) On return, IWORK(1) contains the amount
               of integer workspace required.  On return, the IWORK(2) through
               IWORK(P+2)  indicate the eigenvectors computed by each process.
               Process I  computes  eigenvectors  indexed  IWORK(I+2)+1  thru’
               IWORK(I+3).

       LIWORK  (local input) INTEGER
               Size of array IWORK.  Must be >= 3*N + P + 1

               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   (global output) integer array, dimension (M)
               On normal exit, all elements of IFAIL are zero.  If one or more
               eigenvectors fail to converge after MAXITS  iterations  (as  in
               DSTEIN),  then  INFO > 0 is returned.  If mod(INFO,M+1)>0, then
               for I=1 to mod(INFO,M+1), the eigenvector corresponding to  the
               eigenvalue  W(IFAIL(I))  failed  to  converge ( W refers to the
               array of eigenvalues on output ).

               ICLUSTR (global output) integer  array,  dimension  (2*P)  This
               output  array contains indices of eigenvectors corresponding to
               a cluster of eigenvalues that could not be  orthogonalized  due
               to   insufficient   workspace  (see  LWORK,  ORFAC  and  INFO).
               Eigenvectors corresponding to clusters of  eigenvalues  indexed
               ICLUSTR(2*I-1)  to ICLUSTR(2*I), I = 1 to INFO/(M+1), could not
               be  orthogonalized  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.

       GAP     (global output) DOUBLE PRECISION array, dimension (P)
               This output array contains the gap  between  eigenvalues  whose
               eigenvectors  could  not  be  orthogonalized. The INFO/M output
               values in this array  correspond  to  the  INFO/(M+1)  clusters
               indicated  by  the  array ICLUSTR. As a result, the dot product
               between eigenvectors corresponding to the I^th cluster  may  be
               as high as ( O(n)*macheps ) / GAP(I).

       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 = -I, the I-th argument had an illegal value
               > 0 :  if mod(INFO,M+1) = I,  then  I  eigenvectors  failed  to
               converge  in MAXITS iterations. Their indices are stored in the
               array  IFAIL.    if   INFO/(M+1)   =   I,   then   eigenvectors
               corresponding  to  I  clusters  of  eigenvalues  could  not  be
               orthogonalized due to insufficient workspace.  The  indices  of
               the clusters are stored in the array ICLUSTR.