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NAME

       PDGEQL2   -   compute   a   QL   factorization   of   a  real  distributed  M-by-N  matrix  sub(  A  )  =
       A(IA:IA+M-1,JA:JA+N-1) = Q * L

SYNOPSIS

       SUBROUTINE PDGEQL2( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

           INTEGER         IA, INFO, JA, LWORK, M, N

           INTEGER         DESCA( * )

           DOUBLE          PRECISION A( * ), TAU( * ), WORK( * )

PURPOSE

       PDGEQL2 computes a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1)
       = Q * L.

       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

       M       (global input) INTEGER
               The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A
               ). M >= 0.

       N       (global input) INTEGER
               The number of columns to be operated on, i.e. the number of columns of the distributed  submatrix
               sub( A ). 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, the local pieces of the
               M-by-N distributed matrix sub( A ) which is to be factored.  On  exit,  if  M  >=  N,  the  lower
               triangle  of  the  distributed  submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower
               triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain  the
               M  by  N  lower  trapezoidal  matrix L; the remaining elements, with the array TAU, represent the
               orthogonal matrix Q as a product of elementary reflectors (see Further Details).  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.

       TAU     (local output) DOUBLE PRECISION, array, dimension LOCc(JA+N-1)
               This  array  contains  the  scalar  factors  of  the  elementary  reflectors.  TAU is tied to the
               distributed matrix A.

       WORK    (local workspace/local output) DOUBLE PRECISION array,
               dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK.

       LWORK   (local or global input) INTEGER
               The dimension of the array WORK.  LWORK is local input and must be at least LWORK >= Mp0  +  MAX(
               1, Nq0 ), where

               IROFF  =  MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A,
               NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0   = NUMROC( M+IROFF, MB_A, MYROW,
               IAROW, NPROW ), Nq0   = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ),

               and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined
               by calling the subroutine BLACS_GRIDINFO.

               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.

       INFO    (local 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.

FURTHER DETAILS

       The matrix Q is represented as a product of elementary reflectors

          Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n).

       Each H(i) has the form

          H(i) = I - tau * v * v'

       where tau is a real scalar, and v is a real vector with
       v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau
       in TAU(ja+n-k+i-1).