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NAME

       PSGEQLF  -  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 PSGEQLF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

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

           INTEGER         DESCA( * )

           REAL            A( * ), TAU( * ), WORK( * )

PURPOSE

       PSGEQLF 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) REAL 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) REAL, 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) REAL 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
               >= NB_A * ( Mp0 + Nq0 + NB_A ), 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    (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.

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).