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NAME

       PZLATRZ  -  reduce  the  M-by-N  (  M<=N  )  complex  upper  trapezoidal matrix sub( A ) =
       [A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1)]

SYNOPSIS

       SUBROUTINE PZLATRZ( M, N, L, A, IA, JA, DESCA, TAU, WORK )

           INTEGER         IA, JA, L, M, N

           INTEGER         DESCA( * )

           COMPLEX*16      A( * ), TAU( * ), WORK( * )

PURPOSE

       PZLATRZ reduces the M-by-N  (  M<=N  )  complex  upper  trapezoidal  matrix  sub(  A  )  =
       [A(IA:IA+M-1,JA:JA+M-1)  A(IA:IA+M-1,JA+N-L:JA+N-1)]  to upper triangular form by means of
       unitary transformations.

       The upper trapezoidal matrix sub( A ) is factored as

          sub( A ) = ( R  0 ) * Z,

       where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix.

       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.

       L       (global input) INTEGER
               The  columns  of the distributed submatrix sub( A ) containing the meaningful part
               of the Householder reflectors. L > 0.

       A       (local input/local output) COMPLEX*16 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,
               the leading M-by-M upper triangular part of sub( A )  contains  the  upper  trian-
               gular  matrix R, and elements N-L+1 to N of the first M rows of sub( A ), with the
               array TAU, represent the unitary matrix Z as a product of M elementary reflectors.

       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) COMPLEX*16, array, dimension LOCr(IA+M-1)
               This array contains the scalar factors of the elementary reflectors. TAU  is  tied
               to the distributed matrix A.

       WORK    (local workspace) COMPLEX*16 array, dimension (LWORK)
               LWORK >= Nq0 + MAX( 1, Mp0 ), 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.

FURTHER DETAILS

       The  factorization is obtained by Householder's method.  The kth transformation matrix, Z(
       k ), whose conjugate transpose is used to introduce zeros into the (m - k +  1)th  row  of
       sub( A ), is given in the form

          Z( k ) = ( I     0   ),
                   ( 0  T( k ) )

       where

          T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                                      (   0    )
                                                      ( z( k ) )

       tau  is  a scalar and z( k ) is an ( n - m ) element vector.  tau and z( k ) are chosen to
       annihilate the elements of the kth row of sub( A ).

       The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth  row
       of  sub(  A ), such that the elements of z( k ) are in  a( k, m + 1 ), ..., a( k, n ). The
       elements of R are returned in the upper triangular part of sub( A ).

       Z is given by

          Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).