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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 ).