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NAME

       PCTZRZF  -  reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix
       sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means  of
       unitary transformations

SYNOPSIS

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

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

           INTEGER         DESCA( * )

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

PURPOSE

       PCTZRZF  reduces  the  M-by-N ( M<=N ) complex upper trapezoidal matrix
       sub( A ) = A(IA:IA+M-1,JA: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.

       A       (local input/local output) COMPLEX 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 M+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, 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/local output) COMPLEX 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 >= MB_A * ( Mp0 + Nq0 + MB_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  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 ).