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NAME

       PSLATRZ  -  reduce  the  M-by-N  (  M<=N  )  real  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
       orthogonal transformations

SYNOPSIS

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

           INTEGER         IA, JA, L, M, N

           INTEGER         DESCA( * )

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

PURPOSE

       PSLATRZ  reduces  the  M-by-N  (  M<=N  )  real  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
       orthogonal transformations.

       The upper trapezoidal matrix sub( A ) is factored as

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

       where Z is an N-by-N orthogonal 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) 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,
               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  orthogonal  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) REAL, 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) REAL 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 ), which 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 ).