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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 ).