SYNOPSIS
 SUBROUTINE PSTZRZF(
 M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )
 INTEGER IA, INFO, JA, LWORK, M, N
 INTEGER DESCA( * )
 REAL A( * ), TAU( * ), WORK( * )
PURPOSE
PSTZRZF reduces the MbyN ( M<=N ) real upper trapezoidal matrix sub( A ) = A(IA:IA+M1,JA:JA+N1) 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 NbyN orthogonal matrix and R is an MbyM 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) REAL pointer into the
 local memory to an array of dimension (LLD_A, LOCc(JA+N1)). On entry, the local pieces of the MbyN distributed matrix sub( A ) which is to be factored. On exit, the leading MbyM 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 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+M1)
 This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A.
 WORK (local workspace/local output) REAL 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( IA1, MB_A ), ICOFF = MOD( JA1, 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 ith argument is an array and the jentry had an illegal value, then INFO = (i*100+j), if the ith 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 ), 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 ).