Provided by: scalapack-doc_1.5-11_all 
NAME
PCLATRZ - 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 PCLATRZ( M, N, L, A, IA, JA, DESCA, TAU, WORK )
INTEGER IA, JA, L, M, N
INTEGER DESCA( * )
COMPLEX A( * ), TAU( * ), WORK( * )
PURPOSE
PCLATRZ 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 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, 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 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 ).
LAPACK version 1.5 12 May 1997 PCLATRZ(l)