PZLATRZ(3) ScaLAPACK routine of NEC Numeric Library Collection PZLATRZ(3)
NAME
PZLATRZ - 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 PZLATRZ( M, N, L, A, IA, JA, DESCA, TAU, WORK )
INTEGER IA, JA, L, M, N
INTEGER DESCA( * )
COMPLEX*16 A( * ), TAU( * ), WORK( * )
PURPOSE
PZLATRZ 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 vec-
tor. This vector stores the information required to establish the map-
ping 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 col-
umn.
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*16 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*16, 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*16 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 trans-
formation matrix, Z( k ), whose conjugate transpose is used to intro-
duce 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 ).
ScaLAPACK routine 31 October 2017 PZLATRZ(3)