NAME
PCTZRZF - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix
sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper triangular form by means of
unitary transformations
SYNOPSIS
SUBROUTINE PCTZRZF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )
INTEGER IA, INFO, JA, LWORK, M, N
INTEGER DESCA( * )
COMPLEX A( * ), TAU( * ), WORK( * )
PURPOSE
PCTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix
sub( A ) = A(IA:IA+M-1,JA: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.
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 M+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/local output) COMPLEX 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( 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.
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 i-th argument is an array and the j-entry had an
illegal value, then INFO = -(i*100+j), if the i-th 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 ), 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 ).