Provided by: scalapack-doc_1.5-10_all 
NAME
PZTZRZF - 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 PZTZRZF( M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )
INTEGER IA, INFO, JA, LWORK, M, N
INTEGER DESCA( * )
COMPLEX*16 A( * ), TAU( * ), WORK( * )
PURPOSE
PZTZRZF 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*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 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*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/local output) COMPLEX*16 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 ).