NAME
PDLARFT - form the triangular factor T of a real block reflector H of
order n, which is defined as a product of k elementary reflectors
SYNOPSIS
SUBROUTINE PDLARFT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T,
WORK )
CHARACTER DIRECT, STOREV
INTEGER IV, JV, K, N
INTEGER DESCV( * )
DOUBLE PRECISION TAU( * ), T( * ), V( * ), WORK( * )
PURPOSE
PDLARFT forms the triangular factor T of a real block reflector H of
order n, which is defined as a product of k elementary reflectors.
If DIRECT = ’F’, H = H(1) H(2) . . . H(k) and T is upper triangular;
If DIRECT = ’B’, H = H(k) . . . H(2) H(1) and T is lower triangular.
If STOREV = ’C’, the vector which defines the elementary reflector H(i)
is stored in the i-th column of the distributed matrix V, and
H = I - V * T * V’
If STOREV = ’R’, the vector which defines the elementary reflector H(i)
is stored in the i-th row of the distributed matrix V, and
H = I - V’ * T * V
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
DIRECT (global input) CHARACTER*1
Specifies the order in which the elementary reflectors are
multiplied to form the block reflector:
= ’F’: H = H(1) H(2) . . . H(k) (Forward)
= ’B’: H = H(k) . . . H(2) H(1) (Backward)
STOREV (global input) CHARACTER*1
Specifies how the vectors which define the elementary
reflectors are stored (see also Further Details):
= ’R’: rowwise
N (global input) INTEGER
The order of the block reflector H. N >= 0.
K (global input) INTEGER
The order of the triangular factor T (= the number of
elementary reflectors). 1 <= K <= MB_V (= NB_V).
V (input/output) DOUBLE PRECISION pointer into the local memory
to an array of local dimension (LOCr(IV+N-1),LOCc(JV+K-1)) if
STOREV = ’C’, and (LOCr(IV+K-1),LOCc(JV+N-1)) if STOREV = ’R’.
The distributed matrix V contains the Householder vectors. See
further details.
IV (global input) INTEGER
The row index in the global array V indicating the first row of
sub( V ).
JV (global input) INTEGER
The column index in the global array V indicating the first
column of sub( V ).
DESCV (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix V.
TAU (local input) DOUBLE PRECISION, array, dimension LOCr(IV+K-1)
if INCV = M_V, and LOCc(JV+K-1) otherwise. This array contains
the Householder scalars related to the Householder vectors.
TAU is tied to the distributed matrix V.
T (local output) DOUBLE PRECISION array, dimension (NB_V,NB_V)
if STOREV = ’Col’, and (MB_V,MB_V) otherwise. It contains the
k-by-k triangular factor of the block reflector asso- ciated
with V. If DIRECT = ’F’, T is upper triangular; if DIRECT =
’B’, T is lower triangular.
WORK (local workspace) DOUBLE PRECISION array,
dimension (K*(K-1)/2)
FURTHER DETAILS
The shape of the matrix V and the storage of the vectors which define
the H(i) is best illustrated by the following example with n = 5 and k
= 3. The elements equal to 1 are not stored; the corresponding array
elements are modified but restored on exit. The rest of the array is
not used.
DIRECT = ’F’ and STOREV = ’C’: DIRECT = ’F’ and STOREV = ’R’:
V( IV:IV+N-1, ( 1 ) V( IV:IV+K-1, ( 1 v1 v1 v1 v1 )
JV:JV+K-1 ) = ( v1 1 ) JV:JV+N-1 ) = ( 1 v2 v2 v2 )
( v1 v2 1 ) ( 1 v3 v3 )
( v1 v2 v3 )
( v1 v2 v3 )
DIRECT = ’B’ and STOREV = ’C’: DIRECT = ’B’ and STOREV = ’R’:
V( IV:IV+N-1, ( v1 v2 v3 ) V( IV:IV+K-1, ( v1 v1 1 )
JV:JV+K-1 ) = ( v1 v2 v3 ) JV:JV+N-1 ) = ( v2 v2 v2 1 )
( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
( 1 v3 )
( 1 )