PCLARZT(3) ScaLAPACK routine of NEC Numeric Library Collection PCLARZT(3)
NAME
PCLARZT - form the triangular factor T of a complex block reflector H
of order > n, which is defined as a product of k elementary reflectors
as returned by PCTZRZF
SYNOPSIS
SUBROUTINE PCLARZT( DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T,
WORK )
CHARACTER DIRECT, STOREV
INTEGER IV, JV, K, N
INTEGER DESCV( * )
COMPLEX TAU( * ), T( * ), V( * ), WORK( * )
PURPOSE
PCLARZT forms the triangular factor T of a complex block reflector H of
order > n, which is defined as a product of k elementary reflectors as
returned by PCTZRZF. 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 array 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 array V, and
H = I - V' * T * V
Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
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
DIRECT (global input) CHARACTER
Specifies the order in which the elementary reflectors are mul-
tiplied to form the block reflector:
= 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
STOREV (global input) CHARACTER
Specifies how the vectors which define the elementary reflec-
tors are stored (see also Further Details):
= 'R': rowwise
N (global input) INTEGER
The number of meaningful entries of the block reflector H. N
>= 0.
K (global input) INTEGER
The order of the triangular factor T (= the number of elemen-
tary reflectors). 1 <= K <= MB_V (= NB_V).
V (input/output) COMPLEX pointer into the local memory
to an array of local dimension (LOCr(IV+K-1),LOCc(JV+N-1)).
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) COMPLEX, 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) COMPLEX array, dimension (MB_V,MB_V)
It contains the k-by-k triangular factor of the block reflector
associated with V. T is lower triangular.
WORK (local workspace) COMPLEX 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_____
( v1 v2 v3 ) | |
( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )
V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )
( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )
( v1 v2 v3 )
. . .
. . .
1 . .
1 .
1
DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
______V_____
1 | |
. 1 ( 1 . . . . v1 v1 v1 v1 v1 )
. . 1 ( . 1 . . . v2 v2 v2 v2 v2 )
. . . ( . . 1 . . v3 v3 v3 v3 v3 )
. . .
( v1 v2 v3 )
( v1 v2 v3 )
V = ( v1 v2 v3 )
( v1 v2 v3 )
( v1 v2 v3 )
ScaLAPACK routine 31 October 2017 PCLARZT(3)