PZLARZT - 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 PZTZRZF
- SUBROUTINE PZLARZT(
- DIRECT, STOREV, N, K, V, IV, JV, DESCV, TAU, T, WORK )
CHARACTER DIRECT, STOREV INTEGER IV, JV, K, N INTEGER DESCV( * ) COMPLEX*16 TAU(
* ), T( * ), V( * ), WORK( * )
PZLARZT 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 PZTZRZF. 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 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
- DIRECT (global input) CHARACTER
- Specifies the order in which the elementary reflectors are multiplied 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 reflectors 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 elementary
reflectors). 1 <= K <= MB_V (= NB_V).
- V (input/output) COMPLEX*16 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*16, 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*16 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*16 array,
- dimension (K*(K-1)/2)
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 )