NAME¶

PDLARZT - form the triangular factor T of a real block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PDTZRZF

SYNOPSIS¶

SUBROUTINE PDLARZT(

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¶

PDLARZT forms the triangular factor T of a real block reflector H of order > n, which is defined as a product of k elementary reflectors as returned by PDTZRZF.

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

ARGUMENTS¶

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) DOUBLE PRECISION 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) 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 (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) 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_____
( 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 )