PZLARZB(1) applie a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right

SYNOPSIS

SUBROUTINE PZLARZB(
SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, IV, JV, DESCV, T, C, IC, JC, DESCC, WORK )

    
CHARACTER DIRECT, SIDE, STOREV, TRANS

    
INTEGER IC, IV, JC, JV, K, L, M, N

    
INTEGER DESCC( * ), DESCV( * )

    
COMPLEX*16 C( * ), T( * ), V( * ), WORK( * )

PURPOSE

PZLARZB applies a complex block reflector Q or its conjugate transpose Q**H to a complex M-by-N distributed matrix sub( C ) denoting C(IC:IC+M-1,JC:JC+N-1), from the left or the right.

Q is a product of k elementary reflectors as returned by PZTZRZF.

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

SIDE (global input) CHARACTER
= 'L': apply Q or Q**H from the Left;
= 'R': apply Q or Q**H from the Right.
TRANS (global input) CHARACTER

= 'N': No transpose, apply Q;
= 'C': Conjugate transpose, apply Q**H.
DIRECT (global input) CHARACTER
Indicates how H is formed from a product of elementary reflectors = '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
Indicates how the vectors which define the elementary reflectors are stored:
= 'C': Columnwise (not supported yet)
= 'R': Rowwise
M (global input) INTEGER
The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( C ). 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( C ). N >= 0.
K (global input) INTEGER
The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector).
L (global input) INTEGER
The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
V (local input) COMPLEX*16 pointer into the local memory
to an array of dimension (LLD_V, LOCc(JV+M-1)) if SIDE = 'L', (LLD_V, LOCc(JV+N-1)) if SIDE = 'R'. It contains the local pieces of the distributed vectors V representing the Householder transformation as returned by PZTZRZF. LLD_V >= LOCr(IV+K-1).
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.
T (local input) COMPLEX*16 array, dimension MB_V by MB_V
The lower triangular matrix T in the representation of the block reflector.
C (local input/local output) COMPLEX*16 pointer into the
local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the M-by-N distributed matrix sub( C ). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q or sub( C )*Q'.
IC (global input) INTEGER
The row index in the global array C indicating the first row of sub( C ).
JC (global input) INTEGER
The column index in the global array C indicating the first column of sub( C ).
DESCC (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix C.
WORK (local workspace) COMPLEX*16 array, dimension (LWORK)
If STOREV = 'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC( NUMROC( N+ICOFFC, NB_V, 0, 0, NPCOL ), NB_V, 0, 0, LCMQ ), MpC0 ) ) * K end if else if STOREV = 'R', if SIDE = 'L', LWORK >= ( MpC0 + MAX( MqV0 + NUMROC( NUMROC( M+IROFFC, MB_V, 0, 0, NPROW ), MB_V, 0, 0, LCMP ), NqC0 ) ) * K else if SIDE = 'R', LWORK >= ( MpC0 + NqC0 ) * K end if end if

where LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ),

IROFFV = MOD( IV-1, MB_V ), ICOFFV = MOD( JV-1, NB_V ), IVROW = INDXG2P( IV, MB_V, MYROW, RSRC_V, NPROW ), IVCOL = INDXG2P( JV, NB_V, MYCOL, CSRC_V, NPCOL ), MqV0 = NUMROC( M+ICOFFV, NB_V, MYCOL, IVCOL, NPCOL ), NpV0 = NUMROC( N+IROFFV, MB_V, MYROW, IVROW, NPROW ),

IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NpC0 = NUMROC( N+ICOFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ),

ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

Alignment requirements ======================

The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true:

If STOREV = 'Columnwise' If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if STOREV = 'Rowwise' If SIDE = 'Left', ( NB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if

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