NAME
PCLARZ - applie a complex elementary reflector Q to a complex M-by-N
distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the
left or the right
SYNOPSIS
SUBROUTINE PCLARZ( SIDE, M, N, L, V, IV, JV, DESCV, INCV, TAU, C, IC,
JC, DESCC, WORK )
CHARACTER SIDE
INTEGER IC, INCV, IV, JC, JV, L, M, N
INTEGER DESCC( * ), DESCV( * )
COMPLEX C( * ), TAU( * ), V( * ), WORK( * )
PURPOSE
PCLARZ applies a complex elementary reflector Q to a complex M-by-N
distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the
left or the right. Q is represented in the form
Q = I - tau * v * v’
where tau is a complex scalar and v is a complex vector.
If tau = 0, then Q is taken to be the unit matrix.
Q is a product of k elementary reflectors as returned by PCTZRZF.
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
Because vectors may be viewed as a subclass of matrices, a distributed
vector is considered to be a distributed matrix.
Restrictions
============
If SIDE = ’Left’ and INCV = 1, then the row process having the first
entry V(IV,JV) must also own C(IC+M-L,JC:JC+N-1). Moreover,
MOD(IV-1,MB_V) must be equal to MOD(IC+N-L-1,MB_C), if INCV=M_V, only
the last equality must be satisfied.
If SIDE = ’Right’ and INCV = M_V then the column process having the
first entry V(IV,JV) must also own C(IC:IC+M-1,JC+N-L) and
MOD(JV-1,NB_V) must be equal to MOD(JC+N-L-1,NB_C), if INCV = 1 only
the last equality must be satisfied.
ARGUMENTS
SIDE (global input) CHARACTER
= ’L’: form Q * sub( C ),
= ’R’: form sub( C ) * Q.
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.
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 pointer into the local memory
to an array of dimension (LLD_V,*) containing the local pieces
of the distributed vectors V representing the Householder
transformation Q, V(IV:IV+L-1,JV) if SIDE = ’L’ and INCV = 1,
V(IV,JV:JV+L-1) if SIDE = ’L’ and INCV = M_V,
V(IV:IV+L-1,JV) if SIDE = ’R’ and INCV = 1,
V(IV,JV:JV+L-1) if SIDE = ’R’ and INCV = M_V,
The vector v in the representation of Q. V is not used if TAU =
0.
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.
INCV (global input) INTEGER
The global increment for the elements of V. Only two values of
INCV are supported in this version, namely 1 and M_V. INCV
must not be zero.
TAU (local input) COMPLEX, array, dimension LOCc(JV) if
INCV = 1, and LOCr(IV) otherwise. This array contains the
Householder scalars related to the Householder vectors. TAU is
tied to the distributed matrix V.
C (local input/local output) COMPLEX pointer into the
local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ),
containing the local pieces of sub( C ). On exit, sub( C ) is
overwritten by the Q * sub( C ) if SIDE = ’L’, or sub( C ) * Q
if SIDE = ’R’.
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 array, dimension (LWORK)
If INCV = 1, if SIDE = ’L’, if IVCOL = ICCOL, LWORK >= NqC0
else LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = ’R’,
LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC(
N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV
= M_V, if SIDE = ’L’, LWORK >= MpC0 + MAX( MAX( 1, NqC0 ),
NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) )
else if SIDE = ’R’, if IVROW = ICROW, LWORK >= MpC0 else LWORK
>= NqC0 + MAX( 1, MpC0 ) end if end if end if
where LCM is the least common multiple of NPROW and NPCOL and
LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM /
NPCOL,
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 ), 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:
MB_V = NB_V,
If INCV = 1, If SIDE = ’Left’, ( MB_V.EQ.MB_C .AND.
IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = ’Right’, (
MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else
if INCV = M_V, If SIDE = ’Left’, ( MB_V.EQ.NB_V .AND.
MB_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