PCLARZB(3)    ScaLAPACK routine of NEC Numeric Library Collection   PCLARZB(3)



NAME
       PCLARZB - 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 PCLARZB( 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         C( * ), T( * ), V( * ), WORK( * )

PURPOSE
       PCLARZB 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 PCTZRZF.

       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
       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 reflec-
               tors = '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 reflec-
               tors 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  reflec-
               tors 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 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 House-
               holder  transformation  as  returned  by  PCTZRZF.   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 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 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 array, dimension (LWORK)
               If  STOREV  =  'C', if SIDE = 'L', LWORK >= ( NqC0 + MpC0 ) * K
               else if SIDE = 'R', LWORK >= ( NqC0 + MAX( NpV0 + NUMROC(  NUM-
               ROC(  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 subrou-
               tine 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



ScaLAPACK routine               31 October 2017                     PCLARZB(3)