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NAME

       PDLARFB  -  applie  a real block reflector Q or its transpose Q**T to a
       real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1)

SYNOPSIS

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

           CHARACTER       SIDE, TRANS, DIRECT, STOREV

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

           INTEGER         DESCC( * ), DESCV( * )

           DOUBLE          PRECISION C( * ), T( * ), V( * ), WORK( * )

PURPOSE

       PDLARFB  applies  a  real  block reflector Q or its transpose Q**T to a
       real distributed M-by-N matrix sub( C ) =  C(IC:IC+M-1,JC:JC+N-1)  from
       the left or the right.

       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**T from the Left;
               = ’R’: apply Q or Q**T from the Right.

       TRANS   (global input) CHARACTER
               = ’N’:  No transpose, apply Q;
               = ’T’:  Transpose, apply Q**T.

       DIRECT  (global input) CHARACTER
               Indicates  how  Q  is  formed  from  a  product  of  elementary
               reflectors = ’F’: Q = H(1) H(2) . . . H(k) (Forward)
               = ’B’: Q = H(k) . . . H(2) H(1) (Backward)

       STOREV  (global input) CHARACTER
               Indicates   how   the   vectors  which  define  the  elementary
               reflectors are stored:
               = ’C’: Columnwise
               = ’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).

       V       (local input) DOUBLE PRECISION pointer into the local memory
               to  an  array  of dimension ( LLD_V, LOCc(JV+K-1) ) if STOREV =
               ’C’, ( LLD_V, LOCc(JV+M-1)) if STOREV = ’R’ and SIDE =  ’L’,  (
               LLD_V,  LOCc(JV+N-1)  )  if  STOREV  =  ’R’  and SIDE = ’R’. It
               contains  the  local  pieces  of  the  distributed  vectors   V
               representing   the  Householder  transformation.   See  further
               details.   If  STOREV  =  ’C’  and  SIDE  =   ’L’,   LLD_V   >=
               MAX(1,LOCr(IV+M-1));  if  STOREV = ’C’ and SIDE = ’R’, LLD_V >=
               MAX(1,LOCr(IV+N-1)); if STOREV = ’R’, 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) DOUBLE PRECISION array, dimension MB_V by MB_V
               if STOREV = ’R’ and NB_V by NB_V if STOREV =  ’C’.  The  trian-
               gular matrix T in the representation of the block reflector.

       C       (local input/local output) DOUBLE PRECISION 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) DOUBLE PRECISION 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