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NAME

       PSLARZB  -  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 PSLARZB( 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( * )

           REAL            C( * ), T( * ), V( * ), WORK( * )

PURPOSE

       PSLARZB  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.

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

       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**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  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) REAL 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 PSTZRZF.  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) REAL array, dimension MB_V by MB_V
               The lower triangular matrix T  in  the  representation  of  the
               block reflector.

       C       (local input/local output) REAL 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) REAL 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