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NAME

       PDORMBR - VECT = 'Q', PDORMBR overwrites the general real distributed M-by-N matrix sub( C
       ) = C(IC:IC+M-1,JC:JC+N-1) with   SIDE = 'L' SIDE = 'R' TRANS = 'N'

SYNOPSIS

       SUBROUTINE PDORMBR( VECT, SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC,  JC,  DESCC,
                           WORK, LWORK, INFO )

           CHARACTER       SIDE, TRANS, VECT

           INTEGER         IA, IC, INFO, JA, JC, K, LWORK, M, N

           INTEGER         DESCA( * ), DESCC( * )

           DOUBLE          PRECISION A( * ), C( * ), TAU( * ), WORK( * )

PURPOSE

       If  VECT  =  'Q', PDORMBR overwrites the general real distributed M-by-N matrix sub( C ) =
       C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T':      Q**T * sub( C )       sub( C ) * Q**T

       If VECT = 'P', PDORMBR overwrites sub( C ) with

                            SIDE = 'L'           SIDE = 'R'
       TRANS = 'N':      P * sub( C )          sub( C ) * P
       TRANS = 'T':      P**T * sub( C )       sub( C ) * P**T

       Here Q and P**T are  the  orthogonal  distributed  matrices  determined  by  PDGEBRD  when
       reducing a real distributed matrix A(IA:*,JA:*) to bidiagonal form: A(IA:*,JA:*) = Q * B *
       P**T. Q and  P**T  are  defined  as  products  of  elementary  reflectors  H(i)  and  G(i)
       respectively.

       Let  nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the order of the orthogonal
       matrix Q or P**T that is applied.

       If VECT = 'Q', A(IA:*,JA:*) is assumed to have been an NQ-by-K matrix:
       if nq >= k, Q = H(1) H(2) . . . H(k);
       if nq < k, Q = H(1) H(2) . . . H(nq-1).

       If VECT = 'P', A(IA:*,JA:*) is assumed to have been a K-by-NQ matrix:
       if k < nq, P = G(1) G(2) . . . G(k);
       if k >= nq, P = G(1) G(2) . . . G(nq-1).

       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

       VECT    (global input) CHARACTER
               = 'Q': apply Q or Q**T;
               = 'P': apply P or P**T.

       SIDE    (global input) CHARACTER
               = 'L': apply Q, Q**T, P or P**T from the Left;
               = 'R': apply Q, Q**T, P or P**T from the Right.

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

       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
               If VECT = 'Q', the number of columns in the original distributed matrix reduced by
               PDGEBRD.  If VECT = 'P', the number of rows in  the  original  distributed  matrix
               reduced by PDGEBRD.  K >= 0.

       A       (local input) DOUBLE PRECISION pointer into the local memory
               to   an   array   of   dimension  (LLD_A,LOCc(JA+MIN(NQ,K)-1))  if  VECT='Q',  and
               (LLD_A,LOCc(JA+NQ-1)) if VECT = 'P'. NQ = M if SIDE = 'L', and NQ =  N  otherwise.
               The  vectors  which define the elementary reflectors H(i) and G(i), whose products
               determine the matrices Q and P, as returned by PDGEBRD.  If VECT = 'Q',  LLD_A  >=
               max(1,LOCr(IA+NQ-1)); if VECT = 'P', LLD_A >= max(1,LOCr(IA+MIN(NQ,K)-1)).

       IA      (global input) INTEGER
               The row index in the global array A indicating the first row of sub( A ).

       JA      (global input) INTEGER
               The column index in the global array A indicating the first column of sub( A ).

       DESCA   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix A.

       TAU     (local input) DOUBLE PRECISION array, dimension
               LOCc(JA+MIN(NQ,K)-1)  if  VECT  =  'Q', LOCr(IA+MIN(NQ,K)-1) if VECT = 'P', TAU(i)
               must contain the scalar factor of the elementary  reflector H(i)  or  G(i),  which
               determines Q or P, as returned by PDGEBRD in its array argument TAUQ or TAUP.  TAU
               is tied to the distributed matrix A.

       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  local
               pieces  of  the  distributed  matrix  sub(C).   On  exit, if VECT='Q', sub( C ) is
               overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q; if VECT='P,
               sub( C ) is overwritten by P*sub( C ) or P'*sub( C ) or sub( C )*P or sub( C )*P'.

       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/local output) DOUBLE PRECISION array,
               dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK.

       LWORK   (local or global input) INTEGER
               The  dimension  of  the  array WORK.  LWORK is local input and must be at least If
               SIDE = 'L', NQ = M; if( (VECT = 'Q' and NQ >= K) or (VECT <> 'Q' and NQ  >  K)  ),
               IAA=IA;  JAA=JA;  MI=M; NI=N; ICC=IC; JCC=JC; else IAA=IA+1; JAA=JA; MI=M-1; NI=N;
               ICC=IC+1; JCC=JC; end if else if SIDE = 'R', NQ = N; if( (VECT = 'Q' and NQ >=  K)
               or  (VECT  <>  'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N; ICC=IC; JCC=JC; else
               IAA=IA; JAA=JA+1; MI=M; NI=N-1; ICC=IC; JCC=JC+1; end if end if

               If VECT = 'Q', If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2, (NqC0 +  MpC0)*NB_A
               ) + NB_A * NB_A else if SIDE = 'R', LWORK >= MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX(
               NpA0 + NUMROC( NUMROC( NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0  )
               )*NB_A  )  +  NB_A * NB_A end if else if VECT <> 'Q', if SIDE = 'L', LWORK >= MAX(
               (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 + NUMROC( NUMROC(  MI+IROFFC,  MB_A,  0,  0,
               NPROW  ),  MB_A,  0,  0, LCMP ), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R',
               LWORK >= MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end if end if

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

               IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW  =  INDXG2P(  IAA,
               MB_A,  MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JAA, NB_A, MYCOL, CSRC_A, NPCOL ),
               MqA0 = NUMROC( MI+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ), NpA0 =  NUMROC(  NI+IROFFA,
               MB_A, MYROW, IAROW, NPROW ),

               IROFFC  =  MOD(  ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW = INDXG2P( ICC,
               MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JCC, NB_C, MYCOL, CSRC_C, NPCOL  ),
               MpC0  =  NUMROC( MI+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC,
               NB_C, MYCOL, ICCOL, NPCOL ),

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

               If  LWORK  =  -1, then LWORK is global input and a workspace query is assumed; the
               routine only calculates the minimum and optimal size for all work arrays. Each  of
               these  values  is returned in the first entry of the corresponding work array, and
               no error message is issued by PXERBLA.

       INFO    (global output) INTEGER
               = 0:  successful exit
               < 0:  If the i-th argument is an array and the j-entry had an illegal value,  then
               INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then
               INFO = -i.

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

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

               If  VECT  =  'Q',  If  SIDE  =  'L',  (  MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND.
               IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC  )  else  If
               SIDE  = 'L', ( MB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', ( NB_A.EQ.NB_C
               .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) end if