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NAME

       PDGGRQF  -  compute  a generalized RQ factorization of an M-by-N matrix
       sub( A ) = A(IA:IA+M-1,JA:JA+N-1)

SYNOPSIS

       SUBROUTINE PDGGRQF( M, P, N, A, IA, JA, DESCA, TAUA, B, IB, JB,  DESCB,
                           TAUB, WORK, LWORK, INFO )

           INTEGER         IA, IB, INFO, JA, JB, LWORK, M, N, P

           INTEGER         DESCA( * ), DESCB( * )

           DOUBLE          PRECISION  A(  *  ),  B( * ), TAUA( * ), TAUB( * ),
                           WORK( * )

PURPOSE

       PDGGRQF computes a generalized RQ factorization  of  an  M-by-N  matrix
       sub(  A  )  =  A(IA:IA+M-1,JA:JA+N-1)  and  a  P-by-N matrix sub( B ) =
       B(IB:IB+P-1,JB:JB+N-1):

                   sub( A ) = R*Q,        sub( B ) = Z*T*Q,

       where Q is an N-by-N  orthogonal  matrix,  Z  is  a  P-by-P  orthogonal
       matrix, and R and T assume one of the forms:

       if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
                        N-M  M                           ( R21 ) N
                                                            N

       where R12 or R21 is upper triangular, and

       if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
                       (  0  ) P-N                         P   N-P
                          N

       where T11 is upper triangular.

       In  particular,  if  sub(  B  )  is  square  and  nonsingular,  the GRQ
       factorization of sub( A  )  and  sub(  B  )  implicitly  gives  the  RQ
       factorization of sub( A )*inv( sub( B ) ):

                    sub( A )*inv( sub( B ) ) = (R*inv(T))*Z’

       where  inv(  sub( B ) ) denotes the inverse of the matrix sub( B ), and
       Z’ denotes the transpose of matrix Z.

       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

       M       (global input) INTEGER
               The  number of rows to be operated on i.e the number of rows of
               the distributed submatrix sub( A ).  M >= 0.

       P       (global input) INTEGER
               The number of rows to be operated on i.e the number of rows  of
               the distributed submatrix sub( B ).  P >= 0.

       N       (global input) INTEGER
               The  number  of  columns  to  be  operated on i.e the number of
               columns of the distributed submatrices sub( A ) and sub(  B  ).
               N >= 0.

       A       (local input/local output) DOUBLE PRECISION pointer into the
               local  memory  to  an array of dimension (LLD_A, LOCc(JA+N-1)).
               On entry, the local pieces of  the  M-by-N  distributed  matrix
               sub( A ) which is to be factored. On exit, if M <= N, the upper
               triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) contains the M  by  M
               upper triangular matrix R; if M >= N, the elements on and above
               the (M-N)-th subdiagonal contain the M by N  upper  trapezoidal
               matrix   R;  the  remaining  elements,  with  the  array  TAUA,
               represent the orthogonal matrix Q as a  product  of  elementary
               reflectors  (see  Further  Details).   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.

       TAUA    (local output) DOUBLE PRECISION, array, dimension LOCr(IA+M-1)
               This array  contains  the  scalar  factors  of  the  elementary
               reflectors  which  represent  the  orthogonal unitary matrix Q.
               TAUA is tied to the distributed matrix A (see Further Details).

       B       (local input/local output) DOUBLE PRECISION pointer into the
               local  memory  to  an array of dimension (LLD_B, LOCc(JB+N-1)).
               On entry, the local pieces of  the  P-by-N  distributed  matrix
               sub( B ) which is to be factored.  On exit, the elements on and
               above the diagonal of sub( B ) contain the min(P,N) by N  upper
               trapezoidal  matrix  T  (T  is upper triangular if P >= N); the
               elements below the diagonal, with the array TAUB, represent the
               orthogonal  matrix Z as a product of elementary reflectors (see
               Further Details).  IB      (global input) INTEGER The row index
               in the global array B indicating the first row of sub( B ).

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

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

       TAUB    (local output) DOUBLE PRECISION, array, dimension
               LOCc(JB+MIN(P,N)-1). This array  contains  the  scalar  factors
               TAUB   of   the   elementary  reflectors  which  represent  the
               orthogonal matrix Z. TAUB is tied to the distributed  matrix  B
               (see  Further Details).  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 LWORK >= MAX( MB_A * ( MpA0 + NqA0 + MB_A  ),  MAX(
               (MB_A*(MB_A-1))/2, (PpB0 + NqB0)*MB_A ) + MB_A * MB_A, NB_B * (
               PpB0 + NqB0 + NB_B ) ), where

               IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A  ),  IAROW
               =  INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL  = INDXG2P(
               JA, NB_A, MYCOL, CSRC_A, NPCOL ), MpA0    =  NUMROC(  M+IROFFA,
               MB_A,  MYROW,  IAROW, NPROW ), NqA0   = NUMROC( N+ICOFFA, NB_A,
               MYCOL, IACOL, NPCOL ),

               IROFFB = MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B  ),  IBROW
               =  INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL  = INDXG2P(
               JB, NB_B, MYCOL, CSRC_B, NPCOL ), PpB0    =  NUMROC(  P+IROFFB,
               MB_B,  MYROW,  IBROW, NPROW ), NqB0   = NUMROC( N+ICOFFB, NB_B,
               MYCOL, IBCOL, NPCOL ),

               and NUMROC, INDXG2P 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.

FURTHER DETAILS

       The matrix Q is represented as a product of elementary reflectors

          Q = H(ia) H(ia+1) . . . H(ia+k-1), where k = min(m,n).

       Each H(i) has the form

          H(i) = I - taua * v * v’

       where taua is a real scalar, and v is a real vector with
       v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored  on  exit  in
       A(ia+m-k+i-1,ja:ja+n-k+i-2),  and  taua in TAUA(ia+m-k+i-1).  To form Q
       explicitly, use ScaLAPACK subroutine PDORGRQ.
       To use Q to update another matrix, use ScaLAPACK subroutine PDORMRQ.

       The matrix Z is represented as a product of elementary reflectors

          Z = H(jb) H(jb+1) . . . H(jb+k-1), where k = min(p,n).

       Each H(i) has the form

          H(i) = I - taub * v * v’

       where taub is a real scalar, and v is a real vector with
       v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in
       B(ib+i:ib+p-1,jb+i-1), and taub in TAUB(jb+i-1).
       To form Z explicitly, use ScaLAPACK subroutine PDORGQR.
       To use Z to update another matrix, use ScaLAPACK subroutine PDORMQR.

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

       The distributed submatrices sub( A ) and sub(  B  )  must  verify  some
       alignment properties, namely the following expression should be true:

       ( NB_A.EQ.NB_B .AND. ICOFFA.EQ.ICOFFB .AND. IACOL.EQ.IBCOL )