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NAME

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

SYNOPSIS

       SUBROUTINE PZGGRQF( 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( * )

           COMPLEX*16      A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * )

PURPOSE

       PZGGRQF  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 unitary matrix, Z is a P-by-P unitary 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
       conjugate 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) COMPLEX*16 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 unitary 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) COMPLEX*16, array, dimension LOCr(IA+M-1)
               This array  contains  the  scalar  factors  of  the  elementary  reflectors  which
               represent  the  unitary  matrix  Q.  TAUA is tied to the distributed matrix A (see
               Further Details).  B       (local input/local output) COMPLEX*16 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 unitary 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) COMPLEX*16, array, dimension
               LOCc(JB+MIN(P,N)-1). This array contains the scalar factors TAUB of the elementary
               reflectors which represent the unitary matrix Z. TAUB is tied to  the  distributed
               matrix B (see Further Details).  WORK    (local workspace/local output) COMPLEX*16
               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 complex scalar, and v is a complex vector with v(n-k+i+1:n) = 0  and  v(n-
       k+i)  =  1; conjg(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 PZUNGRQ.
       To use Q to update another matrix, use ScaLAPACK subroutine PZUNMRQ.

       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 complex scalar, and v is a complex 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 PZUNGQR.
       To use Z to update another matrix, use ScaLAPACK subroutine PZUNMQR.

       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 )