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NAME

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

SYNOPSIS

       SUBROUTINE PCGGQRF( N,  M, P, 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         A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * )

PURPOSE

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

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

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

       where R11 is upper triangular, and

       if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                        P-N  N                           ( T21 ) P
                                                            P

       where T12 or T21 is upper triangular.

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

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

       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

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

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

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

       A       (local input/local output) COMPLEX pointer into the
               local memory to an array of  dimension  (LLD_A,  LOCc(JA+M-1)).
               On  entry,  the  local  pieces of the N-by-M distributed matrix
               sub( A ) which is to be factored.  On exit, the elements on and
               above  the diagonal of sub( A ) contain the min(N,M) by M upper
               trapezoidal matrix R (R is upper triangular if  N  >=  M);  the
               elements below the diagonal, with the array TAUA, represent the
               unitary matrix Q as a product of min(N,M) 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, array, dimension
               LOCc(JA+MIN(N,M)-1). This array  contains  the  scalar  factors
               TAUA  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
               pointer into the local memory to an array of dimension  (LLD_B,
               LOCc(JB+P-1)).   On  entry,  the  local  pieces  of  the N-by-P
               distributed matrix sub( B ) which is to be factored.  On  exit,
               if  N  <=  P,  the upper triangle of B(IB:IB+N-1,JB+P-N:JB+P-1)
               contains the N by N upper triangular matrix T; if N  >  P,  the
               elements on and above the (N-P)-th subdiagonal contain the N by
               P upper trapezoidal matrix T; the remaining elements, 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, array, dimension LOCr(IB+N-1)
               This  array  contains  the  scalar  factors  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 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( NB_A * ( NpA0 + MqA0 + NB_A  ),  MAX(
               (NB_A*(NB_A-1))/2, (PqB0 + NpB0)*NB_A ) + NB_A * NB_A, MB_B * (
               NpB0 + PqB0 + MB_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 ), NpA0    =  NUMROC(  N+IROFFA,
               MB_A,  MYROW,  IAROW, NPROW ), MqA0   = NUMROC( M+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 ), NpB0    =  NUMROC(  N+IROFFB,
               MB_B,  MYROW,  IBROW, NPROW ), PqB0   = NUMROC( P+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(ja) H(ja+1) . . . H(ja+k-1), where k = min(n,m).

       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(1:i-1)
       = 0 and v(i) = 1; v(i+1:n) is stored on exit in
       A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1).
       To form Q explicitly, use ScaLAPACK subroutine PCUNGQR.
       To use Q to update another matrix, use ScaLAPACK subroutine PCUNMQR.

       The matrix Z is represented as a product of elementary reflectors

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

       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(p-
       k+i+1:p) = 0 and v(p-k+i) = 1; conjg(v(1:p-k+i-1)) is stored on exit in
       B(ib+n-k+i-1,jb:jb+p-k+i-2),  and  taub in TAUB(ib+n-k+i-1).  To form Z
       explicitly, use ScaLAPACK subroutine PCUNGRQ.
       To use Z to update another matrix, use ScaLAPACK subroutine PCUNMRQ.

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

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

       ( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW )