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NAME

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

SYNOPSIS

       SUBROUTINE PSGGRQF( 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( * )

           REAL            A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * )

PURPOSE

       PSGGRQF  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) REAL 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) REAL, 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) REAL 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) REAL, 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)  REAL  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
       PSORGRQ.
       To use Q to update another matrix, use ScaLAPACK subroutine PSORMRQ.

       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 PSORGQR.
       To use Z to update another matrix, use ScaLAPACK subroutine PSORMQR.

       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 )