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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 )