          SUBROUTINE SGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
         $                   LWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDB, LWORK, M, N, P
    *     ..
    *     .. Array Arguments ..
          REAL               A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
         $                   WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SGGRQF computes a generalized RQ factorization of an M-by-N matrix A
    *  and a P-by-N matrix B:
    *
    *              A = R*Q,        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 B is square and nonsingular, the GRQ factorization
    *  of A and B implicitly gives the RQ factorization of A*inv(B):
    *
    *               A*inv(B) = (R*inv(T))*Z'
    *
    *  where inv(B) denotes the inverse of the matrix B, and Z' denotes the
    *  transpose of the matrix Z.
    *
    *  Arguments
    *  =========
    *
    *  M       (input) INTEGER
    *          The number of rows of the matrix A.  M >= 0.
    *
    *  P       (input) INTEGER
    *          The number of rows of the matrix B.  P >= 0.
    *
    *  N       (input) INTEGER
    *          The number of columns of the matrices A and B. N >= 0.
    *
    *  A       (input/output) REAL array, dimension (LDA,N)
    *          On entry, the M-by-N matrix A.
    *          On exit, if M <= N, the upper triangle of the subarray
    *          A(1:M,N-M+1:N) 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).
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A. LDA >= max(1,M).
    *
    *  TAUA    (output) REAL array, dimension (min(M,N))
    *          The scalar factors of the elementary reflectors which
    *          represent the orthogonal matrix Q (see Further Details).
    *
    *  B       (input/output) REAL array, dimension (LDB,N)
    *          On entry, the P-by-N matrix B.
    *          On exit, the elements on and above the diagonal of the array
    *          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).
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B. LDB >= max(1,P).
    *
    *  TAUB    (output) REAL array, dimension (min(P,N))
    *          The scalar factors of the elementary reflectors which
    *          represent the orthogonal matrix Z (see Further Details).
    *
    *  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
    *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK   (input) INTEGER
    *          The dimension of the array WORK. LWORK >= max(1,N,M,P).
    *          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
    *          where NB1 is the optimal blocksize for the RQ factorization
    *          of an M-by-N matrix, NB2 is the optimal blocksize for the
    *          QR factorization of a P-by-N matrix, and NB3 is the optimal
    *          blocksize for a call of SORMRQ.
    *
    *          If LWORK = -1, then a workspace query is assumed; the routine
    *          only calculates the optimal size of the WORK array, returns
    *          this value as the first entry of the WORK array, and no error
    *          message related to LWORK is issued by XERBLA.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INF0= -i, the i-th argument had an illegal value.
    *
    *  Further Details
    *  ===============
    *
    *  The matrix Q is represented as a product of elementary reflectors
    *
    *     Q = H(1) H(2) . . . H(k), 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(m-k+i,1:n-k+i-1), and taua in TAUA(i).
    *  To form Q explicitly, use LAPACK subroutine SORGRQ.
    *  To use Q to update another matrix, use LAPACK subroutine SORMRQ.
    *
    *  The matrix Z is represented as a product of elementary reflectors
    *
    *     Z = H(1) H(2) . . . H(k), 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(i+1:p,i),
    *  and taub in TAUB(i).
    *  To form Z explicitly, use LAPACK subroutine SORGQR.
    *  To use Z to update another matrix, use LAPACK subroutine SORMQR.
    *
    *  =====================================================================
    *
    *     .. Local Scalars ..
          LOGICAL            LQUERY
          INTEGER            LOPT, LWKOPT, NB, NB1, NB2, NB3
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SGEQRF, SGERQF, SORMRQ, XERBLA
    *     ..
    *     .. External Functions ..
          INTEGER            ILAENV
          EXTERNAL           ILAENV
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          INT, MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters
    *
          INFO = 0
          NB1 = ILAENV( 1, 'SGERQF', ' ', M, N, -1, -1 )
          NB2 = ILAENV( 1, 'SGEQRF', ' ', P, N, -1, -1 )
          NB3 = ILAENV( 1, 'SORMRQ', ' ', M, N, P, -1 )
          NB = MAX( NB1, NB2, NB3 )
          LWKOPT = MAX( N, M, P)*NB
          WORK( 1 ) = LWKOPT
          LQUERY = ( LWORK.EQ.-1 )
          IF( M.LT.0 ) THEN
             INFO = -1
          ELSE IF( P.LT.0 ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
             INFO = -5
          ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
             INFO = -8
          ELSE IF( LWORK.LT.MAX( 1, M, P, N ) .AND. .NOT.LQUERY ) THEN
             INFO = -11
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'SGGRQF', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     RQ factorization of M-by-N matrix A: A = R*Q
    *
          CALL SGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO )
          LOPT = WORK( 1 )
    *
    *     Update B := B*Q'
    *
          CALL SORMRQ( 'Right', 'Transpose', P, N, MIN( M, N ),
         $             A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK,
         $             LWORK, INFO )
          LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
    *
    *     QR factorization of P-by-N matrix B: B = Z*T
    *
          CALL SGEQRF( P, N, B, LDB, TAUB, WORK, LWORK, INFO )
          WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
    *
          RETURN
    *
    *     End of SGGRQF
    *
          END