```      SUBROUTINE DGGRQF( 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 ..
DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
\$                   WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGGRQF 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors which
*          represent the orthogonal matrix Q (see Further Details).
*
*  B       (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (min(P,N))
*          The scalar factors of the elementary reflectors which
*          represent the orthogonal matrix Z (see Further Details).
*
*  WORK    (workspace/output) DOUBLE PRECISION 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 DORMRQ.
*
*          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 DORGRQ.
*  To use Q to update another matrix, use LAPACK subroutine DORMRQ.
*
*  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 DORGQR.
*  To use Z to update another matrix, use LAPACK subroutine DORMQR.
*
*  =====================================================================
*
*     .. Local Scalars ..
LOGICAL            LQUERY
INTEGER            LOPT, LWKOPT, NB, NB1, NB2, NB3
*     ..
*     .. External Subroutines ..
EXTERNAL           DGEQRF, DGERQF, DORMRQ, XERBLA
*     ..
*     .. External Functions ..
INTEGER            ILAENV
EXTERNAL           ILAENV
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          INT, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
INFO = 0
NB1 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'DGEQRF', ' ', P, N, -1, -1 )
NB3 = ILAENV( 1, 'DORMRQ', ' ', 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( 'DGGRQF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
*     RQ factorization of M-by-N matrix A: A = R*Q
*
CALL DGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO )
LOPT = WORK( 1 )
*
*     Update B := B*Q'
*
CALL DORMRQ( '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 DGEQRF( P, N, B, LDB, TAUB, WORK, LWORK, INFO )
WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
*
RETURN
*
*     End of DGGRQF
*
END

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