*> \brief \b DGERQF * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DGERQF + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LWORK, M, N * .. * .. Array Arguments .. * DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DGERQF computes an RQ factorization of a real M-by-N matrix A: *> A = R * Q. *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is 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 TAU, represent the *> orthogonal matrix Q as a product of min(m,n) elementary *> reflectors (see Further Details). *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[out] TAU *> \verbatim *> TAU is DOUBLE PRECISION array, dimension (min(M,N)) *> The scalar factors of the elementary reflectors (see Further *> Details). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. *> LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise. *> For optimum performance LWORK >= M*NB, where NB is *> the optimal blocksize. *> *> 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. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup doubleGEcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> 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 - tau * v * v**T *> *> where tau 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 tau in TAU(i). *> \endverbatim *> * ===================================================================== SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) * .. * * ===================================================================== * * .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT, \$ MU, NB, NBMIN, NU, NX * .. * .. External Subroutines .. EXTERNAL DGERQ2, DLARFB, DLARFT, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF * IF( INFO.EQ.0 ) THEN K = MIN( M, N ) IF( K.EQ.0 ) THEN LWKOPT = 1 ELSE NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 ) LWKOPT = M*NB END IF WORK( 1 ) = LWKOPT * IF ( .NOT.LQUERY ) THEN IF( LWORK.LE.0 .OR. ( N.GT.0 .AND. LWORK.LT.MAX( 1, M ) ) ) \$ INFO = -7 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGERQF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( K.EQ.0 ) THEN RETURN END IF * NBMIN = 2 NX = 1 IWS = M IF( NB.GT.1 .AND. NB.LT.K ) THEN * * Determine when to cross over from blocked to unblocked code. * NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) ) IF( NX.LT.K ) THEN * * Determine if workspace is large enough for blocked code. * LDWORK = M IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN * * Not enough workspace to use optimal NB: reduce NB and * determine the minimum value of NB. * NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1, \$ -1 ) ) END IF END IF END IF * IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN * * Use blocked code initially. * The last kk rows are handled by the block method. * KI = ( ( K-NX-1 ) / NB )*NB KK = MIN( K, KI+NB ) * DO 10 I = K - KK + KI + 1, K - KK + 1, -NB IB = MIN( K-I+1, NB ) * * Compute the RQ factorization of the current block * A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1) * CALL DGERQ2( IB, N-K+I+IB-1, A( M-K+I, 1 ), LDA, TAU( I ), \$ WORK, IINFO ) IF( M-K+I.GT.1 ) THEN * * Form the triangular factor of the block reflector * H = H(i+ib-1) . . . H(i+1) H(i) * CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB, \$ A( M-K+I, 1 ), LDA, TAU( I ), WORK, LDWORK ) * * Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right * CALL DLARFB( 'Right', 'No transpose', 'Backward', \$ 'Rowwise', M-K+I-1, N-K+I+IB-1, IB, \$ A( M-K+I, 1 ), LDA, WORK, LDWORK, A, LDA, \$ WORK( IB+1 ), LDWORK ) END IF 10 CONTINUE MU = M - K + I + NB - 1 NU = N - K + I + NB - 1 ELSE MU = M NU = N END IF * * Use unblocked code to factor the last or only block * IF( MU.GT.0 .AND. NU.GT.0 ) \$ CALL DGERQ2( MU, NU, A, LDA, TAU, WORK, IINFO ) * WORK( 1 ) = IWS RETURN * * End of DGERQF * END