dgerqf()

subroutine dgerqf	(	integer	m,
		integer	n,
		double precision, dimension( lda, * )	a,
		integer	lda,
		double precision, dimension( * )	tau,
		double precision, dimension( * )	work,
		integer	lwork,
		integer	info
	)

DGERQF

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Purpose:

 DGERQF computes an RQ factorization of a real M-by-N matrix A:
 A = R * Q.

Parameters

[in]	M	M is INTEGER The number of rows of the matrix A. M >= 0.
[in]	N	N is INTEGER The number of columns of the matrix A. N >= 0.
[in,out]	A	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).
[in]	LDA	LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
[out]	TAU	TAU is DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details).
[out]	WORK	WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]	LWORK	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.
[out]	INFO	INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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 - 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).

Definition at line 138 of file dgerqf.f.

*
*  -- 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
*

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