sgerq2.f

*> \brief \b SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE SGERQ2( M, N, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SGERQ2 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 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 TAU, represent the orthogonal matrix
*>          Q as a product of 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 REAL array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (M)
*> \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 gerq2
*
*> \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 sgerq2( M, N, A, LDA, TAU, WORK, 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, M, N
*     ..
*     .. Array Arguments ..
      REAL               A( lda, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE
      parameter( one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
      REAL               AII
*     ..
*     .. External Subroutines ..
      EXTERNAL           slarf, slarfg, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      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.NE.0 ) THEN
         CALL xerbla( 'SGERQ2', -info )
         RETURN
      END IF
*
      k = min( m, n )
*
      DO 10 i = k, 1, -1
*
*        Generate elementary reflector H(i) to annihilate
*        A(m-k+i,1:n-k+i-1)
*
         CALL slarfg( n-k+i, a( m-k+i, n-k+i ), a( m-k+i, 1 ), lda,
     $                tau( i ) )
*
*        Apply H(i) to A(1:m-k+i-1,1:n-k+i) from the right
*
         aii = a( m-k+i, n-k+i )
         a( m-k+i, n-k+i ) = one
         CALL slarf( 'Right', m-k+i-1, n-k+i, a( m-k+i, 1 ), lda,
     $               tau( i ), a, lda, work )
         a( m-k+i, n-k+i ) = aii
   10 CONTINUE
      RETURN
*
*     End of SGERQ2
*
      END