text: *> \brief \b SGEQR2 computes the QR 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 SGEQR2( M, N, A, LDA, TAU, WORK, INFO )
  *
  *       .. Scalar Arguments ..
  *       INTEGER            INFO, LDA, M, N
  *       ..
  *       .. Array Arguments ..
  *       REAL               A( LDA, * ), TAU( * ), WORK( * )
  *       ..
  *
  *
  *> \par Purpose:
  *  =============
  *>
  *> \verbatim
  *>
  *> SGEQR2 computes a QR factorization of a real m-by-n matrix A:
  *>
  *>    A = Q * ( R ),
  *>            ( 0 )
  *>
  *> where:
  *>
  *>    Q is a m-by-m orthogonal matrix;
  *>    R is an upper-triangular n-by-n matrix;
  *>    0 is a (m-n)-by-n zero matrix, if m > n.
  *>
  *> \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, the elements on and above the diagonal of the array
  *>          contain the min(m,n) by n upper trapezoidal matrix R (R is
  *>          upper triangular if m >= n); the elements below the diagonal,
  *>          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 (N)
  *> \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.
  *
  *> \date November 2019
  *
  *> \ingroup realGEcomputational
  *
  *> \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(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
  *>  and tau in TAU(i).
  *> \endverbatim
  *>
  *  =====================================================================
        SUBROUTINE SGEQR2( M, N, A, LDA, TAU, WORK, INFO )
  *
  *  -- LAPACK computational routine (version 3.9.0) --
  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  *     November 2019
  *
  *     .. 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( 'SGEQR2', -INFO )
           RETURN
        END IF
  *
        K = MIN( M, N )
  *
        DO 10 I = 1, K
  *
  *        Generate elementary reflector H(i) to annihilate A(i+1:m,i)
  *
           CALL SLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
       $                TAU( I ) )
           IF( I.LT.N ) THEN
  *
  *           Apply H(i) to A(i:m,i+1:n) from the left
  *
              AII = A( I, I )
              A( I, I ) = ONE
              CALL SLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
       $                  A( I, I+1 ), LDA, WORK )
              A( I, I ) = AII
           END IF
     10 CONTINUE
        RETURN
  *
  *     End of SGEQR2
  *
        END
  