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    1:       SUBROUTINE DGEQL2( M, N, A, LDA, TAU, WORK, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       INTEGER            INFO, LDA, M, N
   10: *     ..
   11: *     .. Array Arguments ..
   12:       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
   13: *     ..
   14: *
   15: *  Purpose
   16: *  =======
   17: *
   18: *  DGEQL2 computes a QL factorization of a real m by n matrix A:
   19: *  A = Q * L.
   20: *
   21: *  Arguments
   22: *  =========
   23: *
   24: *  M       (input) INTEGER
   25: *          The number of rows of the matrix A.  M >= 0.
   26: *
   27: *  N       (input) INTEGER
   28: *          The number of columns of the matrix A.  N >= 0.
   29: *
   30: *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
   31: *          On entry, the m by n matrix A.
   32: *          On exit, if m >= n, the lower triangle of the subarray
   33: *          A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
   34: *          if m <= n, the elements on and below the (n-m)-th
   35: *          superdiagonal contain the m by n lower trapezoidal matrix L;
   36: *          the remaining elements, with the array TAU, represent the
   37: *          orthogonal matrix Q as a product of elementary reflectors
   38: *          (see Further Details).
   39: *
   40: *  LDA     (input) INTEGER
   41: *          The leading dimension of the array A.  LDA >= max(1,M).
   42: *
   43: *  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
   44: *          The scalar factors of the elementary reflectors (see Further
   45: *          Details).
   46: *
   47: *  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
   48: *
   49: *  INFO    (output) INTEGER
   50: *          = 0: successful exit
   51: *          < 0: if INFO = -i, the i-th argument had an illegal value
   52: *
   53: *  Further Details
   54: *  ===============
   55: *
   56: *  The matrix Q is represented as a product of elementary reflectors
   57: *
   58: *     Q = H(k) . . . H(2) H(1), where k = min(m,n).
   59: *
   60: *  Each H(i) has the form
   61: *
   62: *     H(i) = I - tau * v * v'
   63: *
   64: *  where tau is a real scalar, and v is a real vector with
   65: *  v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
   66: *  A(1:m-k+i-1,n-k+i), and tau in TAU(i).
   67: *
   68: *  =====================================================================
   69: *
   70: *     .. Parameters ..
   71:       DOUBLE PRECISION   ONE
   72:       PARAMETER          ( ONE = 1.0D+0 )
   73: *     ..
   74: *     .. Local Scalars ..
   75:       INTEGER            I, K
   76:       DOUBLE PRECISION   AII
   77: *     ..
   78: *     .. External Subroutines ..
   79:       EXTERNAL           DLARF, DLARFP, XERBLA
   80: *     ..
   81: *     .. Intrinsic Functions ..
   82:       INTRINSIC          MAX, MIN
   83: *     ..
   84: *     .. Executable Statements ..
   85: *
   86: *     Test the input arguments
   87: *
   88:       INFO = 0
   89:       IF( M.LT.0 ) THEN
   90:          INFO = -1
   91:       ELSE IF( N.LT.0 ) THEN
   92:          INFO = -2
   93:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
   94:          INFO = -4
   95:       END IF
   96:       IF( INFO.NE.0 ) THEN
   97:          CALL XERBLA( 'DGEQL2', -INFO )
   98:          RETURN
   99:       END IF
  100: *
  101:       K = MIN( M, N )
  102: *
  103:       DO 10 I = K, 1, -1
  104: *
  105: *        Generate elementary reflector H(i) to annihilate
  106: *        A(1:m-k+i-1,n-k+i)
  107: *
  108:          CALL DLARFP( M-K+I, A( M-K+I, N-K+I ), A( 1, N-K+I ), 1,
  109:      $                TAU( I ) )
  110: *
  111: *        Apply H(i) to A(1:m-k+i,1:n-k+i-1) from the left
  112: *
  113:          AII = A( M-K+I, N-K+I )
  114:          A( M-K+I, N-K+I ) = ONE
  115:          CALL DLARF( 'Left', M-K+I, N-K+I-1, A( 1, N-K+I ), 1, TAU( I ),
  116:      $               A, LDA, WORK )
  117:          A( M-K+I, N-K+I ) = AII
  118:    10 CONTINUE
  119:       RETURN
  120: *
  121: *     End of DGEQL2
  122: *
  123:       END