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    1: *> \brief \b DLAPLL measures the linear dependence of two vectors.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download DLAPLL + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapll.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapll.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapll.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            INCX, INCY, N
   25: *       DOUBLE PRECISION   SSMIN
   26: *       ..
   27: *       .. Array Arguments ..
   28: *       DOUBLE PRECISION   X( * ), Y( * )
   29: *       ..
   30: *
   31: *
   32: *> \par Purpose:
   33: *  =============
   34: *>
   35: *> \verbatim
   36: *>
   37: *> Given two column vectors X and Y, let
   38: *>
   39: *>                      A = ( X Y ).
   40: *>
   41: *> The subroutine first computes the QR factorization of A = Q*R,
   42: *> and then computes the SVD of the 2-by-2 upper triangular matrix R.
   43: *> The smaller singular value of R is returned in SSMIN, which is used
   44: *> as the measurement of the linear dependency of the vectors X and Y.
   45: *> \endverbatim
   46: *
   47: *  Arguments:
   48: *  ==========
   49: *
   50: *> \param[in] N
   51: *> \verbatim
   52: *>          N is INTEGER
   53: *>          The length of the vectors X and Y.
   54: *> \endverbatim
   55: *>
   56: *> \param[in,out] X
   57: *> \verbatim
   58: *>          X is DOUBLE PRECISION array,
   59: *>                         dimension (1+(N-1)*INCX)
   60: *>          On entry, X contains the N-vector X.
   61: *>          On exit, X is overwritten.
   62: *> \endverbatim
   63: *>
   64: *> \param[in] INCX
   65: *> \verbatim
   66: *>          INCX is INTEGER
   67: *>          The increment between successive elements of X. INCX > 0.
   68: *> \endverbatim
   69: *>
   70: *> \param[in,out] Y
   71: *> \verbatim
   72: *>          Y is DOUBLE PRECISION array,
   73: *>                         dimension (1+(N-1)*INCY)
   74: *>          On entry, Y contains the N-vector Y.
   75: *>          On exit, Y is overwritten.
   76: *> \endverbatim
   77: *>
   78: *> \param[in] INCY
   79: *> \verbatim
   80: *>          INCY is INTEGER
   81: *>          The increment between successive elements of Y. INCY > 0.
   82: *> \endverbatim
   83: *>
   84: *> \param[out] SSMIN
   85: *> \verbatim
   86: *>          SSMIN is DOUBLE PRECISION
   87: *>          The smallest singular value of the N-by-2 matrix A = ( X Y ).
   88: *> \endverbatim
   89: *
   90: *  Authors:
   91: *  ========
   92: *
   93: *> \author Univ. of Tennessee
   94: *> \author Univ. of California Berkeley
   95: *> \author Univ. of Colorado Denver
   96: *> \author NAG Ltd.
   97: *
   98: *> \ingroup doubleOTHERauxiliary
   99: *
  100: *  =====================================================================
  101:       SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
  102: *
  103: *  -- LAPACK auxiliary routine --
  104: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  105: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  106: *
  107: *     .. Scalar Arguments ..
  108:       INTEGER            INCX, INCY, N
  109:       DOUBLE PRECISION   SSMIN
  110: *     ..
  111: *     .. Array Arguments ..
  112:       DOUBLE PRECISION   X( * ), Y( * )
  113: *     ..
  114: *
  115: *  =====================================================================
  116: *
  117: *     .. Parameters ..
  118:       DOUBLE PRECISION   ZERO, ONE
  119:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
  120: *     ..
  121: *     .. Local Scalars ..
  122:       DOUBLE PRECISION   A11, A12, A22, C, SSMAX, TAU
  123: *     ..
  124: *     .. External Functions ..
  125:       DOUBLE PRECISION   DDOT
  126:       EXTERNAL           DDOT
  127: *     ..
  128: *     .. External Subroutines ..
  129:       EXTERNAL           DAXPY, DLARFG, DLAS2
  130: *     ..
  131: *     .. Executable Statements ..
  132: *
  133: *     Quick return if possible
  134: *
  135:       IF( N.LE.1 ) THEN
  136:          SSMIN = ZERO
  137:          RETURN
  138:       END IF
  139: *
  140: *     Compute the QR factorization of the N-by-2 matrix ( X Y )
  141: *
  142:       CALL DLARFG( N, X( 1 ), X( 1+INCX ), INCX, TAU )
  143:       A11 = X( 1 )
  144:       X( 1 ) = ONE
  145: *
  146:       C = -TAU*DDOT( N, X, INCX, Y, INCY )
  147:       CALL DAXPY( N, C, X, INCX, Y, INCY )
  148: *
  149:       CALL DLARFG( N-1, Y( 1+INCY ), Y( 1+2*INCY ), INCY, TAU )
  150: *
  151:       A12 = Y( 1 )
  152:       A22 = Y( 1+INCY )
  153: *
  154: *     Compute the SVD of 2-by-2 Upper triangular matrix.
  155: *
  156:       CALL DLAS2( A11, A12, A22, SSMIN, SSMAX )
  157: *
  158:       RETURN
  159: *
  160: *     End of DLAPLL
  161: *
  162:       END