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    1: *> \brief \b ZLAPLL 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 ZLAPLL + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlapll.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlapll.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlapll.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZLAPLL( N, X, INCX, Y, INCY, SSMIN )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            INCX, INCY, N
   25: *       DOUBLE PRECISION   SSMIN
   26: *       ..
   27: *       .. Array Arguments ..
   28: *       COMPLEX*16         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 COMPLEX*16 array, dimension (1+(N-1)*INCX)
   59: *>          On entry, X contains the N-vector X.
   60: *>          On exit, X is overwritten.
   61: *> \endverbatim
   62: *>
   63: *> \param[in] INCX
   64: *> \verbatim
   65: *>          INCX is INTEGER
   66: *>          The increment between successive elements of X. INCX > 0.
   67: *> \endverbatim
   68: *>
   69: *> \param[in,out] Y
   70: *> \verbatim
   71: *>          Y is COMPLEX*16 array, dimension (1+(N-1)*INCY)
   72: *>          On entry, Y contains the N-vector Y.
   73: *>          On exit, Y is overwritten.
   74: *> \endverbatim
   75: *>
   76: *> \param[in] INCY
   77: *> \verbatim
   78: *>          INCY is INTEGER
   79: *>          The increment between successive elements of Y. INCY > 0.
   80: *> \endverbatim
   81: *>
   82: *> \param[out] SSMIN
   83: *> \verbatim
   84: *>          SSMIN is DOUBLE PRECISION
   85: *>          The smallest singular value of the N-by-2 matrix A = ( X Y ).
   86: *> \endverbatim
   87: *
   88: *  Authors:
   89: *  ========
   90: *
   91: *> \author Univ. of Tennessee
   92: *> \author Univ. of California Berkeley
   93: *> \author Univ. of Colorado Denver
   94: *> \author NAG Ltd.
   95: *
   96: *> \date December 2016
   97: *
   98: *> \ingroup complex16OTHERauxiliary
   99: *
  100: *  =====================================================================
  101:       SUBROUTINE ZLAPLL( N, X, INCX, Y, INCY, SSMIN )
  102: *
  103: *  -- LAPACK auxiliary routine (version 3.7.0) --
  104: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  105: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  106: *     December 2016
  107: *
  108: *     .. Scalar Arguments ..
  109:       INTEGER            INCX, INCY, N
  110:       DOUBLE PRECISION   SSMIN
  111: *     ..
  112: *     .. Array Arguments ..
  113:       COMPLEX*16         X( * ), Y( * )
  114: *     ..
  115: *
  116: *  =====================================================================
  117: *
  118: *     .. Parameters ..
  119:       DOUBLE PRECISION   ZERO
  120:       PARAMETER          ( ZERO = 0.0D+0 )
  121:       COMPLEX*16         CONE
  122:       PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
  123: *     ..
  124: *     .. Local Scalars ..
  125:       DOUBLE PRECISION   SSMAX
  126:       COMPLEX*16         A11, A12, A22, C, TAU
  127: *     ..
  128: *     .. Intrinsic Functions ..
  129:       INTRINSIC          ABS, DCONJG
  130: *     ..
  131: *     .. External Functions ..
  132:       COMPLEX*16         ZDOTC
  133:       EXTERNAL           ZDOTC
  134: *     ..
  135: *     .. External Subroutines ..
  136:       EXTERNAL           DLAS2, ZAXPY, ZLARFG
  137: *     ..
  138: *     .. Executable Statements ..
  139: *
  140: *     Quick return if possible
  141: *
  142:       IF( N.LE.1 ) THEN
  143:          SSMIN = ZERO
  144:          RETURN
  145:       END IF
  146: *
  147: *     Compute the QR factorization of the N-by-2 matrix ( X Y )
  148: *
  149:       CALL ZLARFG( N, X( 1 ), X( 1+INCX ), INCX, TAU )
  150:       A11 = X( 1 )
  151:       X( 1 ) = CONE
  152: *
  153:       C = -DCONJG( TAU )*ZDOTC( N, X, INCX, Y, INCY )
  154:       CALL ZAXPY( N, C, X, INCX, Y, INCY )
  155: *
  156:       CALL ZLARFG( N-1, Y( 1+INCY ), Y( 1+2*INCY ), INCY, TAU )
  157: *
  158:       A12 = Y( 1 )
  159:       A22 = Y( 1+INCY )
  160: *
  161: *     Compute the SVD of 2-by-2 Upper triangular matrix.
  162: *
  163:       CALL DLAS2( ABS( A11 ), ABS( A12 ), ABS( A22 ), SSMIN, SSMAX )
  164: *
  165:       RETURN
  166: *
  167: *     End of ZLAPLL
  168: *
  169:       END