dlae2.f

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*> \brief \b DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
*       .. Scalar Arguments ..
*       DOUBLE PRECISION   A, B, C, RT1, RT2
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAE2  computes the eigenvalues of a 2-by-2 symmetric matrix
*>    [  A   B  ]
*>    [  B   C  ].
*> On return, RT1 is the eigenvalue of larger absolute value, and RT2
*> is the eigenvalue of smaller absolute value.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION
*>          The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION
*>          The (1,2) and (2,1) elements of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is DOUBLE PRECISION
*>          The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] RT1
*> \verbatim
*>          RT1 is DOUBLE PRECISION
*>          The eigenvalue of larger absolute value.
*> \endverbatim
*>
*> \param[out] RT2
*> \verbatim
*>          RT2 is DOUBLE PRECISION
*>          The eigenvalue of smaller absolute value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lae2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  RT1 is accurate to a few ulps barring over/underflow.
*>
*>  RT2 may be inaccurate if there is massive cancellation in the
*>  determinant A*C-B*B; higher precision or correctly rounded or
*>  correctly truncated arithmetic would be needed to compute RT2
*>  accurately in all cases.
*>
*>  Overflow is possible only if RT1 is within a factor of 5 of overflow.
*>  Underflow is harmless if the input data is 0 or exceeds
*>     underflow_threshold / macheps.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE dlae2( A, B, C, RT1, RT2 )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, RT1, RT2
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      parameter( one = 1.0d0 )
      DOUBLE PRECISION   TWO
      parameter( two = 2.0d0 )
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d0 )
      DOUBLE PRECISION   HALF
      parameter( half = 0.5d0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   AB, ACMN, ACMX, ADF, DF, RT, SM, TB
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, sqrt
*     ..
*     .. Executable Statements ..
*
*     Compute the eigenvalues
*
      sm = a + c
      df = a - c
      adf = abs( df )
      tb = b + b
      ab = abs( tb )
      IF( abs( a ).GT.abs( c ) ) THEN
         acmx = a
         acmn = c
      ELSE
         acmx = c
         acmn = a
      END IF
      IF( adf.GT.ab ) THEN
         rt = adf*sqrt( one+( ab / adf )**2 )
      ELSE IF( adf.LT.ab ) THEN
         rt = ab*sqrt( one+( adf / ab )**2 )
      ELSE
*
*        Includes case AB=ADF=0
*
         rt = ab*sqrt( two )
      END IF
      IF( sm.LT.zero ) THEN
         rt1 = half*( sm-rt )
*
*        Order of execution important.
*        To get fully accurate smaller eigenvalue,
*        next line needs to be executed in higher precision.
*
         rt2 = ( acmx / rt1 )*acmn - ( b / rt1 )*b
      ELSE IF( sm.GT.zero ) THEN
         rt1 = half*( sm+rt )
*
*        Order of execution important.
*        To get fully accurate smaller eigenvalue,
*        next line needs to be executed in higher precision.
*
         rt2 = ( acmx / rt1 )*acmn - ( b / rt1 )*b
      ELSE
*
*        Includes case RT1 = RT2 = 0
*
         rt1 = half*rt
         rt2 = -half*rt
      END IF
      RETURN
*
*     End of DLAE2
*
      END