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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/
*
*> \htmlonly
*> Download DLAE2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlae2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlae2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlae2.f">
*> [TXT]</a>
*> \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.
*
*> \date December 2016
*
*> \ingroup OTHERauxiliary
*
*> \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 (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. 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
|
