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*> \brief \b CLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CLAESY + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/claesy.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/claesy.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/claesy.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CLAESY( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
*
* .. Scalar Arguments ..
* COMPLEX A, B, C, CS1, EVSCAL, RT1, RT2, SN1
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix
*> ( ( A, B );( B, C ) )
*> provided the norm of the matrix of eigenvectors is larger than
*> some threshold value.
*>
*> RT1 is the eigenvalue of larger absolute value, and RT2 of
*> smaller absolute value. If the eigenvectors are computed, then
*> on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence
*>
*> [ CS1 SN1 ] . [ A B ] . [ CS1 -SN1 ] = [ RT1 0 ]
*> [ -SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ]
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is COMPLEX
*> The ( 1, 1 ) element of input matrix.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is COMPLEX
*> The ( 1, 2 ) element of input matrix. The ( 2, 1 ) element
*> is also given by B, since the 2-by-2 matrix is symmetric.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is COMPLEX
*> The ( 2, 2 ) element of input matrix.
*> \endverbatim
*>
*> \param[out] RT1
*> \verbatim
*> RT1 is COMPLEX
*> The eigenvalue of larger modulus.
*> \endverbatim
*>
*> \param[out] RT2
*> \verbatim
*> RT2 is COMPLEX
*> The eigenvalue of smaller modulus.
*> \endverbatim
*>
*> \param[out] EVSCAL
*> \verbatim
*> EVSCAL is COMPLEX
*> The complex value by which the eigenvector matrix was scaled
*> to make it orthonormal. If EVSCAL is zero, the eigenvectors
*> were not computed. This means one of two things: the 2-by-2
*> matrix could not be diagonalized, or the norm of the matrix
*> of eigenvectors before scaling was larger than the threshold
*> value THRESH (set below).
*> \endverbatim
*>
*> \param[out] CS1
*> \verbatim
*> CS1 is COMPLEX
*> \endverbatim
*>
*> \param[out] SN1
*> \verbatim
*> SN1 is COMPLEX
*> If EVSCAL .NE. 0, ( CS1, SN1 ) is the unit right eigenvector
*> for RT1.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complexSYauxiliary
*
* =====================================================================
SUBROUTINE CLAESY( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
*
* -- 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 ..
COMPLEX A, B, C, CS1, EVSCAL, RT1, RT2, SN1
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E0 )
REAL ONE
PARAMETER ( ONE = 1.0E0 )
COMPLEX CONE
PARAMETER ( CONE = ( 1.0E0, 0.0E0 ) )
REAL HALF
PARAMETER ( HALF = 0.5E0 )
REAL THRESH
PARAMETER ( THRESH = 0.1E0 )
* ..
* .. Local Scalars ..
REAL BABS, EVNORM, TABS, Z
COMPLEX S, T, TMP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
*
* Special case: The matrix is actually diagonal.
* To avoid divide by zero later, we treat this case separately.
*
IF( ABS( B ).EQ.ZERO ) THEN
RT1 = A
RT2 = C
IF( ABS( RT1 ).LT.ABS( RT2 ) ) THEN
TMP = RT1
RT1 = RT2
RT2 = TMP
CS1 = ZERO
SN1 = ONE
ELSE
CS1 = ONE
SN1 = ZERO
END IF
ELSE
*
* Compute the eigenvalues and eigenvectors.
* The characteristic equation is
* lambda **2 - (A+C) lambda + (A*C - B*B)
* and we solve it using the quadratic formula.
*
S = ( A+C )*HALF
T = ( A-C )*HALF
*
* Take the square root carefully to avoid over/under flow.
*
BABS = ABS( B )
TABS = ABS( T )
Z = MAX( BABS, TABS )
IF( Z.GT.ZERO )
$ T = Z*SQRT( ( T / Z )**2+( B / Z )**2 )
*
* Compute the two eigenvalues. RT1 and RT2 are exchanged
* if necessary so that RT1 will have the greater magnitude.
*
RT1 = S + T
RT2 = S - T
IF( ABS( RT1 ).LT.ABS( RT2 ) ) THEN
TMP = RT1
RT1 = RT2
RT2 = TMP
END IF
*
* Choose CS1 = 1 and SN1 to satisfy the first equation, then
* scale the components of this eigenvector so that the matrix
* of eigenvectors X satisfies X * X**T = I . (No scaling is
* done if the norm of the eigenvalue matrix is less than THRESH.)
*
SN1 = ( RT1-A ) / B
TABS = ABS( SN1 )
IF( TABS.GT.ONE ) THEN
T = TABS*SQRT( ( ONE / TABS )**2+( SN1 / TABS )**2 )
ELSE
T = SQRT( CONE+SN1*SN1 )
END IF
EVNORM = ABS( T )
IF( EVNORM.GE.THRESH ) THEN
EVSCAL = CONE / T
CS1 = EVSCAL
SN1 = SN1*EVSCAL
ELSE
EVSCAL = ZERO
END IF
END IF
RETURN
*
* End of CLAESY
*
END
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