219 lines
6.2 KiB
Fortran
219 lines
6.2 KiB
Fortran
*> \brief \b ZLAESY 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 ZLAESY + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaesy.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaesy.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaesy.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE ZLAESY( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* COMPLEX*16 A, B, C, CS1, EVSCAL, RT1, RT2, SN1
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> ZLAESY 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*16
|
|
*> The ( 1, 1 ) element of input matrix.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] B
|
|
*> \verbatim
|
|
*> B is COMPLEX*16
|
|
*> 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*16
|
|
*> The ( 2, 2 ) element of input matrix.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] RT1
|
|
*> \verbatim
|
|
*> RT1 is COMPLEX*16
|
|
*> The eigenvalue of larger modulus.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] RT2
|
|
*> \verbatim
|
|
*> RT2 is COMPLEX*16
|
|
*> The eigenvalue of smaller modulus.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] EVSCAL
|
|
*> \verbatim
|
|
*> EVSCAL is COMPLEX*16
|
|
*> 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*16
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] SN1
|
|
*> \verbatim
|
|
*> SN1 is COMPLEX*16
|
|
*> 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.
|
|
*
|
|
*> \ingroup complex16SYauxiliary
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE ZLAESY( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
|
|
*
|
|
* -- 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 ..
|
|
COMPLEX*16 A, B, C, CS1, EVSCAL, RT1, RT2, SN1
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
DOUBLE PRECISION ZERO
|
|
PARAMETER ( ZERO = 0.0D0 )
|
|
DOUBLE PRECISION ONE
|
|
PARAMETER ( ONE = 1.0D0 )
|
|
COMPLEX*16 CONE
|
|
PARAMETER ( CONE = ( 1.0D0, 0.0D0 ) )
|
|
DOUBLE PRECISION HALF
|
|
PARAMETER ( HALF = 0.5D0 )
|
|
DOUBLE PRECISION THRESH
|
|
PARAMETER ( THRESH = 0.1D0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
DOUBLE PRECISION BABS, EVNORM, TABS, Z
|
|
COMPLEX*16 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 ZLAESY
|
|
*
|
|
END
|