*> \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 *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \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. * *> \date December 2016 * *> \ingroup complex16SYauxiliary * * ===================================================================== SUBROUTINE ZLAESY( 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*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