zlaesy.f

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*> \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/
*
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*
*  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 laesy
*
*  =====================================================================
      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