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    1: *> \brief \b ZLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download ZLAESY + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaesy.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaesy.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaesy.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZLAESY( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       COMPLEX*16         A, B, C, CS1, EVSCAL, RT1, RT2, SN1
   25: *       ..
   26: *
   27: *
   28: *> \par Purpose:
   29: *  =============
   30: *>
   31: *> \verbatim
   32: *>
   33: *> ZLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix
   34: *>    ( ( A, B );( B, C ) )
   35: *> provided the norm of the matrix of eigenvectors is larger than
   36: *> some threshold value.
   37: *>
   38: *> RT1 is the eigenvalue of larger absolute value, and RT2 of
   39: *> smaller absolute value.  If the eigenvectors are computed, then
   40: *> on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence
   41: *>
   42: *> [  CS1     SN1   ] . [ A  B ] . [ CS1    -SN1   ] = [ RT1  0  ]
   43: *> [ -SN1     CS1   ]   [ B  C ]   [ SN1     CS1   ]   [  0  RT2 ]
   44: *> \endverbatim
   45: *
   46: *  Arguments:
   47: *  ==========
   48: *
   49: *> \param[in] A
   50: *> \verbatim
   51: *>          A is COMPLEX*16
   52: *>          The ( 1, 1 ) element of input matrix.
   53: *> \endverbatim
   54: *>
   55: *> \param[in] B
   56: *> \verbatim
   57: *>          B is COMPLEX*16
   58: *>          The ( 1, 2 ) element of input matrix.  The ( 2, 1 ) element
   59: *>          is also given by B, since the 2-by-2 matrix is symmetric.
   60: *> \endverbatim
   61: *>
   62: *> \param[in] C
   63: *> \verbatim
   64: *>          C is COMPLEX*16
   65: *>          The ( 2, 2 ) element of input matrix.
   66: *> \endverbatim
   67: *>
   68: *> \param[out] RT1
   69: *> \verbatim
   70: *>          RT1 is COMPLEX*16
   71: *>          The eigenvalue of larger modulus.
   72: *> \endverbatim
   73: *>
   74: *> \param[out] RT2
   75: *> \verbatim
   76: *>          RT2 is COMPLEX*16
   77: *>          The eigenvalue of smaller modulus.
   78: *> \endverbatim
   79: *>
   80: *> \param[out] EVSCAL
   81: *> \verbatim
   82: *>          EVSCAL is COMPLEX*16
   83: *>          The complex value by which the eigenvector matrix was scaled
   84: *>          to make it orthonormal.  If EVSCAL is zero, the eigenvectors
   85: *>          were not computed.  This means one of two things:  the 2-by-2
   86: *>          matrix could not be diagonalized, or the norm of the matrix
   87: *>          of eigenvectors before scaling was larger than the threshold
   88: *>          value THRESH (set below).
   89: *> \endverbatim
   90: *>
   91: *> \param[out] CS1
   92: *> \verbatim
   93: *>          CS1 is COMPLEX*16
   94: *> \endverbatim
   95: *>
   96: *> \param[out] SN1
   97: *> \verbatim
   98: *>          SN1 is COMPLEX*16
   99: *>          If EVSCAL .NE. 0,  ( CS1, SN1 ) is the unit right eigenvector
  100: *>          for RT1.
  101: *> \endverbatim
  102: *
  103: *  Authors:
  104: *  ========
  105: *
  106: *> \author Univ. of Tennessee
  107: *> \author Univ. of California Berkeley
  108: *> \author Univ. of Colorado Denver
  109: *> \author NAG Ltd.
  110: *
  111: *> \ingroup complex16SYauxiliary
  112: *
  113: *  =====================================================================
  114:       SUBROUTINE ZLAESY( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
  115: *
  116: *  -- LAPACK auxiliary routine --
  117: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  118: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  119: *
  120: *     .. Scalar Arguments ..
  121:       COMPLEX*16         A, B, C, CS1, EVSCAL, RT1, RT2, SN1
  122: *     ..
  123: *
  124: * =====================================================================
  125: *
  126: *     .. Parameters ..
  127:       DOUBLE PRECISION   ZERO
  128:       PARAMETER          ( ZERO = 0.0D0 )
  129:       DOUBLE PRECISION   ONE
  130:       PARAMETER          ( ONE = 1.0D0 )
  131:       COMPLEX*16         CONE
  132:       PARAMETER          ( CONE = ( 1.0D0, 0.0D0 ) )
  133:       DOUBLE PRECISION   HALF
  134:       PARAMETER          ( HALF = 0.5D0 )
  135:       DOUBLE PRECISION   THRESH
  136:       PARAMETER          ( THRESH = 0.1D0 )
  137: *     ..
  138: *     .. Local Scalars ..
  139:       DOUBLE PRECISION   BABS, EVNORM, TABS, Z
  140:       COMPLEX*16         S, T, TMP
  141: *     ..
  142: *     .. Intrinsic Functions ..
  143:       INTRINSIC          ABS, MAX, SQRT
  144: *     ..
  145: *     .. Executable Statements ..
  146: *
  147: *
  148: *     Special case:  The matrix is actually diagonal.
  149: *     To avoid divide by zero later, we treat this case separately.
  150: *
  151:       IF( ABS( B ).EQ.ZERO ) THEN
  152:          RT1 = A
  153:          RT2 = C
  154:          IF( ABS( RT1 ).LT.ABS( RT2 ) ) THEN
  155:             TMP = RT1
  156:             RT1 = RT2
  157:             RT2 = TMP
  158:             CS1 = ZERO
  159:             SN1 = ONE
  160:          ELSE
  161:             CS1 = ONE
  162:             SN1 = ZERO
  163:          END IF
  164:       ELSE
  165: *
  166: *        Compute the eigenvalues and eigenvectors.
  167: *        The characteristic equation is
  168: *           lambda **2 - (A+C) lambda + (A*C - B*B)
  169: *        and we solve it using the quadratic formula.
  170: *
  171:          S = ( A+C )*HALF
  172:          T = ( A-C )*HALF
  173: *
  174: *        Take the square root carefully to avoid over/under flow.
  175: *
  176:          BABS = ABS( B )
  177:          TABS = ABS( T )
  178:          Z = MAX( BABS, TABS )
  179:          IF( Z.GT.ZERO )
  180:      $      T = Z*SQRT( ( T / Z )**2+( B / Z )**2 )
  181: *
  182: *        Compute the two eigenvalues.  RT1 and RT2 are exchanged
  183: *        if necessary so that RT1 will have the greater magnitude.
  184: *
  185:          RT1 = S + T
  186:          RT2 = S - T
  187:          IF( ABS( RT1 ).LT.ABS( RT2 ) ) THEN
  188:             TMP = RT1
  189:             RT1 = RT2
  190:             RT2 = TMP
  191:          END IF
  192: *
  193: *        Choose CS1 = 1 and SN1 to satisfy the first equation, then
  194: *        scale the components of this eigenvector so that the matrix
  195: *        of eigenvectors X satisfies  X * X**T = I .  (No scaling is
  196: *        done if the norm of the eigenvalue matrix is less than THRESH.)
  197: *
  198:          SN1 = ( RT1-A ) / B
  199:          TABS = ABS( SN1 )
  200:          IF( TABS.GT.ONE ) THEN
  201:             T = TABS*SQRT( ( ONE / TABS )**2+( SN1 / TABS )**2 )
  202:          ELSE
  203:             T = SQRT( CONE+SN1*SN1 )
  204:          END IF
  205:          EVNORM = ABS( T )
  206:          IF( EVNORM.GE.THRESH ) THEN
  207:             EVSCAL = CONE / T
  208:             CS1 = EVSCAL
  209:             SN1 = SN1*EVSCAL
  210:          ELSE
  211:             EVSCAL = ZERO
  212:          END IF
  213:       END IF
  214:       RETURN
  215: *
  216: *     End of ZLAESY
  217: *
  218:       END