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    1:       SUBROUTINE ZLAESY( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
    2: *
    3: *  -- LAPACK auxiliary routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       COMPLEX*16         A, B, C, CS1, EVSCAL, RT1, RT2, SN1
   10: *     ..
   11: *
   12: *  Purpose
   13: *  =======
   14: *
   15: *  ZLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix
   16: *     ( ( A, B );( B, C ) )
   17: *  provided the norm of the matrix of eigenvectors is larger than
   18: *  some threshold value.
   19: *
   20: *  RT1 is the eigenvalue of larger absolute value, and RT2 of
   21: *  smaller absolute value.  If the eigenvectors are computed, then
   22: *  on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence
   23: *
   24: *  [  CS1     SN1   ] . [ A  B ] . [ CS1    -SN1   ] = [ RT1  0  ]
   25: *  [ -SN1     CS1   ]   [ B  C ]   [ SN1     CS1   ]   [  0  RT2 ]
   26: *
   27: *  Arguments
   28: *  =========
   29: *
   30: *  A       (input) COMPLEX*16
   31: *          The ( 1, 1 ) element of input matrix.
   32: *
   33: *  B       (input) COMPLEX*16
   34: *          The ( 1, 2 ) element of input matrix.  The ( 2, 1 ) element
   35: *          is also given by B, since the 2-by-2 matrix is symmetric.
   36: *
   37: *  C       (input) COMPLEX*16
   38: *          The ( 2, 2 ) element of input matrix.
   39: *
   40: *  RT1     (output) COMPLEX*16
   41: *          The eigenvalue of larger modulus.
   42: *
   43: *  RT2     (output) COMPLEX*16
   44: *          The eigenvalue of smaller modulus.
   45: *
   46: *  EVSCAL  (output) COMPLEX*16
   47: *          The complex value by which the eigenvector matrix was scaled
   48: *          to make it orthonormal.  If EVSCAL is zero, the eigenvectors
   49: *          were not computed.  This means one of two things:  the 2-by-2
   50: *          matrix could not be diagonalized, or the norm of the matrix
   51: *          of eigenvectors before scaling was larger than the threshold
   52: *          value THRESH (set below).
   53: *
   54: *  CS1     (output) COMPLEX*16
   55: *  SN1     (output) COMPLEX*16
   56: *          If EVSCAL .NE. 0,  ( CS1, SN1 ) is the unit right eigenvector
   57: *          for RT1.
   58: *
   59: * =====================================================================
   60: *
   61: *     .. Parameters ..
   62:       DOUBLE PRECISION   ZERO
   63:       PARAMETER          ( ZERO = 0.0D0 )
   64:       DOUBLE PRECISION   ONE
   65:       PARAMETER          ( ONE = 1.0D0 )
   66:       COMPLEX*16         CONE
   67:       PARAMETER          ( CONE = ( 1.0D0, 0.0D0 ) )
   68:       DOUBLE PRECISION   HALF
   69:       PARAMETER          ( HALF = 0.5D0 )
   70:       DOUBLE PRECISION   THRESH
   71:       PARAMETER          ( THRESH = 0.1D0 )
   72: *     ..
   73: *     .. Local Scalars ..
   74:       DOUBLE PRECISION   BABS, EVNORM, TABS, Z
   75:       COMPLEX*16         S, T, TMP
   76: *     ..
   77: *     .. Intrinsic Functions ..
   78:       INTRINSIC          ABS, MAX, SQRT
   79: *     ..
   80: *     .. Executable Statements ..
   81: *
   82: *
   83: *     Special case:  The matrix is actually diagonal.
   84: *     To avoid divide by zero later, we treat this case separately.
   85: *
   86:       IF( ABS( B ).EQ.ZERO ) THEN
   87:          RT1 = A
   88:          RT2 = C
   89:          IF( ABS( RT1 ).LT.ABS( RT2 ) ) THEN
   90:             TMP = RT1
   91:             RT1 = RT2
   92:             RT2 = TMP
   93:             CS1 = ZERO
   94:             SN1 = ONE
   95:          ELSE
   96:             CS1 = ONE
   97:             SN1 = ZERO
   98:          END IF
   99:       ELSE
  100: *
  101: *        Compute the eigenvalues and eigenvectors.
  102: *        The characteristic equation is
  103: *           lambda **2 - (A+C) lambda + (A*C - B*B)
  104: *        and we solve it using the quadratic formula.
  105: *
  106:          S = ( A+C )*HALF
  107:          T = ( A-C )*HALF
  108: *
  109: *        Take the square root carefully to avoid over/under flow.
  110: *
  111:          BABS = ABS( B )
  112:          TABS = ABS( T )
  113:          Z = MAX( BABS, TABS )
  114:          IF( Z.GT.ZERO )
  115:      $      T = Z*SQRT( ( T / Z )**2+( B / Z )**2 )
  116: *
  117: *        Compute the two eigenvalues.  RT1 and RT2 are exchanged
  118: *        if necessary so that RT1 will have the greater magnitude.
  119: *
  120:          RT1 = S + T
  121:          RT2 = S - T
  122:          IF( ABS( RT1 ).LT.ABS( RT2 ) ) THEN
  123:             TMP = RT1
  124:             RT1 = RT2
  125:             RT2 = TMP
  126:          END IF
  127: *
  128: *        Choose CS1 = 1 and SN1 to satisfy the first equation, then
  129: *        scale the components of this eigenvector so that the matrix
  130: *        of eigenvectors X satisfies  X * X' = I .  (No scaling is
  131: *        done if the norm of the eigenvalue matrix is less than THRESH.)
  132: *
  133:          SN1 = ( RT1-A ) / B
  134:          TABS = ABS( SN1 )
  135:          IF( TABS.GT.ONE ) THEN
  136:             T = TABS*SQRT( ( ONE / TABS )**2+( SN1 / TABS )**2 )
  137:          ELSE
  138:             T = SQRT( CONE+SN1*SN1 )
  139:          END IF
  140:          EVNORM = ABS( T )
  141:          IF( EVNORM.GE.THRESH ) THEN
  142:             EVSCAL = CONE / T
  143:             CS1 = EVSCAL
  144:             SN1 = SN1*EVSCAL
  145:          ELSE
  146:             EVSCAL = ZERO
  147:          END IF
  148:       END IF
  149:       RETURN
  150: *
  151: *     End of ZLAESY
  152: *
  153:       END