claesy.f

          SUBROUTINE CLAESY( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          COMPLEX            A, B, C, CS1, EVSCAL, RT1, RT2, SN1
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CLAESY 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 ]
    *
    *  Arguments
    *  =========
    *
    *  A       (input) COMPLEX
    *          The ( 1, 1 ) element of input matrix.
    *
    *  B       (input) COMPLEX
    *          The ( 1, 2 ) element of input matrix.  The ( 2, 1 ) element
    *          is also given by B, since the 2-by-2 matrix is symmetric.
    *
    *  C       (input) COMPLEX
    *          The ( 2, 2 ) element of input matrix.
    *
    *  RT1     (output) COMPLEX
    *          The eigenvalue of larger modulus.
    *
    *  RT2     (output) COMPLEX
    *          The eigenvalue of smaller modulus.
    *
    *  EVSCAL  (output) COMPLEX
    *          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).
    *
    *  CS1     (output) COMPLEX
    *  SN1     (output) COMPLEX
    *          If EVSCAL .NE. 0,  ( CS1, SN1 ) is the unit right eigenvector
    *          for RT1.
    *
    * =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO
          PARAMETER          ( ZERO = 0.0E0 )
          REAL               ONE
          PARAMETER          ( ONE = 1.0E0 )
          COMPLEX            CONE
          PARAMETER          ( CONE = ( 1.0E0, 0.0E0 ) )
          REAL               HALF
          PARAMETER          ( HALF = 0.5E0 )
          REAL               THRESH
          PARAMETER          ( THRESH = 0.1E0 )
    *     ..
    *     .. Local Scalars ..
          REAL               BABS, EVNORM, TABS, Z
          COMPLEX            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' = 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 CLAESY
    *
          END