zlaesy.f.html

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<pre>
      SUBROUTINE <a name="ZLAESY.1"></a><a href="zlaesy.f.html#ZLAESY.1">ZLAESY</a>( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      COMPLEX*16         A, B, C, CS1, EVSCAL, RT1, RT2, SN1
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="ZLAESY.14"></a><a href="zlaesy.f.html#ZLAESY.1">ZLAESY</a> computes the eigendecomposition of a 2-by-2 symmetric matrix
</span><span class="comment">*</span><span class="comment">     ( ( A, B );( B, C ) )
</span><span class="comment">*</span><span class="comment">  provided the norm of the matrix of eigenvectors is larger than
</span><span class="comment">*</span><span class="comment">  some threshold value.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RT1 is the eigenvalue of larger absolute value, and RT2 of
</span><span class="comment">*</span><span class="comment">  smaller absolute value.  If the eigenvectors are computed, then
</span><span class="comment">*</span><span class="comment">  on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  [  CS1     SN1   ] . [ A  B ] . [ CS1    -SN1   ] = [ RT1  0  ]
</span><span class="comment">*</span><span class="comment">  [ -SN1     CS1   ]   [ B  C ]   [ SN1     CS1   ]   [  0  RT2 ]
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input) COMPLEX*16
</span><span class="comment">*</span><span class="comment">          The ( 1, 1 ) element of input matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B       (input) COMPLEX*16
</span><span class="comment">*</span><span class="comment">          The ( 1, 2 ) element of input matrix.  The ( 2, 1 ) element
</span><span class="comment">*</span><span class="comment">          is also given by B, since the 2-by-2 matrix is symmetric.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  C       (input) COMPLEX*16
</span><span class="comment">*</span><span class="comment">          The ( 2, 2 ) element of input matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RT1     (output) COMPLEX*16
</span><span class="comment">*</span><span class="comment">          The eigenvalue of larger modulus.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RT2     (output) COMPLEX*16
</span><span class="comment">*</span><span class="comment">          The eigenvalue of smaller modulus.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  EVSCAL  (output) COMPLEX*16
</span><span class="comment">*</span><span class="comment">          The complex value by which the eigenvector matrix was scaled
</span><span class="comment">*</span><span class="comment">          to make it orthonormal.  If EVSCAL is zero, the eigenvectors
</span><span class="comment">*</span><span class="comment">          were not computed.  This means one of two things:  the 2-by-2
</span><span class="comment">*</span><span class="comment">          matrix could not be diagonalized, or the norm of the matrix
</span><span class="comment">*</span><span class="comment">          of eigenvectors before scaling was larger than the threshold
</span><span class="comment">*</span><span class="comment">          value THRESH (set below).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CS1     (output) COMPLEX*16
</span><span class="comment">*</span><span class="comment">  SN1     (output) COMPLEX*16
</span><span class="comment">*</span><span class="comment">          If EVSCAL .NE. 0,  ( CS1, SN1 ) is the unit right eigenvector
</span><span class="comment">*</span><span class="comment">          for RT1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      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 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      DOUBLE PRECISION   BABS, EVNORM, TABS, Z
      COMPLEX*16         S, T, TMP
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, MAX, SQRT
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Special case:  The matrix is actually diagonal.
</span><span class="comment">*</span><span class="comment">     To avoid divide by zero later, we treat this case separately.
</span><span class="comment">*</span><span class="comment">
</span>      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
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Compute the eigenvalues and eigenvectors.
</span><span class="comment">*</span><span class="comment">        The characteristic equation is
</span><span class="comment">*</span><span class="comment">           lambda **2 - (A+C) lambda + (A*C - B*B)
</span><span class="comment">*</span><span class="comment">        and we solve it using the quadratic formula.
</span><span class="comment">*</span><span class="comment">
</span>         S = ( A+C )*HALF
         T = ( A-C )*HALF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Take the square root carefully to avoid over/under flow.
</span><span class="comment">*</span><span class="comment">
</span>         BABS = ABS( B )
         TABS = ABS( T )
         Z = MAX( BABS, TABS )
         IF( Z.GT.ZERO )
     $      T = Z*SQRT( ( T / Z )**2+( B / Z )**2 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Compute the two eigenvalues.  RT1 and RT2 are exchanged
</span><span class="comment">*</span><span class="comment">        if necessary so that RT1 will have the greater magnitude.
</span><span class="comment">*</span><span class="comment">
</span>         RT1 = S + T
         RT2 = S - T
         IF( ABS( RT1 ).LT.ABS( RT2 ) ) THEN
            TMP = RT1
            RT1 = RT2
            RT2 = TMP
         END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Choose CS1 = 1 and SN1 to satisfy the first equation, then
</span><span class="comment">*</span><span class="comment">        scale the components of this eigenvector so that the matrix
</span><span class="comment">*</span><span class="comment">        of eigenvectors X satisfies  X * X' = I .  (No scaling is
</span><span class="comment">*</span><span class="comment">        done if the norm of the eigenvalue matrix is less than THRESH.)
</span><span class="comment">*</span><span class="comment">
</span>         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
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="ZLAESY.150"></a><a href="zlaesy.f.html#ZLAESY.1">ZLAESY</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

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