slaev2.f.html

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<pre>
      SUBROUTINE <a name="SLAEV2.1"></a><a href="slaev2.f.html#SLAEV2.1">SLAEV2</a>( A, B, C, RT1, RT2, 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>      REAL               A, B, C, CS1, 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="SLAEV2.14"></a><a href="slaev2.f.html#SLAEV2.1">SLAEV2</a> computes the eigendecomposition of a 2-by-2 symmetric matrix
</span><span class="comment">*</span><span class="comment">     [  A   B  ]
</span><span class="comment">*</span><span class="comment">     [  B   C  ].
</span><span class="comment">*</span><span class="comment">  On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
</span><span class="comment">*</span><span class="comment">  eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
</span><span class="comment">*</span><span class="comment">  eigenvector for RT1, giving the decomposition
</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) REAL
</span><span class="comment">*</span><span class="comment">          The (1,1) element of the 2-by-2 matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B       (input) REAL
</span><span class="comment">*</span><span class="comment">          The (1,2) element and the conjugate of the (2,1) element of
</span><span class="comment">*</span><span class="comment">          the 2-by-2 matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  C       (input) REAL
</span><span class="comment">*</span><span class="comment">          The (2,2) element of the 2-by-2 matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RT1     (output) REAL
</span><span class="comment">*</span><span class="comment">          The eigenvalue of larger absolute value.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RT2     (output) REAL
</span><span class="comment">*</span><span class="comment">          The eigenvalue of smaller absolute value.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CS1     (output) REAL
</span><span class="comment">*</span><span class="comment">  SN1     (output) REAL
</span><span class="comment">*</span><span class="comment">          The vector (CS1, SN1) is a unit right eigenvector for RT1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RT1 is accurate to a few ulps barring over/underflow.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RT2 may be inaccurate if there is massive cancellation in the
</span><span class="comment">*</span><span class="comment">  determinant A*C-B*B; higher precision or correctly rounded or
</span><span class="comment">*</span><span class="comment">  correctly truncated arithmetic would be needed to compute RT2
</span><span class="comment">*</span><span class="comment">  accurately in all cases.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CS1 and SN1 are accurate to a few ulps barring over/underflow.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Overflow is possible only if RT1 is within a factor of 5 of overflow.
</span><span class="comment">*</span><span class="comment">  Underflow is harmless if the input data is 0 or exceeds
</span><span class="comment">*</span><span class="comment">     underflow_threshold / macheps.
</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>      REAL               ONE
      PARAMETER          ( ONE = 1.0E0 )
      REAL               TWO
      PARAMETER          ( TWO = 2.0E0 )
      REAL               ZERO
      PARAMETER          ( ZERO = 0.0E0 )
      REAL               HALF
      PARAMETER          ( HALF = 0.5E0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      INTEGER            SGN1, SGN2
      REAL               AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
     $                   TB, TN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, 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">     Compute the eigenvalues
</span><span class="comment">*</span><span class="comment">
</span>      SM = A + C
      DF = A - C
      ADF = ABS( DF )
      TB = B + B
      AB = ABS( TB )
      IF( ABS( A ).GT.ABS( C ) ) THEN
         ACMX = A
         ACMN = C
      ELSE
         ACMX = C
         ACMN = A
      END IF
      IF( ADF.GT.AB ) THEN
         RT = ADF*SQRT( ONE+( AB / ADF )**2 )
      ELSE IF( ADF.LT.AB ) THEN
         RT = AB*SQRT( ONE+( ADF / AB )**2 )
      ELSE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Includes case AB=ADF=0
</span><span class="comment">*</span><span class="comment">
</span>         RT = AB*SQRT( TWO )
      END IF
      IF( SM.LT.ZERO ) THEN
         RT1 = HALF*( SM-RT )
         SGN1 = -1
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Order of execution important.
</span><span class="comment">*</span><span class="comment">        To get fully accurate smaller eigenvalue,
</span><span class="comment">*</span><span class="comment">        next line needs to be executed in higher precision.
</span><span class="comment">*</span><span class="comment">
</span>         RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
      ELSE IF( SM.GT.ZERO ) THEN
         RT1 = HALF*( SM+RT )
         SGN1 = 1
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Order of execution important.
</span><span class="comment">*</span><span class="comment">        To get fully accurate smaller eigenvalue,
</span><span class="comment">*</span><span class="comment">        next line needs to be executed in higher precision.
</span><span class="comment">*</span><span class="comment">
</span>         RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
      ELSE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Includes case RT1 = RT2 = 0
</span><span class="comment">*</span><span class="comment">
</span>         RT1 = HALF*RT
         RT2 = -HALF*RT
         SGN1 = 1
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Compute the eigenvector
</span><span class="comment">*</span><span class="comment">
</span>      IF( DF.GE.ZERO ) THEN
         CS = DF + RT
         SGN2 = 1
      ELSE
         CS = DF - RT
         SGN2 = -1
      END IF
      ACS = ABS( CS )
      IF( ACS.GT.AB ) THEN
         CT = -TB / CS
         SN1 = ONE / SQRT( ONE+CT*CT )
         CS1 = CT*SN1
      ELSE
         IF( AB.EQ.ZERO ) THEN
            CS1 = ONE
            SN1 = ZERO
         ELSE
            TN = -CS / TB
            CS1 = ONE / SQRT( ONE+TN*TN )
            SN1 = TN*CS1
         END IF
      END IF
      IF( SGN1.EQ.SGN2 ) THEN
         TN = CS1
         CS1 = -SN1
         SN1 = TN
      END IF
      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="SLAEV2.167"></a><a href="slaev2.f.html#SLAEV2.1">SLAEV2</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

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