dlaev2.f

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      SUBROUTINE dlaev2( A, B, C, RT1, RT2, CS1, SN1 )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1992
*
*     .. Scalar Arguments ..
      DOUBLE PRECISION   A, B, C, CS1, RT1, RT2, SN1
*     ..
*
*  Purpose
*  =======
*
*  DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
*     [  A   B  ]
*     [  B   C  ].
*  On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
*  eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
*  eigenvector for RT1, giving the decomposition
*
*     [ CS1  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]
*     [-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].
*
*  Arguments
*  =========
*
*  A       (input) DOUBLE PRECISION
*          The (1,1) element of the 2-by-2 matrix.
*
*  B       (input) DOUBLE PRECISION
*          The (1,2) element and the conjugate of the (2,1) element of
*          the 2-by-2 matrix.
*
*  C       (input) DOUBLE PRECISION
*          The (2,2) element of the 2-by-2 matrix.
*
*  RT1     (output) DOUBLE PRECISION
*          The eigenvalue of larger absolute value.
*
*  RT2     (output) DOUBLE PRECISION
*          The eigenvalue of smaller absolute value.
*
*  CS1     (output) DOUBLE PRECISION
*  SN1     (output) DOUBLE PRECISION
*          The vector (CS1, SN1) is a unit right eigenvector for RT1.
*
*  Further Details
*  ===============
*
*  RT1 is accurate to a few ulps barring over/underflow.
*
*  RT2 may be inaccurate if there is massive cancellation in the
*  determinant A*C-B*B; higher precision or correctly rounded or
*  correctly truncated arithmetic would be needed to compute RT2
*  accurately in all cases.
*
*  CS1 and SN1 are accurate to a few ulps barring over/underflow.
*
*  Overflow is possible only if RT1 is within a factor of 5 of overflow.
*  Underflow is harmless if the input data is 0 or exceeds
*     underflow_threshold / macheps.
*
* =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      parameter( one = 1.0d0 )
      DOUBLE PRECISION   TWO
      parameter( two = 2.0d0 )
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d0 )
      DOUBLE PRECISION   HALF
      parameter( half = 0.5d0 )
*     ..
*     .. Local Scalars ..
      INTEGER            SGN1, SGN2
      DOUBLE PRECISION   AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
     $                   TB, TN
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, sqrt
*     ..
*     .. Executable Statements ..
*
*     Compute the eigenvalues
*
      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
*
*        Includes case AB=ADF=0
*
         rt = ab*sqrt( two )
      END IF
      IF( sm.LT.zero ) THEN
         rt1 = half*( sm-rt )
         sgn1 = -1
*
*        Order of execution important.
*        To get fully accurate smaller eigenvalue,
*        next line needs to be executed in higher precision.
*
         rt2 = ( acmx / rt1 )*acmn - ( b / rt1 )*b
      ELSE IF( sm.GT.zero ) THEN
         rt1 = half*( sm+rt )
         sgn1 = 1
*
*        Order of execution important.
*        To get fully accurate smaller eigenvalue,
*        next line needs to be executed in higher precision.
*
         rt2 = ( acmx / rt1 )*acmn - ( b / rt1 )*b
      ELSE
*
*        Includes case RT1 = RT2 = 0
*
         rt1 = half*rt
         rt2 = -half*rt
         sgn1 = 1
      END IF
*
*     Compute the eigenvector
*
      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
*
*     End of DLAEV2
*
      END