dlae2.f

          SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          DOUBLE PRECISION   A, B, C, RT1, RT2
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLAE2  computes the eigenvalues of a 2-by-2 symmetric matrix
    *     [  A   B  ]
    *     [  B   C  ].
    *  On return, RT1 is the eigenvalue of larger absolute value, and RT2
    *  is the eigenvalue of smaller absolute value.
    *
    *  Arguments
    *  =========
    *
    *  A       (input) DOUBLE PRECISION
    *          The (1,1) element of the 2-by-2 matrix.
    *
    *  B       (input) DOUBLE PRECISION
    *          The (1,2) and (2,1) elements 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.
    *
    *  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.
    *
    *  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 ..
          DOUBLE PRECISION   AB, ACMN, ACMX, ADF, DF, RT, SM, TB
    *     ..
    *     .. 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 )
    *
    *        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 )
    *
    *        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
          END IF
          RETURN
    *
    *     End of DLAE2
    *
          END