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    1:       SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
    2: *
    3: *  -- LAPACK auxiliary routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       DOUBLE PRECISION   A, B, C, CS1, RT1, RT2, SN1
   10: *     ..
   11: *
   12: *  Purpose
   13: *  =======
   14: *
   15: *  DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
   16: *     [  A   B  ]
   17: *     [  B   C  ].
   18: *  On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
   19: *  eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
   20: *  eigenvector for RT1, giving the decomposition
   21: *
   22: *     [ CS1  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]
   23: *     [-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].
   24: *
   25: *  Arguments
   26: *  =========
   27: *
   28: *  A       (input) DOUBLE PRECISION
   29: *          The (1,1) element of the 2-by-2 matrix.
   30: *
   31: *  B       (input) DOUBLE PRECISION
   32: *          The (1,2) element and the conjugate of the (2,1) element of
   33: *          the 2-by-2 matrix.
   34: *
   35: *  C       (input) DOUBLE PRECISION
   36: *          The (2,2) element of the 2-by-2 matrix.
   37: *
   38: *  RT1     (output) DOUBLE PRECISION
   39: *          The eigenvalue of larger absolute value.
   40: *
   41: *  RT2     (output) DOUBLE PRECISION
   42: *          The eigenvalue of smaller absolute value.
   43: *
   44: *  CS1     (output) DOUBLE PRECISION
   45: *  SN1     (output) DOUBLE PRECISION
   46: *          The vector (CS1, SN1) is a unit right eigenvector for RT1.
   47: *
   48: *  Further Details
   49: *  ===============
   50: *
   51: *  RT1 is accurate to a few ulps barring over/underflow.
   52: *
   53: *  RT2 may be inaccurate if there is massive cancellation in the
   54: *  determinant A*C-B*B; higher precision or correctly rounded or
   55: *  correctly truncated arithmetic would be needed to compute RT2
   56: *  accurately in all cases.
   57: *
   58: *  CS1 and SN1 are accurate to a few ulps barring over/underflow.
   59: *
   60: *  Overflow is possible only if RT1 is within a factor of 5 of overflow.
   61: *  Underflow is harmless if the input data is 0 or exceeds
   62: *     underflow_threshold / macheps.
   63: *
   64: * =====================================================================
   65: *
   66: *     .. Parameters ..
   67:       DOUBLE PRECISION   ONE
   68:       PARAMETER          ( ONE = 1.0D0 )
   69:       DOUBLE PRECISION   TWO
   70:       PARAMETER          ( TWO = 2.0D0 )
   71:       DOUBLE PRECISION   ZERO
   72:       PARAMETER          ( ZERO = 0.0D0 )
   73:       DOUBLE PRECISION   HALF
   74:       PARAMETER          ( HALF = 0.5D0 )
   75: *     ..
   76: *     .. Local Scalars ..
   77:       INTEGER            SGN1, SGN2
   78:       DOUBLE PRECISION   AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
   79:      $                   TB, TN
   80: *     ..
   81: *     .. Intrinsic Functions ..
   82:       INTRINSIC          ABS, SQRT
   83: *     ..
   84: *     .. Executable Statements ..
   85: *
   86: *     Compute the eigenvalues
   87: *
   88:       SM = A + C
   89:       DF = A - C
   90:       ADF = ABS( DF )
   91:       TB = B + B
   92:       AB = ABS( TB )
   93:       IF( ABS( A ).GT.ABS( C ) ) THEN
   94:          ACMX = A
   95:          ACMN = C
   96:       ELSE
   97:          ACMX = C
   98:          ACMN = A
   99:       END IF
  100:       IF( ADF.GT.AB ) THEN
  101:          RT = ADF*SQRT( ONE+( AB / ADF )**2 )
  102:       ELSE IF( ADF.LT.AB ) THEN
  103:          RT = AB*SQRT( ONE+( ADF / AB )**2 )
  104:       ELSE
  105: *
  106: *        Includes case AB=ADF=0
  107: *
  108:          RT = AB*SQRT( TWO )
  109:       END IF
  110:       IF( SM.LT.ZERO ) THEN
  111:          RT1 = HALF*( SM-RT )
  112:          SGN1 = -1
  113: *
  114: *        Order of execution important.
  115: *        To get fully accurate smaller eigenvalue,
  116: *        next line needs to be executed in higher precision.
  117: *
  118:          RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
  119:       ELSE IF( SM.GT.ZERO ) THEN
  120:          RT1 = HALF*( SM+RT )
  121:          SGN1 = 1
  122: *
  123: *        Order of execution important.
  124: *        To get fully accurate smaller eigenvalue,
  125: *        next line needs to be executed in higher precision.
  126: *
  127:          RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
  128:       ELSE
  129: *
  130: *        Includes case RT1 = RT2 = 0
  131: *
  132:          RT1 = HALF*RT
  133:          RT2 = -HALF*RT
  134:          SGN1 = 1
  135:       END IF
  136: *
  137: *     Compute the eigenvector
  138: *
  139:       IF( DF.GE.ZERO ) THEN
  140:          CS = DF + RT
  141:          SGN2 = 1
  142:       ELSE
  143:          CS = DF - RT
  144:          SGN2 = -1
  145:       END IF
  146:       ACS = ABS( CS )
  147:       IF( ACS.GT.AB ) THEN
  148:          CT = -TB / CS
  149:          SN1 = ONE / SQRT( ONE+CT*CT )
  150:          CS1 = CT*SN1
  151:       ELSE
  152:          IF( AB.EQ.ZERO ) THEN
  153:             CS1 = ONE
  154:             SN1 = ZERO
  155:          ELSE
  156:             TN = -CS / TB
  157:             CS1 = ONE / SQRT( ONE+TN*TN )
  158:             SN1 = TN*CS1
  159:          END IF
  160:       END IF
  161:       IF( SGN1.EQ.SGN2 ) THEN
  162:          TN = CS1
  163:          CS1 = -SN1
  164:          SN1 = TN
  165:       END IF
  166:       RETURN
  167: *
  168: *     End of DLAEV2
  169: *
  170:       END