CLAEV2 (3lapack)

compute the eigendecomposition of a 2-by-2 Hermitian matrix [ A B ] [ CONJG(B) C ]

SYNOPSIS

  SUBROUTINE CLAEV2( A, B, C, RT1, RT2, CS1, SN1 )

      REAL           CS1, RT1, RT2

      COMPLEX        A, B, C, SN1

PURPOSE

  CLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
     [  A         B  ]
     [  CONJG(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  CONJG(SN1) ] [    A     B ] [ CS1 -CONJG(SN1) ] = [ RT1  0  ] [-SN1
  CS1     ] [ CONJG(B) C ] [ SN1     CS1     ]   [  0  RT2 ].

ARGUMENTS

  A      (input) COMPLEX
         The (1,1) element of the 2-by-2 matrix.

  B      (input) COMPLEX
         The (1,2) element and the conjugate of the (2,1) element of the 2-
         by-2 matrix.

  C      (input) COMPLEX
         The (2,2) element of the 2-by-2 matrix.

  RT1    (output) REAL
         The eigenvalue of larger absolute value.

  RT2    (output) REAL
         The eigenvalue of smaller absolute value.

  CS1    (output) REAL
         SN1    (output) COMPLEX 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.