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analytic expression for eigenvalues of 3x3 matrix

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Martin Diehl 2016-02-01 07:48:42 +01:00
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@ -2051,6 +2051,40 @@ function math_eigenvaluesSym33(m)
end function math_eigenvaluesSym33
!--------------------------------------------------------------------------------------------------
!> @brief Eigenvalues of symmetric 3X3 matrix m
! will return NaN on error
!--------------------------------------------------------------------------------------------------
!function math_eigenvaluesSym33(m)
! use prec, only: &
! DAMASK_NaN
!
! implicit none
! real(pReal), dimension(3,3), intent(in) :: m
! real(pReal), dimension(3) :: math_eigenvaluesSym33
!
! real(pReal) :: C1, C0
! real(pReal) :: P, Q, C, S, PHI
!
!! Determine coefficients of characteristic poynomial. We write
!
! C1 = m(1,1)*m(2,2) + m(1,1)*m(3,3) + m(2,2)*m(3,3) &
! -(m(1,2)**2 + m(2,3)**2 +m(1,3)**2)
! C0 = m(3,3)*m(1,2)**2 + m(1,1)*m(2,3)**2 + m(2,2)*m(2,3)**2 &
! - m(1,1)*m(2,2)*m(3,3) - 2.0_pReal * m(1,3)*m(1,2)*m(2,3)
!
! P = math_trace33(m)**2 - 3.0_pReal * C1
! Q = math_trace33(m)*(P - (3.0_pReal/2.0_pReal)*C1) - (27.0_pReal/2.0_pReal)*C0
!
! PHI = 27.0_pReal * ( 0.25_pReal * C1**2 * (P - C1) + C0 * (Q + (27.0_pReal/4.0_pReal)*C0) )
! PHI = (1.0_pReal/3.0_pReal) * ATAN2(SQRT(ABS(PHI)), Q)
!
! C = sqrt(abs(P)) * COS(PHI)
! S = (1.0_pReal/sqrt(3.0_pReal)) * sqrt(abs(P)) * SIN(PHI)
!
! math_eigenvaluesSym33 = [C,-S,C] + (1.0_pReal/3.0_pReal) * (math_trace33(m) - C)
!
!end function math_eigenvaluesSym33
!--------------------------------------------------------------------------------------------------
!> @brief invariants of matrix m