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cosolver/src/numerical_coeffs.h
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numerical_coeffs.h
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!-----------------------------------------------------------
!> @brief
!> Computes the associated-Laguerre/Laguerre/x to Laguerre basis transformation coefficient \f$ d^{|m|}_{nks} \f$,
!>
!> \f[ L_n^{|m|}(x) L_k(x) x^{|m|} =\sum_{s=0}^{n+|m| +k} d^{|m|}_{nks} L_s(x),\f]
!> where
!> \f[d^{|m|}_{nks} = \int_0^\infty d x e^{-x} L_n^{|m|}(x) L_s(x) L_k(x) x^{|m|}. \f]
!> yielding the closed analyticalform,
!> \f[ d_{nks}^{|m|} =\sum_{n_1=0}^{n} \sum_{k_1=0}^k \sum_{s_1=0}^s L_{kk_1}^{-1/2} L_{nn_1}^{|m|-1/2} L_{ss_1}^{-1/2} (n_1 + k_1 + s_1 + |m|)!, \f]
!
!> @param[in] n
!> @param[in] k
!> @param[in] s
!> @param[in] m
!>
!> @param[out] XOUT Output result
!-----------------------------------------------------------
FUNCTION ALLX2L(n,k,s,m) RESULT(XOUT)
!
! Computes the associated-Laguerre/Laguerre/x to Laguerre coefficient, i.e.
!
! L_n^m L_k x^m = \sum_{s=0}^{n+m+k}d_{nks}^m L_s
!
!
!
IMPLICIT NONE
!
INTEGER, intent(in) :: n,k,s,m ! ... degree and orders of Laguerre polynomials
TYPE(FM) :: XOUT ! ... coefficient values
TYPE(FM),SAVE:: AUXN,AUXK,AUXS
INTEGER :: n1,k1,s1
!
CALL FM_ENTER_USER_FUNCTION(XOUT)
!
! Compute coefficients
!
XOUT = TO_FM('0.0')
!
DO n1=0,n
AUXN =Lpjl(REAL(m,dp)-0.5_dp,REAL(n,dp),REAL(n1,dp))
DO k1=0,k
AUXK =Lpjl(-0.5_dp,REAL(k,dp),REAL(k1,dp))
DO s1=0,s
AUXS = Lpjl(-0.5_dp,REAL(s,dp),REAL(s1,dp))
XOUT = XOUT + FACTORIAL(TO_FM(n1 + k1 + s1 + m))*AUXN*AUXK*AUXS
ENDDO
ENDDO
ENDDO
!
CALL FM_EXIT_USER_FUNCTION(XOUT)
!
END FUNCTION ALLX2L
!-----------------------------------------------------------
!> @brief
!> Computes the associated-Laguerre/Laguerre to Laguerre basis transformation coefficient \f$ \overline{d}^{|m|}_{nkf} \f$,
!>
!> \f[ \overline{d}^{|m|}_{nkf} = \sum_{n_1 = 0}^n \sum_{k_1 =0}^k \sum_{f_1=0}^f L_{kk_1}^{-1/2} L_{nn_1}^{|m|-1/2} L_{f f_1}^{-1/2} (n_1 + k_1 + f_1)!. \f]
!> @todo Checked against Mathematica
!> @param[in] n
!> @param[in] k
!> @param[in] s
!> @param[in] m
!> @param[out] XOUT
!-----------------------------------------------------------
FUNCTION ALL2L(n,k,s,m) RESULT(XOUT)
!
! Computes the associated-Laguerre/Laguerre to Laguerre coefficient, i.e.
!
! L_n^m L_k = sum_{s=0}^{n+k}\overline{d}_{nks}^m L_s
!
!
!
IMPLICIT NONE
!
INTEGER, intent(in) :: n,k,s,m ! ... degree and orders of Laguerre polynomials
TYPE(FM) :: XOUT ! ... coefficient values
TYPE(FM), SAVE :: AUXN,AUXK,AUXS
INTEGER :: n1,k1,s1
!
CALL FM_ENTER_USER_FUNCTION(XOUT)
! Compute coefficients
!
XOUT = TO_FM('0.0')
!
DO n1=0,n
AUXN =Lpjl(REAL(m,dp)-0.5_dp,REAL(n,dp),REAL(n1,dp))
DO k1=0,k
AUXK =Lpjl(-0.5_dp,REAL(k,dp),REAL(k1,dp))
DO s1=0,s
AUXS = Lpjl(-0.5_dp,REAL(s,dp),REAL(s1,dp))
XOUT = XOUT + FACTORIAL(TO_FM(n1 + k1 + s1 ))*AUXN*AUXK*AUXS
ENDDO
ENDDO
ENDDO
!
CALL FM_EXIT_USER_FUNCTION(XOUT)
!
END FUNCTION ALL2L
!-----------------------------------------------------------
!> @brief
!> Computes the associated Laguerre serie coefficient \f$ L_{jl}^p \f$
!>
!> \f[ L_{jl}^{p} = \frac{(-1)^l (p + j + 1/2)!}{(j - l)!(l +p + 1/2 )! l!}. \f]
!> such that
!> \f[ L^{p+1/2}_j(x) = \sum_{l=0}^j L^p_{jl} x^l \f]
!> @param[in] XOUT
!-----------------------------------------------------------
FUNCTION Lpjl(p,j,l) RESULT(XOUT)
!
! Computes the associated Laguerre serie coefficient,
!
IMPLICIT NONE
!
REAL(dp), intent(in) :: p,j,l
TYPE(FM) :: XOUT
!
CALL FM_ENTER_USER_FUNCTION(XOUT)
! compute coeff
XOUT = TO_FM('0.0')
XOUT = TO_FM(((-1)**l))*FACTORIAL(TO_FM(2*p + 2*j + 1)/2)/&
(FACTORIAL(TO_FM( j -l ))*FACTORIAL(TO_FM(2*l + 2*p + 1)/2)*FACTORIAL(TO_FM(l)))
CALL FM_EXIT_USER_FUNCTION(XOUT)
END FUNCTION Lpjl
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