biosspheres.helmholtz.crossinteractions#

Routine listings#

sj_pre_single_layer_semi_analytic_v1d v_0_sj_semi_analytic_v1d v_0_sj_semi_analytic_v2d v_0_js_from_v_0_sj k_0_sj_semi_analytic_v1d k_0_sj_semi_analytic_v2d jey_array k_0_sj_from_v_0_sj ka_0_sj_semi_analytic_recurrence_v1d ka_0_sj_semi_analytic_recurrence_v2d ka_0_sj_from_k_js w_0_sj_from_ka_sj w_0_sj_semi_analytic_recurrence_v1d w_0_sj_semi_analytic_recurrence_v2d a_0_sj_and_js_from_v_sj v_k_w_0_sj_from_quadratures_1d v_k_w_0_sj_from_quadratures_2d a_0_sj_and_js_from_v_k_w all_cross_interactions_n_spheres_from_v_1d all_cross_interactions_n_spheres_from_v_2d all_cross_interactions_n_spheres_1d all_cross_interactions_n_spheres_2d

Functions

`a_0_sj_and_js_from_v_k_w`(data_v_sj, ...)	Returns two numpy arrays that represents a numerical approximation of two matrices formed by the following boundary integral operators: a_sj = [-K_{s,j}^0 , V_{s,j}^0 ] [ W_{s,j}^0 , K_{s,j}^{0}] a_js = [-K_{j,s}^0 , V_{j,s}^0 ] [ W_{j,s}^0 , K_{j,s}^{0}] with Helmholtz kernel evaluated and tested with complex spherical harmonics. These are the Calderon operators for the spheres s and j.
`a_0_sj_and_js_from_v_sj`(big_l, data_v_sj, ...)	Returns two numpy arrays that represents a numerical approximation of two matrices formed by the following boundary integral operators: a_sj = [-K_{s,j}^0 , V_{s,j}^0 ] [ W_{s,j}^0 , K_{s,j}^{0}] a_js = [-K_{j,s}^0 , V_{j,s}^0 ] [ W_{j,s}^0 , K_{j,s}^{0}] with Helmholtz kernel evaluated and tested with complex spherical harmonics.
`all_cross_interactions_n_spheres_1d`(n, ...)
`all_cross_interactions_n_spheres_2d`(n, ...)
`all_cross_interactions_n_spheres_from_v_1d`(n, ...)
`all_cross_interactions_n_spheres_from_v_2d`(n, ...)
`jey_array`(big_l, k0, r_j)
`k_0_sj_from_v_0_sj`(data_v, k0, r_j)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics.
`k_0_sj_semi_analytic_v1d`(big_l, k0, r_j, ...)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics.
`k_0_sj_semi_analytic_v2d`(big_l, k0, r_j, ...)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics.
`ka_0_sj_from_k_js`(data_kjs)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^{*0} with Helmholtz kernel evaluated and tested with complex spherical harmonics. This routine needs the numpy array corresponding to K_{j,s}^0 (notice the change of the order of the indexes indicating the spheres).
`ka_0_sj_semi_analytic_recurrence_v1d`(big_l, ...)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^{*0} with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of one dimension. It is a SLOW routine, because it does not use any symmetry or properties of the spherical harmonics.
`ka_0_sj_semi_analytic_recurrence_v2d`(big_l, ...)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^{*0} with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of two dimensions, given by the function from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d of the module biosspheres.quadratures.spheres.
`sj_pre_single_layer_semi_analytic_v1d`(big_l, ...)
`v_0_js_from_v_0_sj`(data_v_sj)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator V_{j,s}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics.
`v_0_sj_semi_analytic_v1d`(big_l, k0, r_j, ...)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator V_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics.
`v_0_sj_semi_analytic_v2d`(big_l, k0, r_j, ...)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator V_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics.
`v_k_w_0_sj_from_quadratures_1d`(big_l, k0, ...)
`v_k_w_0_sj_from_quadratures_2d`(big_l, k0, ...)
`w_0_sj_from_ka_sj`(data_ka_sj, k0, r_j)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator W_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics. This routine needs the numpy array corresponding to the testing of K_{s,j}^{*0}.
`w_0_sj_semi_analytic_recurrence_v1d`(big_l, ...)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator W_{s,j}^{0} with Helmholtz kernel evaluated and tested with complex spherical harmonics.
`w_0_sj_semi_analytic_recurrence_v2d`(big_l, ...)	Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator W_{s,j}^{0} with Helmholtz kernel evaluated and tested with complex spherical harmonics.

biosspheres.helmholtz.crossinteractions.a_0_sj_and_js_from_v_k_w(data_v_sj: ndarray, data_k_sj: ndarray, data_ka_sj: ndarray, data_w_sj: ndarray, giro_sign: ndarray) → tuple[ndarray, ndarray][source]#

Returns two numpy arrays that represents a numerical approximation of two matrices formed by the following boundary integral operators: a_sj = [-K_{s,j}^0 , V_{s,j}^0 ]

[ W_{s,j}^0 , K_{s,j}^{*0}]

a_js = [-K_{j,s}^0 , V_{j,s}^0 ]: [ W_{j,s}^0 , K_{j,s}^{*0}]

with Helmholtz kernel evaluated and tested with complex spherical harmonics. These are the Calderon operators for the spheres s and j.

Notes

Parameters:

data_v_sj (numpy array) – of complex values.
data_k_sj (numpy array) – of complex values.
data_ka_sj (numpy array) – of complex values.
data_w_sj (numpy array) – of complex values.
giro_sign (np.ndarray) – of floats.

Returns:

a_sj (numpy array) – Calderon operator s j. Shape (2 * (big_l+1)**2, 2 * (big_l+1)**2).
a_js (numpy array) – Calderon operator j s. Shape (2 * (big_l+1)**2, 2 * (big_l+1)**2).

biosspheres.helmholtz.crossinteractions.a_0_sj_and_js_from_v_sj(big_l: int, data_v_sj: ndarray, k0_ratio_j_l_j: ndarray, k0_ratio_j_l_s: ndarray, giro_sign: ndarray) → tuple[ndarray, ndarray][source]#

Returns two numpy arrays that represents a numerical approximation of two matrices formed by the following boundary integral operators: a_sj = [-K_{s,j}^0 , V_{s,j}^0 ]

[ W_{s,j}^0 , K_{s,j}^{*0}]

a_js = [-K_{j,s}^0 , V_{j,s}^0 ]: [ W_{j,s}^0 , K_{j,s}^{*0}]

with Helmholtz kernel evaluated and tested with complex spherical harmonics.

Notes

The only operator computed directly with the numerical quadrature is V_{s,j}^0, which is an input. The others are computed with the same properties used in: v_0_js_from_v_0_sj k_0_sj_from_v_0_sj ka_0_sj_from_k_js w_0_sj_from_ka_sj

Parameters:

big_l (int) – >= 0, max degree.
data_v_sj (np.ndarray) – represents a numerical approximation of the matrix formed by the boundary integral operator V_{s,j}^0 with Helmholtz kernel evaluated and tested with spherical harmonics.
k0_ratio_j_l_j (np.ndarray) – of floats. = k0 * j_lp_j / j_l_j. With j_lp_j the derivative of the spherical Bessel function evaluated in k0 * r_j.
k0_ratio_j_l_s (np.ndarray) – of floats. = k0 * j_lp_s / j_l_s. With j_lp_j the derivative of the spherical Bessel function evaluated in k0 * r_s.
giro_sign (np.ndarray) – of floats and with only zeros, ones or minus ones. Come from the function giro_sign from the module biosspheres.miscella.auxindexes.

Returns:

a_sj (numpy array) – Shape (2 * (big_l+1)**2, 2 * (big_l+1)**2).
a_js (numpy array) – Shape (2 * (big_l+1)**2, 2 * (big_l+1)**2).

biosspheres.helmholtz.crossinteractions.k_0_sj_from_v_0_sj(data_v: ndarray, k0: float, r_j: float) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics. This routine needs the numpy array corresponding to the testing of V_{s,j}^0.

Notes

data_k[p*(2p+1) + q, l*(2l+1) + m] =: ( K_{s,j}^0 Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

This computation uses the following result for this specific case: K_{s,j}^0 Y_{l,m} = -k0 * (j_l’(k0 r_j) / j_l (k0 r_j))

V_{s,j}^0 Y_{l,m}.

With j_l’ the derivative of the spherical Bessel function, and j_l the spherical Bessel function.

It will blow up if k0 r_j is a root of any j_l.

Parameters:

data_v (np.ndarray) – that represents a numerical approximation of the matrix formed by the boundary integral operator V_{s,j}^0 with Helmholtz kernel evaluated and tested with spherical harmonics.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.

Returns:

data_k – of complex numbers. Same shape than data_v. See notes for the indexes ordering.

Return type:

np.ndarray

biosspheres.helmholtz.crossinteractions.k_0_sj_semi_analytic_v1d(big_l: int, k0: float, r_j: float, r_s: float, j_lp_j: ndarray, r_coord: ndarray, phi_coord: ndarray, cos_theta_coord: ndarray, final_length: int, transform: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of one dimension. It is a SLOW routine, because it does not use any symmetry or properties of the spherical harmonics.

Notes

data_k[p*(2p+1) + q, l*(2l+1) + m] =: ( K_{s,j}^0 Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

The expression K_{s,j}^0 Y_l,m,j can be obtained analytically. A quadrature scheme is used to compute the surface integral corresponding to the inner product.

In this routine the quadrature points NEED to be ordered in an array of one dimension. It is a SLOW routine, because it does not use any symmetry or properties of the spherical harmonics.

Parameters:

big_l (int) – >= 0, max degree.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.
r_s (float) – > 0, radius of the sphere s.
j_lp_j (np.ndarray) – of floats. Derivative of the spherical Bessel function evaluated in k0 * r_j.
r_coord (np.ndarray) – Array of floats with the spherical coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from the function from_sphere_s_cartesian_to_j_spherical_1d of the module biosspheres.quadratures.spheres.
phi_coord (np.ndarray) – Array of floats with the phi coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from the function from_sphere_s_cartesian_to_j_spherical_1d of the module biosspheres.quadratures.spheres.
cos_theta_coord (np.ndarray) – Array of floats with the cosine of the spherical coordinate theta of the quadrature points in the coordinate system s. Lengths equal to final_length. Can come from the function from_sphere_s_cartesian_to_j_spherical_1d of the module biosspheres.quadratures.spheres.
final_length (int) – How many points for the surface integral. Can come from the function spherical_harmonic_transform_1d of the module biosspheres.quadratures.spheres.
transform (np.ndarray) – of complex numbers for doing the complex spherical harmonic transform. Can come from the function complex_spherical_harmonic_transform_1d of the module biosspheres.quadratures.spheres.

Returns:

data_k – of complex numbers. Shape ((big_l+1)**2, (big_l+1)**2). See notes for the indexes ordering.

Return type:

numpy array

biosspheres.helmholtz.crossinteractions.k_0_sj_semi_analytic_v2d(big_l: int, k0: float, r_j: float, r_s: float, j_lp_j: ndarray, r_coord: ndarray, phi_coord: ndarray, cos_theta_coord: ndarray, weights: ndarray, zeros: ndarray, quantity_theta_points: int, quantity_phi_points: float, pesykus: ndarray, p2_plus_p_plus_q: ndarray, p2_plus_p_minus_q: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of two dimensions, given by the function from_sphere_s_cartesian_to_j_spherical_2d of the module biosspheres.quadratures.spheres.

Notes

data_k[p(2p + 1) + q, l(2l + 1) + m] =: ( K_{s,j}^0 Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

The expression K_{s,j}^0 Y_l,m,j can be obtained analytically. A quadrature scheme is used to compute the other surface integral. It uses functions from the package pyshtools to compute the spherical harmonic transforms.

Parameters:

big_l (int) – >= 0, max degree of spherical harmonics used.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.
r_s (float) – > 0, radius of the sphere s.
j_lp_j (np.ndarray) – of floats. Derivative of the spherical Bessel function evaluated in k0 * r_j.
r_coord (np.ndarray) – Two dimensional array of floats with the spherical coordinate r of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from the function from_sphere_s_cartesian_to_j_spherical_2d of the module biosspheres.quadratures.spheres.
phi_coord (np.ndarray) – Two dimensional array of floats with the phi coordinate r of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from the function from_sphere_s_cartesian_to_j_spherical_2d of the module biosspheres.quadratures.spheres.
cos_theta_coord (np.ndarray) – Two dimensional array of floats with the cosine of the spherical coordinate theta of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from the function from_sphere_s_cartesian_to_j_spherical_2d of the module biosspheres.quadratures.spheres.
weights (np.ndarray) – of floats. Weights for the integral quadrature in the theta variable. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
zeros (np.ndarray) – of floats. Zeros of the integral quadrature in the theta variable. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
quantity_theta_points (int) – how many points for the integral in theta. Can come from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
quantity_phi_points (int) – how many points for the integral in phi. Can come fron the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
pesykus (np.ndarray) – dtype int, shape ((big_l+1) * big_l // 2, 2). Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)
p2_plus_p_plus_q (np.ndarray) – dtype int, length (big_l+1) * big_l // 2. Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)
p2_plus_p_minus_q (np.ndarray) – dtype int, length (big_l+1) * big_l // 2. Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)

Returns:

data_k – of complex numbers. Shape ((big_l+1)**2, (big_l+1)**2). See notes for the indexes ordering.

Return type:

numpy array

See also

k_0_sj_semi_analytic_v1d, k_0_sj_from_v_0_sj, gauss_legendre_trapezoidal_shtools_2d, biosspheres.miscella.auxindexes.pes_y_kus

biosspheres.helmholtz.crossinteractions.ka_0_sj_from_k_js(data_kjs: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^{*0} with Helmholtz kernel evaluated and tested with complex spherical harmonics. This routine needs the numpy array corresponding to K_{j,s}^0 (notice the change of the order of the indexes indicating the spheres).

Notes

data_ka_sj[p*(2p+1) + q, l*(2l+1) + m] =: ( K_{s,j}^{*0} Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

This computation uses the following result for this specific case: ( K_{s,j}^{*0} Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}

= (-1)**(m+q) ( K_{j,s}^0 Y_{p,-q,s} ; Y_{l,-m,j} )_{L^2(S_j)}.

Parameters:: data_kjs (numpy array.) – represents a numerical approximation of the matrix formed by the boundary integral operator V_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics.
Returns:: data_ka_sj – Same shape than data_kjs. See notes for the indexes ordering.
Return type:: numpy array.

biosspheres.helmholtz.crossinteractions.ka_0_sj_semi_analytic_recurrence_v1d(big_l: int, k0: float, r_j: float, r_s: float, j_l_j: ndarray, r_coord: ndarray, phi_coord: ndarray, cos_theta_coord: ndarray, er_times_n: ndarray, etheta_times_n: ndarray, ephi_times_n: ndarray, final_length: int, transform: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^{*0} with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of one dimension. It is a SLOW routine, because it does not use any symmetry or properties of the spherical harmonics.

Notes

data_ka[p*(2p+1) + q, l*(2l+1) + m] =: ( K_{s,j}^{*0} Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

The expression K_{s,j}^{*0} Y_l,m,j can be obtained analytically. A quadrature scheme is used to compute the other surface integral corresponding to the inner product. For computing the derivative in theta a recurrence formula for Legendre Functions is used.

In this routine the quadrature points NEED to be ordered in an array of one dimension.

Parameters:

big_l (int) – >= 0, max degree.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.
r_s (float) – > 0, radius of the sphere s.
j_l_j (np.ndarray) – of floats. Spherical Bessel function evaluated in k0 * r_j.
r_coord (np.ndarray) – One dimensional array of floats with the spherical coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
phi_coord (np.ndarray) – One dimensional array of floats with the phi coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
cos_theta_coord (np.ndarray) – One dimensional array of floats with the cosine of the spherical coordinate theta of the quadrature points in the coordinate system s. Lenght equals to final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
er_times_n (np.ndarray) – One dimensional array of floats with the dot product of the canonical vector of the spherical coordinate r in the points in the coordinate system s and the normal of a sphere. Length final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
etheta_times_n (np.ndarray) – One dimensional array of floats with the dot product of the canonical vector of the spherical coordinate theta in the points in the coordinate system s and the normal of a sphere. Length final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
ephi_times_n (np.ndarray) – One dimensional array of floats with the dot product of the canonical vector of the spherical coordinate phi in the points in the coordinate system s and the normal of a sphere. Length final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
final_length (int) – how many points for the surface integral, (big_l_c + 1) * (2 * big_l_c + 1).
transform (np.ndarray) – of complex numbers for doing the complex spherical harmonic transform. Shape ((big_l+1)**2, final_length)

Returns:

data_ka – of complex numbers. Shape ((big_l+1)**2, (big_l+1)**2). See notes for the indexes ordering.

Return type:

numpy array

biosspheres.helmholtz.crossinteractions.ka_0_sj_semi_analytic_recurrence_v2d(big_l: int, k0: float, r_j: float, r_s: float, j_l_j: ndarray, r_coord: ndarray, phi_coord: ndarray, cos_theta_coord: ndarray, er_times_n: ndarray, etheta_times_n: ndarray, ephi_times_n: ndarray, weights: ndarray, zeros: ndarray, quantity_theta_points: int, quantity_phi_points: float, pesykus: ndarray, p2_plus_p_plus_q: ndarray, p2_plus_p_minus_q: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^{*0} with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of two dimensions, given by the function from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d of the module biosspheres.quadratures.spheres.

Notes

data_ka[p*(2p+1) + q, l*(2l+1) + m] =: ( K_{s,j}^{*0} Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

In this routine the quadrature points NEED to be ordered in an array of two dimensions, given by the function from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d of the module biosspheres.quadratures.spheres. It uses functions from the package pyshtools to compute the spherical harmonic transforms.

Parameters:

big_l (int) – >= 0, max degree.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.
r_s (float) – > 0, radius of the sphere s.
j_l_j (np.ndarray) – of floats. Spherical Bessel function evaluated in k0 * r_j.
r_coord (np.ndarray) – Two dimensional array of floats with the spherical coordinate r of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
phi_coord (np.ndarray) – Two dimensional array of floats with the phi coordinate r of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
cos_theta_coord (np.ndarray) – Two dimensional array of floats with the cosine of the spherical coordinate theta of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
er_times_n (np.ndarray) – Two dimensional array of floats with the dot product of the canonical vector of the spherical coordinate r in the points in the coordinate system s and the normal of a sphere. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
etheta_times_n (np.ndarray) – Two dimensional array of floats with the dot product of the canonical vector of the spherical coordinate theta in the points in the coordinate system s and the normal of a sphere. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
ephi_times_n (np.ndarray) – Two dimensional array of floats with the dot product of the canonical vector of the spherical coordinate phi in the points in the coordinate system s and the normal of a sphere. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
weights (np.ndarray) – of floats. Weights for the integral quadrature in the theta variable. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
zeros (np.ndarray) – of floats. Zeros of the integral quadrature in the theta variable. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
quantity_theta_points (int) – how many points for the integral in theta. Can come from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
quantity_phi_points (int) – how many points for the integral in phi. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
pesykus (np.ndarray) – dtype int, shape ((big_l+1) * big_l // 2, 2). Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)
p2_plus_p_plus_q (np.ndarray) – dtype int, length (big_l+1) * big_l // 2. Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)
p2_plus_p_minus_q (np.ndarray) – dtype int, length (big_l+1) * big_l // 2. Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)

Returns:

data_ka – of complex numbers. Shape ((big_l+1)**2, (big_l+1)**2). See notes for the indexes ordering.

Return type:

numpy array

biosspheres.helmholtz.crossinteractions.v_0_js_from_v_0_sj(data_v_sj: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator V_{j,s}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics. This routine needs the numpy array corresponding to V_{s,j}^0 (notice the change of the order of the indexes indicating the spheres).

Notes

data_v_js[p*(2p+1) + q, l*(2l+1) + m] =: ( V_{j,s}^0 Y_{l,m,s} ; Y_{p,q,j} )_{L^2(S_j)}.
Y_{l,m,s}spherical harmonic degree l, order m, in the coordinate: system s.

S_j : surface of the sphere j.

This computation uses the following result for this specific case: ( V_{j,s}^0 Y_{l,m,s} ; Y_{p,q,j} )_{L^2(S_j)}.

= (-1)**(m+q) ( V_{s,j}^0 Y_{p,-q,j} ; Y_{l,-m,s} )_{L^2(S_s)}

Parameters:: data_v_sj (np.ndarray) – represents a numerical approximation of the matrix formed by the boundary integral operator V_{s,j}^0 with Helmholtz kernel evaluated and tested with spherical harmonics.
Returns:: data_v_js – of complex numbers. Same shape than data_v_sj. See notes for the indexes ordering.
Return type:: np.ndarray

biosspheres.helmholtz.crossinteractions.v_0_sj_semi_analytic_v1d(big_l: int, k0: float, r_j: float, r_s: float, j_l_j: ndarray, r_coord: ndarray, phi_coord: ndarray, cos_theta_coord: ndarray, final_length: int, transform: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator V_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of one dimension. It is a SLOW routine, because it does not use any symmetry or properties of the spherical harmonics.

Notes

data_v[p(2p + 1) + q, l(2l + 1) + m] =: ( V_{s,j}^0 Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

The expression V_{s,j}^0 Y_{l,m,j} is analytic. A quadrature scheme is used to compute the surface integral corresponding to the inner product.

Parameters:

big_l (int) – >= 0, max degree of spherical harmonics used.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.
r_s (float) – > 0, radius of the sphere s.
j_l_j (np.ndarray) – of floats. Spherical Bessel function evaluated in k0 * r_j.
r_coord (np.ndarray) – Array of floats with the spherical coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from the function from_sphere_s_cartesian_to_j_spherical_1d of the module biosspheres.quadratures.spheres.
phi_coord (np.ndarray) – Array of floats with the phi coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from the function from_sphere_s_cartesian_to_j_spherical_1d of the module biosspheres.quadratures.spheres.
cos_theta_coord (np.ndarray) – Array of floats with the cosine of the spherical coordinate theta of the quadrature points in the coordinate system s. Lengths equal to final_length. Can come from the function from_sphere_s_cartesian_to_j_spherical_1d of the module biosspheres.quadratures.spheres.
final_length (int) – How many points for the surface integral. Can come from the function spherical_harmonic_transform_1d of the module biosspheres.quadratures.spheres.
transform (np.ndarray) – of complex numbers for doing the complex spherical harmonic transform. Can come from the function complex_spherical_harmonic_transform_1d of the module biosspheres.quadratures.spheres.

Returns:

data_v – of complex numbers. Shape ((big_l+1)**2, (big_l+1)**2). See notes for the indexes ordering.

Return type:

numpy array

See also

v_0_sj_semi_analytic_v2d

biosspheres.helmholtz.crossinteractions.v_0_sj_semi_analytic_v2d(big_l: int, k0: float, r_j: float, r_s: float, j_l_j: ndarray, r_coord: ndarray, phi_coord: ndarray, cos_theta_coord: ndarray, weights: ndarray, zeros: ndarray, quantity_theta_points: int, quantity_phi_points: float, pesykus: ndarray, p2_plus_p_plus_q: ndarray, p2_plus_p_minus_q: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator V_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of two dimensions, given by the function from_sphere_s_cartesian_to_j_spherical_2d of the module biosspheres.quadratures.spheres.

Notes

data_v[p(2p + 1) + q, l(2l + 1) + m] =: ( V_{s,j}^0 Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

The expression V_{s,j}^0 Y_{l,m,j} is analytic. A quadrature scheme is used to compute the surface integral corresponding to the inner product.

It uses functions from the package pyshtools to compute the spherical harmonic transforms.

Parameters:

big_l (int) – >= 0, max degree of spherical harmonics used.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.
r_s (float) – > 0, radius of the sphere s.
j_l_j (np.ndarray) – of floats. Spherical Bessel function evaluated in k0 * r_j.
r_coord (np.ndarray) – Two dimensional array of floats with the spherical coordinate r of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from the function from_sphere_s_cartesian_to_j_spherical_2d of the module biosspheres.quadratures.spheres.
phi_coord (np.ndarray) – Two dimensional array of floats with the phi coordinate r of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from the function from_sphere_s_cartesian_to_j_spherical_2d of the module biosspheres.quadratures.spheres.
cos_theta_coord (np.ndarray) – Two dimensional array of floats with the cosine of the spherical coordinate theta of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from the function from_sphere_s_cartesian_to_j_spherical_2d of the module biosspheres.quadratures.spheres.
weights (np.ndarray) – of floats. Weights for the integral quadrature in the theta variable. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
zeros (np.ndarray) – of floats. Zeros of the integral quadrature in the theta variable. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
quantity_theta_points (int) – how many points for the integral in theta. Can come from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
quantity_phi_points (int) – how many points for the integral in phi. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
pesykus (np.ndarray) – dtype int, shape ((big_l+1) * big_l // 2, 2). Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)
p2_plus_p_plus_q (np.ndarray) – dtype int, length (big_l+1) * big_l // 2. Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)
p2_plus_p_minus_q (np.ndarray) – dtype int, length (big_l+1) * big_l // 2. Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)

Returns:

data_v – of complex numbers. Shape ((big_l+1)**2, (big_l+1)**2). See notes for the indexes ordering.

Return type:

numpy array

See also

v_0_sj_semi_analytic_v1d, gauss_legendre_trapezoidal_shtools_2d, biosspheres.miscella.auxindexes.pes_y_kus

biosspheres.helmholtz.crossinteractions.v_k_w_0_sj_from_quadratures_1d(big_l: int, k0: float, r_j: float, r_s: float, j_l_j: ndarray, j_lp_j: ndarray, r_coord: ndarray, phi_coord: ndarray, cos_theta_coord: ndarray, er_times_n: ndarray, etheta_times_n: ndarray, ephi_times_n: ndarray, final_length: int, transform: ndarray) → tuple[ndarray, ndarray, ndarray, ndarray][source]#

Notes

In this routine the quadrature points NEED to be ordered in an array of one dimension. It is a SLOW routine, because it does not use any symmetry or properties of the spherical harmonics.

Parameters:

big_l (int) – >= 0, max degree.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.
r_s (float) – > 0, radius of the sphere s.
j_l_j (np.ndarray) – of floats. Spherical Bessel function evaluated in k0 * r_j.
j_lp_j (np.ndarray) – of floats. Derivative of the spherical Bessel function evaluated in k0 * r_j.
r_coord (np.ndarray) – One dimensional array of floats with the spherical coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
phi_coord (np.ndarray) – One dimensional array of floats with the phi coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
cos_theta_coord (np.ndarray) – One dimensional array of floats with the cosine of the spherical coordinate theta of the quadrature points in the coordinate system s. Lenght equals to final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
er_times_n (np.ndarray) – One dimensional array of floats with the dot product of the canonical vector of the spherical coordinate r in the points in the coordinate system s and the normal of a sphere. Length final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
etheta_times_n (np.ndarray) – One dimensional array of floats with the dot product of the canonical vector of the spherical coordinate theta in the points in the coordinate system s and the normal of a sphere. Length final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
ephi_times_n (np.ndarray) – One dimensional array of floats with the dot product of the canonical vector of the spherical coordinate phi in the points in the coordinate system s and the normal of a sphere. Length final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
final_length (int) – how many points for the surface integral, (big_l_c + 1) * (2 * big_l_c + 1).
transform (np.ndarray) – of complex numbers for doing the complex spherical harmonic transform. Shape ((big_l+1)**2, final_length)

Returns:

data_v_sj (numpy array) – Shape ((big_l+1)**2, (big_l+1)**2).
data_k_sj (numpy array) – Shape ((big_l+1)**2, (big_l+1)**2).
data_ka_sj (numpy array) – Shape ((big_l+1)**2, (big_l+1)**2).
data_w_sj (numpy array) – Shape ((big_l+1)**2, (big_l+1)**2).

biosspheres.helmholtz.crossinteractions.w_0_sj_from_ka_sj(data_ka_sj: ndarray, k0: float, r_j: float) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator W_{s,j}^0 with Helmholtz kernel evaluated and tested with complex spherical harmonics. This routine needs the numpy array corresponding to the testing of K_{s,j}^{*0}.

Notes

data_w[p*(2p+1) + q, l*(2l+1) + m] =: ( W_{s,j}^0 Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

This computation uses the following result for this specific case: W_{s,j}^0 Y_{l,m} = k0 * (j_l’(k0 r_j) / j_l (k0 r_j))

K_{s,j}^{*0} Y_{l,m}.

With j_l’ the derivative of the spherical Bessel function, and j_l the spherical Bessel function.

It will blow up if k0 r_j is a root of any j_l.

Parameters:

data_ka_sj (np.ndarray) – that represents a numerical approximation of the matrix formed by the boundary integral operator K_{s,j}^{*0} with Helmholtz kernel evaluated and tested with spherical harmonics.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.

Returns:

data_w – of complex numbers. Same shape than data_ka_sj. See notes for the indexes ordering.

Return type:

np.ndarray

biosspheres.helmholtz.crossinteractions.w_0_sj_semi_analytic_recurrence_v1d(big_l: int, k0: float, r_j: float, r_s: float, j_lp_j: ndarray, r_coord: ndarray, phi_coord: ndarray, cos_theta_coord: ndarray, er_times_n: ndarray, etheta_times_n: ndarray, ephi_times_n: ndarray, final_length: int, transform: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator W_{s,j}^{0} with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of one dimension. It is a SLOW routine, because it does not use any symmetry or properties of the spherical harmonics.

Notes

data_w[p*(2p+1) + q, l*(2l+1) + m] =: ( W_{s,j}^{0} Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

The expression W_{s,j}^{0} Y_l,m,j can be obtained analytically. A quadrature scheme is used to compute the other surface integral corresponding to the inner product. For computing the derivative in theta a recurrence formula for Legendre Functions is used.

In this routine the quadrature points NEED to be ordered in an array of one dimension.

Parameters:

big_l (int) – >= 0, max degree.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.
r_s (float) – > 0, radius of the sphere s.
j_lp_j (np.ndarray) – of floats. Derivative of the spherical Bessel function evaluated in k0 * r_j.
r_coord (np.ndarray) – One dimensional array of floats with the spherical coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
phi_coord (np.ndarray) – One dimensional array of floats with the phi coordinate r of the quadrature points in the coordinate system s. Length equals to final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
cos_theta_coord (np.ndarray) – One dimensional array of floats with the cosine of the spherical coordinate theta of the quadrature points in the coordinate system s. Lenght equals to final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
er_times_n (np.ndarray) – One dimensional array of floats with the dot product of the canonical vector of the spherical coordinate r in the points in the coordinate system s and the normal of a sphere. Length final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
etheta_times_n (np.ndarray) – One dimensional array of floats with the dot product of the canonical vector of the spherical coordinate theta in the points in the coordinate system s and the normal of a sphere. Length final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
ephi_times_n (np.ndarray) – One dimensional array of floats with the dot product of the canonical vector of the spherical coordinate phi in the points in the coordinate system s and the normal of a sphere. Length final_length. Can come from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_1d
final_length (int) – how many points for the surface integral, (big_l_c + 1) * (2 * big_l_c + 1).
transform (np.ndarray) – of complex numbers for doing the complex spherical harmonic transform. Shape ((big_l+1)**2, final_length)

Returns:

data_w – of complex numbers. Shape ((big_l+1)**2, (big_l+1)**2). See notes for the indexes ordering.

Return type:

numpy array

biosspheres.helmholtz.crossinteractions.w_0_sj_semi_analytic_recurrence_v2d(big_l: int, k0: float, r_j: float, r_s: float, j_lp_j: ndarray, r_coord: ndarray, phi_coord: ndarray, cos_theta_coord: ndarray, er_times_n: ndarray, etheta_times_n: ndarray, ephi_times_n: ndarray, weights: ndarray, zeros: ndarray, quantity_theta_points: int, quantity_phi_points: float, pesykus: ndarray, p2_plus_p_plus_q: ndarray, p2_plus_p_minus_q: ndarray) → ndarray[source]#

Returns a numpy array that represents a numerical approximation of the matrix formed by the boundary integral operator W_{s,j}^{0} with Helmholtz kernel evaluated and tested with complex spherical harmonics. In this routine the quadrature points NEED to be ordered in an array of two dimensions, given by the function from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d of the module biosspheres.quadratures.spheres.

Notes

data_w[p*(2p+1) + q, l*(2l+1) + m] =: ( W_{s,j}^{0} Y_{l,m,j} ; Y_{p,q,s} )_{L^2(S_s)}.
Y_{l,m,j}spherical harmonic degree l, order m, in the coordinate: system j.

S_s : surface of the sphere s.

Parameters:

big_l (int) – >= 0, max degree.
k0 (float) – > 0
r_j (float) – > 0, radius of the sphere j.
r_s (float) – > 0, radius of the sphere s.
j_lp_j (np.ndarray) – of floats. Derivative of the spherical Bessel function evaluated in k0 * r_j.
r_coord (np.ndarray) – Two dimensional array of floats with the spherical coordinate r of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
phi_coord (np.ndarray) – Two dimensional array of floats with the phi coordinate r of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
cos_theta_coord (np.ndarray) – Two dimensional array of floats with the cosine of the spherical coordinate theta of the quadrature points in the coordinate system s. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
er_times_n (np.ndarray) – Two dimensional array of floats with the dot product of the canonical vector of the spherical coordinate r in the points in the coordinate system s and the normal of a sphere. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
etheta_times_n (np.ndarray) – Two dimensional array of floats with the dot product of the canonical vector of the spherical coordinate theta in the points in the coordinate system s and the normal of a sphere. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
ephi_times_n (np.ndarray) – Two dimensional array of floats with the dot product of the canonical vector of the spherical coordinate phi in the points in the coordinate system s and the normal of a sphere. Shape equals to (quantity_theta_points, quantity_phi_points). Comes from biosspheres.quadratures.spheres. from_sphere_s_cartesian_to_j_spherical_and_spherical_vectors_2d
weights (np.ndarray) – of floats. Weights for the integral quadrature in the theta variable. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
zeros (np.ndarray) – of floats. Zeros of the integral quadrature in the theta variable. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
quantity_theta_points (int) – how many points for the integral in theta. Can come from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
quantity_phi_points (int) – how many points for the integral in phi. Comes from the function gauss_legendre_trapezoidal_shtools_2d from the module biosspheres.quadratures.spheres.
pesykus (np.ndarray) – dtype int, shape ((big_l+1) * big_l // 2, 2). Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)
p2_plus_p_plus_q (np.ndarray) – dtype int, length (big_l+1) * big_l // 2. Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)
p2_plus_p_minus_q (np.ndarray) – dtype int, length (big_l+1) * big_l // 2. Used for the vectorization of some computations. Comes from the function biosspheres.miscella.auxindexes.pes_y_kus(big_l)

Returns:

data_w – of complex numbers. Shape ((big_l+1)**2, (big_l+1)**2). See notes for the indexes ordering.

Return type:

numpy array

See also

w_0_sj_semi_analytic_recurrence_v1d, w_0_sj_from_k_js