biKddlZddlZddlZddlmZddlmZmZm Z m Z ddl m Z m Z mZmZgdZGddZee d d d Z e j$d Ze j(d Zeeddd Zej$dZej*dZej,dZej(dZeedddddZej$dZej.dZej(dZee dddddZ e j$dZe j.dZe j*d Ze j,d!Ze j(d"Zeed#d$d Zej$d%Zej(d&Zee d'd(d Z e j$d)Ze j*d*Ze j,d+Ze j(d,Zee d-d.d/d0Z e j$d1Ze j(d2Zeed3d4d/d0Zej$d5Zej*d6Zej,d7Zej(d8Zy)9N_nonneg_int_or_fail) legendre_passoc_legendre_psph_legendre_p sph_harm_y)legendre_p_allassoc_legendre_p_allsph_legendre_p_allsph_harm_y_all)rr rr r r rr cTeZdZd dddZedZdZdZdZd Z d Z d Z d Z y) MultiUFuncNF)force_complex_outputc t|tjst|tjj r|j }n2t|tjjr|}n tdt}|D]U}t|tjstd||jtd|jDWt|dkDr td||_||_||_||_||_d|_d|_d|_d|_d|_y)Nz7ufunc_or_ufuncs should be a ufunc or a ufunc collectionz2All ufuncs must have type `numpy.ufunc`. Received c3DK|]}|jddyw)z->rN)split).0xs U/home/cdr/jupyterlab/.venv/lib/python3.12/site-packages/scipy/special/_multiufuncs.py z&MultiUFunc.__init__..+s.UAqwwt}Q/?.Us rz*All ufuncs must take the same input types.cy)Nrargskwargss rz%MultiUFunc.__init__..7sciSNrrs rrz%MultiUFunc.__init__..8sRr) isinstancenpufunc collectionsabcMappingvaluesIterable ValueErrorsetadd frozensettypeslen__name___ufunc_or_ufuncs_MultiUFunc__doc!_MultiUFunc__force_complex_output_default_kwargs_resolve_out_shapes _finalize_out_key_ufunc_default_args_ufunc_default_kwargs) selfufunc_or_ufuncsnamedocrdefault_kwargs ufuncs_iterseen_input_typesr#s r__init__zMultiUFunc.__init__s,/2884/;??+B+BC-446 O[__-E-EF-  "566 #u $ W!%2$&22A1B&DEE $$Y.U.U%UV  W #$q( !MNN / &:#-#' ! #= %?"rc|jSr )r1)r9s r__doc__zMultiUFunc.__doc__:s zzrc||_y)z3Set `key` method by decorating a function. N)r6r9funcs r _override_keyzMultiUFunc._override_key>s  rc||_yr )r7rDs r_override_ufunc_default_argsz'MultiUFunc._override_ufunc_default_argsCs #' rc||_yr )r8rDs r_override_ufunc_default_kwargsz)MultiUFunc._override_ufunc_default_kwargsFs %)"rcF|jd|_d|_||_y)z9Set `resolve_out_shapes` method by decorating a function.Nz2Resolve to output shapes based on relevant inputs.resolve_out_shapes)rBr/r4rDs r_override_resolve_out_shapesz'MultiUFunc._override_resolve_out_shapesIs% << H L, #' rc||_yr )r5rDs r_override_finalize_outz!MultiUFunc._override_finalize_outQs !rc t|jtjr |jS|jdi|}|j|S)z.Resolve to a ufunc based on keyword arguments.r)r!r0r"r#r6)r9r ufunc_keys r_resolve_ufunczMultiUFunc._resolve_ufuncTsH d++RXX 6(( (DII'' $$Y//rc|j|z}||jdi|z }|jdi|}||j dDcgc]}t j |}}|j di|}|j*td|D}|jg|d|j ||ji|}td|D} t|dr4| |jdzz} |j| } | |j d} nVt j| } t j| tjstj} |j| fz} |j rtd| D} tdt#|| D} | |d<||i|} |j$|j%| } | Scc}w) Nc3FK|]}tj|ywr )r"shaper ufunc_args rrz&MultiUFunc.__call__..js$UYRXXi%8$Us!c3K|]:}t|dr |jntjt|<yw)dtypeN)hasattrrYr"typerVs rrz&MultiUFunc.__call__..os>%B)29@ 78SY__*,((4 ?*C&D%BsAAresolve_dtypesr c3HK|]}tjd|yw)y?N)r" result_type)rufunc_out_dtypes rrz&MultiUFunc.__call__..s%)R-<*,O)L)Rs "c3PK|]\}}tj|| yw))rYN)r"empty)rufunc_out_shaper_s rrz&MultiUFunc.__call__..s,D= 0``. m : ArrayLike[int] Order of the spherical Legendre polynomial. theta : ArrayLike[float] Input value. diff_n : Optional[int] A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- p : ndarray or tuple[ndarray] Spherical Legendre polynomial with ``diff_n`` derivatives. Notes ----- The spherical counterpart of an (unnormalized) associated Legendre polynomial has the additional factor .. math:: \sqrt{\frac{(2 n + 1) (n - m)!}{4 \pi (n + m)!}} It is the same as the spherical harmonic :math:`Y_{n}^{m}(\theta, \phi)` with :math:`\phi = 0`. diff_ncZt|dd}d|cxkrdksntd|d|SNryFstrictrGdiff_n is currently only implemented for orders 0, 1, and 2, received: .rr)rxs r_rB % @F  !    $   Mrc0tj|ddSNrr"moveaxisrcs rrr ;;sB ""rr asph_legendre_p_all(n, m, theta, *, diff_n=0) All spherical Legendre polynomials of the first kind up to the specified degree ``n``, order ``m``, and all derivatives up to order ``diff_n``. Output shape is ``(diff_n + 1, n + 1, 2 * m + 1, ...)``. The entry at ``(i, j, k)`` corresponds to the ``i``-th derivative, degree ``j``, and order ``k`` for all ``0 <= i <= diff_n``, ``0 <= j <= n``, and ``-m <= k <= m``. See Also -------- sph_legendre_p cZt|dd}d|cxkrdksntd|d|Sr{rrxs rrrrrcddgdgziSNaxesr)rrrrrxs rrrs RDJ<' ((rct|tjr|dkr td|dzdt |zdzf|z|dzfzfS)Nr!n must be a non-negative integer.rr~)r!numbersIntegralr)abs)nm theta_shapergrys rrrsR a)) *q1u<== UAAJN #k 1VaZM A CCrc0tj|ddSrrrs rrrrrraassoc_legendre_p(n, m, z, *, branch_cut=2, norm=False, diff_n=0) Associated Legendre polynomial of the first kind. Parameters ---------- n : ArrayLike[int] Degree of the associated Legendre polynomial. Must have ``n >= 0``. m : ArrayLike[int] order of the associated Legendre polynomial. z : ArrayLike[float | complex] Input value. branch_cut : Optional[ArrayLike[int]] Selects branch cut. Must be 2 (default) or 3. 2: cut on the real axis ``|z| > 1`` 3: cut on the real axis ``-1 < z < 1`` norm : Optional[bool] If ``True``, compute the normalized associated Legendre polynomial. Default is ``False``. diff_n : Optional[int] A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- p : ndarray or tuple[ndarray] Associated Legendre polynomial with ``diff_n`` derivatives. Notes ----- The normalized counterpart of an (unnormalized) associated Legendre polynomial has the additional factor .. math:: \sqrt{\frac{(2 n + 1) (n - m)!}{2 (n + m)!}} r~F branch_cutnormryc^t|dd}d|cxkrdksntd|d||fSr{rrs rrr#sG % @F  !    $   <rc|fSr rrs rrr. ;rc0tj|ddSrrrs rrr3rrr aassoc_legendre_p_all(n, m, z, *, branch_cut=2, norm=False, diff_n=0) All associated Legendre polynomials of the first kind up to the specified degree ``n``, order ``m``, and all derivatives up to order ``diff_n``. Output shape is ``(diff_n + 1, n + 1, 2 * m + 1, ...)``. The entry at ``(i, j, k)`` corresponds to the ``i``-th derivative, degree ``j``, and order ``k`` for all ``0 <= i <= diff_n``, ``0 <= j <= n``, and ``-m <= k <= m``. See Also -------- assoc_legendre_p ct|tjr|dk\std|dd|cxkrdksntd|d||fSNrz1diff_n must be a non-negative integer, received: rr~r)r!rrr)rs rrrMsk  0 0 1! ?xq I    !    $   <rc|fSr rrs rrr\rrcdddgdgziSrrrs rrras RH |+ ,,rc |d}t|tjr|dkr tdt|tjr|dkr td|dzdt |zdzft j ||z|dzfzfS)Nryrrz!m must be a non-negative integer.rr~r!rrr)rr"broadcast_shapes)rrz_shapebranch_cut_shapergrrys rrrfs H F a)) *q1u<== a)) *q1u<== UAAJN # G%56 7:@1* G IIrc0tj|ddSrrrs rrrsrrralegendre_p(n, z, *, diff_n=0) Legendre polynomial of the first kind. Parameters ---------- n : ArrayLike[int] Degree of the Legendre polynomial. Must have ``n >= 0``. z : ArrayLike[float] Input value. diff_n : Optional[int] A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- p : ndarray or tuple[ndarray] Legendre polynomial with ``diff_n`` derivatives. See Also -------- legendre References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html ct|tjr|dkrtd|dd|cxkrdksnt d|d|Sr)r!rrr)NotImplementedErrorrxs rrrse vw// 0fqj?xq I    ! !   $   Mrc0tj|ddSrrrs rrrrrr alegendre_p_all(n, z, *, diff_n=0) All Legendre polynomials of the first kind up to the specified degree ``n`` and all derivatives up to order ``diff_n``. Output shape is ``(diff_n + 1, n + 1, ...)``. The entry at ``(i, j)`` corresponds to the ``i``-th derivative and degree ``j`` for all ``0 <= i <= diff_n`` and ``0 <= j <= n``. See Also -------- legendre_p cZt|dd}d|cxkrdksntd|d|Sr{rrxs rrrrrcdddgiS)Nrr)rrrrxs rrrs RM ""rcFt|dd}||dzf|z|dzfzfzS)NrFr|rr)rrrgrys rrrs4As51A AE8g%! 57 77rc0tj|ddSrrrs rrrrrr asph_harm_y(n, m, theta, phi, *, diff_n=0) Spherical harmonics. They are defined as .. math:: Y_n^m(\theta,\phi) = \sqrt{\frac{2 n + 1}{4 \pi} \frac{(n - m)!}{(n + m)!}} P_n^m(\cos(\theta)) e^{i m \phi} where :math:`P_n^m` are the (unnormalized) associated Legendre polynomials. Parameters ---------- n : ArrayLike[int] Degree of the harmonic. Must have ``n >= 0``. This is often denoted by ``l`` (lower case L) in descriptions of spherical harmonics. m : ArrayLike[int] Order of the harmonic. theta : ArrayLike[float] Polar (colatitudinal) coordinate; must be in ``[0, pi]``. phi : ArrayLike[float] Azimuthal (longitudinal) coordinate; must be in ``[0, 2*pi]``. diff_n : Optional[int] A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- y : ndarray[complex] or tuple[ndarray[complex]] Spherical harmonics with ``diff_n`` derivatives. Notes ----- There are different conventions for the meanings of the input arguments ``theta`` and ``phi``. In SciPy ``theta`` is the polar angle and ``phi`` is the azimuthal angle. It is common to see the opposite convention, that is, ``theta`` as the azimuthal angle and ``phi`` as the polar angle. Note that SciPy's spherical harmonics include the Condon-Shortley phase [2]_ because it is part of `sph_legendre_p`. With SciPy's conventions, the first several spherical harmonics are .. math:: Y_0^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{1}{\pi}} \\ Y_1^{-1}(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{-i\phi} \sin(\theta) \\ Y_1^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos(\theta) \\ Y_1^1(\theta, \phi) &= -\frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{i\phi} \sin(\theta). References ---------- .. [1] Digital Library of Mathematical Functions, 14.30. https://dlmf.nist.gov/14.30 .. [2] https://en.wikipedia.org/wiki/Spherical_harmonics#Condon.E2.80.93Shortley_phase T)rrycZt|dd}d|cxkrdksntd|d|Sr{rrxs rrr!rrc|jddk(r|dS|jddk(r|d|dddgddgffS|jddk(r$|d|dddgddgf|dddgddggddgddggffSyNrr).rrr~.rrUrs rrr, " 9~ " 9~s3AA#6777 " IC!Q!Q$7 8 q!fq!f%AA'77 8: : rr asph_harm_y_all(n, m, theta, phi, *, diff_n=0) All spherical harmonics up to the specified degree ``n``, order ``m``, and all derivatives up to order ``diff_n``. Returns a tuple of length ``diff_n + 1`` (if ``diff_n > 0``). The first entry corresponds to the spherical harmonics, the second entry (if ``diff_n >= 1``) to the gradient, and the third entry (if ``diff_n >= 2``) to the Hessian matrix. Each entry is an array of shape ``(n + 1, 2 * m + 1, ...)``, where the entry at ``(i, j)`` corresponds to degree ``i`` and order ``j`` for all ``0 <= i <= n`` and ``-m <= j <= m``. See Also -------- sph_harm_y cZt|dd}d|cxkrdksntd|d|S)NryFr|rr~z=diff_n is currently only implemented for orders 2, received: rrrxs rrrPrrcdddgdgziS)Nrr)rrrrrxs rrr[s RH// 00rc |d}t|tjr|dkr td|dzdt |zdzft j ||z|dz|dzfzfS)Nryrrrr~r)rrr phi_shapergrrys rrr`sw H F a)) *q1u<== UAAJN #b&9&9+y&Q Q !VaZ  ! ##rc|jddk(r|dS|jddk(r|d|dddgddgffS|jddk(r$|d|dddgddgf|dddgddggddgddggffSyrrrs rrrkrr)r$rnumpyr"_input_validationr_special_ufuncsrrrr _gufuncsr r r r __all__rrFrrOrJrMrHrrrrs2::;; ttn @G$N&&#'# #*!!"22)3)00D1D**#+#$HE!O(V ../((#)#"E!#*## $ 22344-5-22 I3 I,,#-#8?  F    ""### &..#/#,,8-8 &&#'#=z#1AA H  "" :# : #1'...1/1,,#-#&& :' :r