gZHddlZddlZddlZddlmZddlmZmZm Z m Z ddl m Z m Z mZmZgdZGddZee d d Z e jd Ze jd Zeed d ZejdZejdZejdZejdZeeddddZejdZejdZejdZee ddddZ e jdZe jdZe jdZe jdZe jdZeedd Zejd Zejd!Zee d"d Z e jd#Ze jd$Ze jd%Ze jd&Zee d'd(d)Z e jd*Ze jd+Zeed,d(d)Zejd-Zejd.Zejd/Zejd0ZdS)1N_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 c\eZdZd dddZedZdZdZdZd Z d Z d Z d Z dS) MultiUFuncNF)force_complex_outputc t|tjst|tjjr|}n1t|tjjr|}ntdt}|D]_}t|tjstd|| td|j D`t|dkrtd||_||_||_||_d|_d|_d|_d|_d|_dS)Nz7ufunc_or_ufuncs should be a ufunc or a ufunc collectionz2All ufuncs must have type `numpy.ufunc`. Received c3LK|]}|ddV dS)z->rN)split).0xs P/granary/edev/quantum/lib/python3.11/site-packages/scipy/special/_multiufuncs.py z&MultiUFunc.__init__..+s1.U.UAqwwt}}Q/?.U.U.U.U.U.Urz*All ufuncs must take the same input types.cdS)Nrargskwargss rz%MultiUFunc.__init__..6s2rciSNrrs rrz%MultiUFunc.__init__..7sRr) isinstancenpufunc collectionsabcMappingvaluesIterable ValueErrorsetadd frozensettypeslen_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_ufuncsdocrdefault_kwargs ufuncs_iterseen_input_typesr#s r__init__zMultiUFunc.__init__sw/2844 O/;?+BCC 6-4466 O[_-EFF 6-  "5666 #uu $ W W!%22E$&D2A&D&DEEE $$Y.U.U.U.U.U%U%UVVVV#$$q(( !MNNN / &:#-#' ! #=#= %?%?"""rc|jSr )r0)r8s r__doc__zMultiUFunc.__doc__9s zrc||_dS)z3Set `key` method by decorating a function. N)r5r8funcs r _override_keyzMultiUFunc._override_key=s rc||_dSr )r6rBs r_override_ufunc_default_argsz'MultiUFunc._override_ufunc_default_argsBs#'   rc||_dSr )r7rBs r_override_ufunc_default_kwargsz)MultiUFunc._override_ufunc_default_kwargsEs%)"""rc>|jd|_d|_||_dS)z9Set `resolve_out_shapes` method by decorating a function.Nz2Resolve to output shapes based on relevant inputs.resolve_out_shapes)r@__name__r3rBs r_override_resolve_out_shapesz'MultiUFunc._override_resolve_out_shapesHs) < H L, #'   rc||_dSr )r4rBs r_override_finalize_outz!MultiUFunc._override_finalize_outPs!rc t|jtjr|jS|jdi|}|j|S)z.Resolve to a ufunc based on keyword arguments.r)r!r/r"r#r5)r8r ufunc_keys r_resolve_ufunczMultiUFunc._resolve_ufuncSsI d+RX 6 6 )( (DI'''' $Y//rcR|j|z}||jd i|z }|jd i|}d||j dD}|jd i|}|j,t d|D}|jg|d|j ||jRi|}t d|D}t|dr3||jdzz} | | } | |j d} nDtj |} tj | tj s tj} |j| fz} |jrt d| D} t dt!|| D} | |d<||i|} |j|| } | S) Nc6g|]}tj|Sr)r"asarray)rargs r z'MultiUFunc.__call__..ds CCC#bjooCCCrc3>K|]}tj|VdSr )r"shaper ufunc_args rrz&MultiUFunc.__call__..is,$U$UYRXi%8%8$U$U$U$U$U$Urc3K|]<}t|dr|jn tjt|V=dS)dtypeN)hasattrr\r"typerYs rrz&MultiUFunc.__call__..nsh%B%B)29@ 78S8S&DY__*,(4 ??*C*C%B%B%B%B%B%Brresolve_dtypesr c3@K|]}tjd|VdS)y?N)r" result_type)rufunc_out_dtypes rrz&MultiUFunc.__call__..~sJ)R)R-<*,O)L)L)R)R)R)R)R)Rrc3HK|]\}}tj||VdS))r\N)r"empty)rufunc_out_shaperbs rrz&MultiUFunc.__call__..sQDDpropertyr@rDrFrHrLrNrQrurrrrrs@&+@@@@@@X (((***((("""000/////rrasph_legendre_p(n, m, theta, *, diff_n=0) Spherical Legendre polynomial of the first kind. Parameters ---------- n : ArrayLike[int] Degree of the spherical Legendre polynomial. Must have ``n >= 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_ncnt|dd}d|cxkrdksntd|d|SNrzFstrictrGdiff_n is currently only implemented for orders 0, 1, and 2, received: .rr)rys r_re % @ @ @F     !     $  $ $ $    Mrc.tj|ddSNrr"moveaxisrfs rrr ;sB " ""ra|sph_legendre_p_all(n, m, theta, *, diff_n=0) All spherical Legendre polynomials of the first kind up to the specified degree ``n`` and order ``m``. Output shape is ``(n + 1, 2 * m + 1, ...)``. The entry at ``(j, i)`` corresponds to degree ``j`` and order ``i`` for all ``0 <= j <= n`` and ``-m <= i <= m``. See Also -------- sph_legendre_p cnt|dd}d|cxkrdksntd|d|Sr|rrys rrrrrcddgdgziSNaxesr)rrrrrys rrrs RDJ<' ((rct|tjr|dkrtd|dzdt |zdzf|z|dzfzfS)Nr!n must be a non-negative integer.rr)r!numbersIntegralr)abs)nm theta_shaperirzs rrrs^ a) * *>q1uu<=== UAAJN #k 1VaZM A CCrc.tj|ddSrrrs rrrrraassoc_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)!}} rF branch_cutnormrzcrt|dd}d|cxkrdksntd|d||fSr|rrs rrrsj % @ @ @F     !     $  $ $ $    <rc|fSr rrs rrr( ;rc.tj|ddSrrrs rrr-rraassoc_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`` and order ``m``. Output shape is ``(n + 1, 2 * m + 1, ...)``. The entry at ``(j, i)`` corresponds to degree ``j`` and order ``i`` for all ``0 <= j <= n`` and ``-m <= i <= m``. See Also -------- assoc_legendre_p ct|tjr|dkstd|dd|cxkrdksntd|d||fSNrz1diff_n must be a non-negative integer, received: rrr)r!rrr)rs rrrDs  0 1 1 !  I I I I        !     $  $ $ $    <rc|fSr rrs rrrSrrcdddgdgziSrrrs rrrXs RH |+ ,,rc >|d}t|tjr|dkrtdt|tjr|dkrtd|dzdt |zdzft j||z|dzfzfS)Nrzrrz!m must be a non-negative integer.rrr!rrr)rr"broadcast_shapes)rrz_shapebranch_cut_shaperirrzs rrr]s H F a) * *>q1uu<=== a) * *>q1uu<=== UAAJN # G%566 7:@1* G IIrc.tj|ddSrrrs rrrjrralegendre_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)NotImplementedErrorrys rrrs vw/ 0 0 fqjj I I I I        !    ! $  $ $ $    Mrc.tj|ddSrrrs rrrrralegendre_p_all(n, z, *, diff_n=0) All Legendre polynomials of the first kind up to the specified degree ``n``. Output shape is ``(n + 1, ...)``. The entry at ``j`` corresponds to degree ``j`` for all ``0 <= j <= n``. See Also -------- legendre_p cnt|dd}d|cxkrdksntd|d|Sr|rrys rrrrrcdddgiS)Nrr)rrrrys rrrs RM ""rcNt|dd}||dzf|z|dzfzfzS)NrFr}rr)rrrirzs rrrs:As5111A AE8g%! 57 77rc.tj|ddSrrrs rrrrrasph_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)rrzcnt|dd}d|cxkrdksntd|d|Sr|rrys rrrrrc|jddkr|dS|jddkr|d|dddgddgffS|jddkr-|d|dddgddgf|dddgddggddgddggffSdSNrr).rrr.rrXrs rrr " 9~ " 9~s3AA#6777 " IC!Q!Q$7 8 q!fq!f%AA'77 8: : raXsph_harm_y_all(n, m, theta, phi, *, diff_n=0) All spherical harmonics up to the specified degree ``n`` and order ``m``. Output shape is ``(n + 1, 2 * m + 1, ...)``. The entry at ``(j, i)`` corresponds to degree ``j`` and order ``i`` for all ``0 <= j <= n`` and ``-m <= i <= m``. See Also -------- sph_harm_y cnt|dd}d|cxkrdksntd|d|S)NrzFr}rrz=diff_n is currently only implemented for orders 2, received: rrrys rrr=rrcdddgdgziS)Nrr)rrrrrys rrrHs RH// 00rc |d}t|tjr|dkrtd|dzdt |zdzft j||z|dz|dzfzfS)Nrzrrrrr)rrr phi_shaperirrzs rrrMs H F a) * *>q1uu<=== UAAJN #b&9+y&Q&Q Q !VaZ  ! ##rc|jddkr|dS|jddkr|d|dddgddgffS|jddkr-|d|dddgddgf|dddgddggddgddggffSdSrrrs rrrXrr)r$rnumpyr"_input_validationr_special_ufuncsrrrr _gufuncsr r r r __all__rrDrrNrHrLrFrrrrsl222222::::::::::::;;;;;;;;;;;;   ssssssssl @E###L&##'&# Z $!"!2))32)0DD10D*##+*#:$HE!M'''T ./.(##)(#"z E!$#  $# 2324--54-2 I I32 I,##-,#Z8= D     "###"# ".##/.#,88-,8 &##'&#Z=z#1@@@ F  " : :#" : #1".11/.1,##-,#& : :'& : : :r