o ?HhZH@sVddlZddlZddlZddlmZddlmZmZm Z m Z ddl m Z m Z mZmZgdZGdddZee d dd Z e jd d Ze jd d Zeeddd Zejdd Zejdd Zejdd Zejdd ZeedddddZejdd Zejdd Zejdd Zee dddddZ e jdd Ze jdd Ze jdd Ze jdd Ze jdd Zeed dd Zejd!d Zejd"d Zee d#dd Z e jd$d Ze jd%d Ze jd&d Ze jd'd Zee d(d)dd*Z e jd+d Ze jd,d Zeed-d)dd*Zejd.d Zejd/d Zejd0d Zejd1d ZdS)2N_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 rr rr c@s`eZdZdddddZeddZdd Zd d Zd d ZddZ ddZ ddZ ddZ dS) MultiUFuncNF)force_complex_outputcKst|tjsLt|tjjr|}nt|tjjr|}ntdt }|D]}t|tjs4td|| t dd|j Dq%t |dkrLtd||_||_||_||_d|_d|_d|_dd|_d d|_dS) Nz7ufunc_or_ufuncs should be a ufunc or a ufunc collectionz2All ufuncs must have type `numpy.ufunc`. Received css|] }|ddVqdS)z->rN)split).0xrZ/home/air/sanwanet/gpt-api/venv/lib/python3.10/site-packages/scipy/special/_multiufuncs.py +sz&MultiUFunc.__init__..rz*All ufuncs must take the same input types.c_sdS)Nrrargskwargsrrr6z%MultiUFunc.__init__..c_siSNrrrrrr7r) 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_typesrrrr__init__s0     zMultiUFunc.__init__cCs|jSr)r*)r2rrr__doc__9zMultiUFunc.__doc__cCs ||_dS)z3Set `key` method by decorating a function. N)r/r2funcrrr _override_key=s zMultiUFunc._override_keycC ||_dSr)r0r;rrr_override_ufunc_default_argsB z'MultiUFunc._override_ufunc_default_argscCr>r)r1r;rrr_override_ufunc_default_kwargsEr@z)MultiUFunc._override_ufunc_default_kwargscCs |jdurd|_d|_||_dS)z9Set `resolve_out_shapes` method by decorating a function.Nz2Resolve to output shapes based on relevant inputs.resolve_out_shapes)r9__name__r-r;rrr_override_resolve_out_shapesHs  z'MultiUFunc._override_resolve_out_shapescCr>r)r.r;rrr_override_finalize_outPr@z!MultiUFunc._override_finalize_outcKs.t|jtjr |jS|jdi|}|j|S)z.Resolve to a ufunc based on keyword arguments.Nr)rr)rrr/)r2r ufunc_keyrrr_resolve_ufuncSs zMultiUFunc._resolve_ufuncc Osn|j|B}||jd i|7}|jd i|}dd||j dD}|jd i|}|jdurtdd|D}|jg|d|j ||jRi|}tdd|D}t|drs||jd} | | } | |j d} nt j |} t | t j st j} |j| f} |jrtdd| D} td dt|| D} | |d <||i|} |jdur|| } | S) NcSsg|]}t|qSr)rasarray)rargrrr dsz'MultiUFunc.__call__..css|]}t|VqdSr)rshaper ufunc_argrrrrisz&MultiUFunc.__call__..css.|]}t|dr |jntt|VqdS)dtypeN)hasattrrNrtyperLrrrrns   resolve_dtypesrcss|] }td|VqdS)y?N)r result_type)rufunc_out_dtyperrrr~scss"|] \}}tj||dVqdS))rNN)rempty)rufunc_out_shaperSrrrrsoutr)r,r0rGninr1r-tuplenoutrOrQrrR issubdtypeinexactfloat64r+zipr.) r2rrr ufunc_args ufunc_kwargsufunc_arg_shapesufunc_out_shapesufunc_arg_dtypes ufunc_dtypesufunc_out_dtypesrSrVrrr__call__\sJ        zMultiUFunc.__call__r) rC __module__ __qualname__r8propertyr9r=r?rArDrErGrerrrrr s   r asph_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_ncC8t|ddd}d|krdksntd|d|SNrjFstrictrGdiff_n is currently only implemented for orders 0, 1, and 2, received: .rr#rirrr_rscCt|ddSNrrmoveaxisrVrrrrsa|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 cCrkrlrrrirrrrsrtcCsddgdgiSNaxesr)rrrwrrirrrrsscCsDt|tjr |dkrtd|ddt|df||dffS)Nr!n must be a non-negative integer.rro)rnumbersIntegralr#abs)nm theta_shaperYrjrrrrss(cCrurvrxrzrrrrsr{aassoc_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)!}} roF branch_cutnormrjcCs<t|ddd}d|krdksntd|d||fSrlrrrrrrrsscC|fSrrrrrrrs(r:cCrurvrxrzrrrrs-r{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`` 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 cCsRt|tjr |dkstd|dd|krdks%ntd|d||fSNrz1diff_n must be a non-negative integer, received: rqrorp)rrrr#rrrrrsDs  cCrrrrrrrrsSr:cCdddgdgiSr|rrrrrrsXcKsp|d}t|tjr|dkrtdt|tjr|dkr td|ddt|dft|||dffS)Nrjrr~z!m must be a non-negative integer.rrorrrr#rrbroadcast_shapes)rrz_shapebranch_cut_shaperYrrjrrrrs]s cCrurvrxrzrrrrsjr{alegendre_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 cCsNt|tjr |dkrtd|dd|krdks%ntd|d|Sr)rrrr#NotImplementedErrorrirrrrss cCrurvrxrzrrrrsr{alegendre_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 cCrkrlrrrirrrrsrtcCs dddgiS)Nr}r)rrwrrirrrrss cCs,t|ddd}||df||dffS)NrFrmrr)rrrYrjrrrrsscCrurvrxrzrrrrsr{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)rrjcCrkrlrrrirrrrsrtcC|jddkr |dS|jddkr!|d|dddgddgffS|jddkrI|d|dddgddgf|dddgddggddgddggffSdSNrwr).rrro.rrKrzrrrrs$aXsph_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 cCrk)NrjFrmrroz=diff_n is currently only implemented for orders 2, received: rqrrrirrrrs=rtcCr)Nr}r)rrrwrrirrrrsHrcKsZ|d}t|tjr|dkrtd|ddt|dft|||d|dffS)Nrjrr~rror)rrr phi_shaperYrrjrrrrsMs "cCrrrrzrrrrsXr)rrnumpyr_input_validationr_special_ufuncsrrrr_gufuncsr r r r __all__r r=rsrErArDr?rrrrs  v &       $*         "       =C