895 lines
32 KiB
Python
895 lines
32 KiB
Python
|
"""
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========================================
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Special functions (:mod:`scipy.special`)
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========================================
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.. currentmodule:: scipy.special
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Almost all of the functions below accept NumPy arrays as input
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arguments as well as single numbers. This means they follow
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broadcasting and automatic array-looping rules. Technically,
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they are `NumPy universal functions
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<https://numpy.org/doc/stable/user/basics.ufuncs.html#ufuncs-basics>`_.
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Functions which do not accept NumPy arrays are marked by a warning
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in the section description.
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.. seealso::
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`scipy.special.cython_special` -- Typed Cython versions of special functions
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Error handling
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==============
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Errors are handled by returning NaNs or other appropriate values.
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Some of the special function routines can emit warnings or raise
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exceptions when an error occurs. By default this is disabled; to
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query and control the current error handling state the following
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functions are provided.
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.. autosummary::
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:toctree: generated/
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geterr -- Get the current way of handling special-function errors.
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seterr -- Set how special-function errors are handled.
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errstate -- Context manager for special-function error handling.
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SpecialFunctionWarning -- Warning that can be emitted by special functions.
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SpecialFunctionError -- Exception that can be raised by special functions.
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Available functions
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===================
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Airy functions
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--------------
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.. autosummary::
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:toctree: generated/
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airy -- Airy functions and their derivatives.
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airye -- Exponentially scaled Airy functions and their derivatives.
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ai_zeros -- Compute `nt` zeros and values of the Airy function Ai and its derivative.
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bi_zeros -- Compute `nt` zeros and values of the Airy function Bi and its derivative.
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itairy -- Integrals of Airy functions
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Elliptic functions and integrals
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--------------------------------
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.. autosummary::
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:toctree: generated/
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ellipj -- Jacobian elliptic functions.
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ellipk -- Complete elliptic integral of the first kind.
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ellipkm1 -- Complete elliptic integral of the first kind around `m` = 1.
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ellipkinc -- Incomplete elliptic integral of the first kind.
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ellipe -- Complete elliptic integral of the second kind.
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ellipeinc -- Incomplete elliptic integral of the second kind.
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elliprc -- Degenerate symmetric integral RC.
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elliprd -- Symmetric elliptic integral of the second kind.
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elliprf -- Completely-symmetric elliptic integral of the first kind.
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elliprg -- Completely-symmetric elliptic integral of the second kind.
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elliprj -- Symmetric elliptic integral of the third kind.
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Bessel functions
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----------------
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.. autosummary::
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:toctree: generated/
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jv -- Bessel function of the first kind of real order and \
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complex argument.
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jve -- Exponentially scaled Bessel function of order `v`.
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yn -- Bessel function of the second kind of integer order and \
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real argument.
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yv -- Bessel function of the second kind of real order and \
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complex argument.
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yve -- Exponentially scaled Bessel function of the second kind \
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of real order.
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kn -- Modified Bessel function of the second kind of integer \
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order `n`
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kv -- Modified Bessel function of the second kind of real order \
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`v`
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kve -- Exponentially scaled modified Bessel function of the \
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second kind.
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iv -- Modified Bessel function of the first kind of real order.
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ive -- Exponentially scaled modified Bessel function of the \
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first kind.
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hankel1 -- Hankel function of the first kind.
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hankel1e -- Exponentially scaled Hankel function of the first kind.
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hankel2 -- Hankel function of the second kind.
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hankel2e -- Exponentially scaled Hankel function of the second kind.
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wright_bessel -- Wright's generalized Bessel function.
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log_wright_bessel -- Logarithm of Wright's generalized Bessel function.
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The following function does not accept NumPy arrays (it is not a
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universal function):
|
||
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.. autosummary::
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||
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:toctree: generated/
|
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lmbda -- Jahnke-Emden Lambda function, Lambdav(x).
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Zeros of Bessel functions
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^^^^^^^^^^^^^^^^^^^^^^^^^
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The following functions do not accept NumPy arrays (they are not
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||
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universal functions):
|
||
|
|
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.. autosummary::
|
||
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:toctree: generated/
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jnjnp_zeros -- Compute zeros of integer-order Bessel functions Jn and Jn'.
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jnyn_zeros -- Compute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x).
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jn_zeros -- Compute zeros of integer-order Bessel function Jn(x).
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jnp_zeros -- Compute zeros of integer-order Bessel function derivative Jn'(x).
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yn_zeros -- Compute zeros of integer-order Bessel function Yn(x).
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ynp_zeros -- Compute zeros of integer-order Bessel function derivative Yn'(x).
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y0_zeros -- Compute nt zeros of Bessel function Y0(z), and derivative at each zero.
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y1_zeros -- Compute nt zeros of Bessel function Y1(z), and derivative at each zero.
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y1p_zeros -- Compute nt zeros of Bessel derivative Y1'(z), and value at each zero.
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Faster versions of common Bessel functions
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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.. autosummary::
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||
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:toctree: generated/
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j0 -- Bessel function of the first kind of order 0.
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j1 -- Bessel function of the first kind of order 1.
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y0 -- Bessel function of the second kind of order 0.
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y1 -- Bessel function of the second kind of order 1.
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i0 -- Modified Bessel function of order 0.
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i0e -- Exponentially scaled modified Bessel function of order 0.
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i1 -- Modified Bessel function of order 1.
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i1e -- Exponentially scaled modified Bessel function of order 1.
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k0 -- Modified Bessel function of the second kind of order 0, :math:`K_0`.
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k0e -- Exponentially scaled modified Bessel function K of order 0
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k1 -- Modified Bessel function of the second kind of order 1, :math:`K_1(x)`.
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k1e -- Exponentially scaled modified Bessel function K of order 1.
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||
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Integrals of Bessel functions
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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.. autosummary::
|
||
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:toctree: generated/
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itj0y0 -- Integrals of Bessel functions of order 0.
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it2j0y0 -- Integrals related to Bessel functions of order 0.
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iti0k0 -- Integrals of modified Bessel functions of order 0.
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it2i0k0 -- Integrals related to modified Bessel functions of order 0.
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besselpoly -- Weighted integral of a Bessel function.
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Derivatives of Bessel functions
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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.. autosummary::
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:toctree: generated/
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jvp -- Compute nth derivative of Bessel function Jv(z) with respect to `z`.
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yvp -- Compute nth derivative of Bessel function Yv(z) with respect to `z`.
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kvp -- Compute nth derivative of real-order modified Bessel function Kv(z)
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ivp -- Compute nth derivative of modified Bessel function Iv(z) with respect to `z`.
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h1vp -- Compute nth derivative of Hankel function H1v(z) with respect to `z`.
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h2vp -- Compute nth derivative of Hankel function H2v(z) with respect to `z`.
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Spherical Bessel functions
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^^^^^^^^^^^^^^^^^^^^^^^^^^
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.. autosummary::
|
||
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:toctree: generated/
|
||
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spherical_jn -- Spherical Bessel function of the first kind or its derivative.
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||
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spherical_yn -- Spherical Bessel function of the second kind or its derivative.
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||
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spherical_in -- Modified spherical Bessel function of the first kind or its derivative.
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||
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spherical_kn -- Modified spherical Bessel function of the second kind or its derivative.
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||
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Riccati-Bessel functions
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^^^^^^^^^^^^^^^^^^^^^^^^
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The following functions do not accept NumPy arrays (they are not
|
||
|
|
universal functions):
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
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||
|
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riccati_jn -- Compute Ricatti-Bessel function of the first kind and its derivative.
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riccati_yn -- Compute Ricatti-Bessel function of the second kind and its derivative.
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Struve functions
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----------------
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.. autosummary::
|
||
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:toctree: generated/
|
||
|
|
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struve -- Struve function.
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modstruve -- Modified Struve function.
|
||
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itstruve0 -- Integral of the Struve function of order 0.
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it2struve0 -- Integral related to the Struve function of order 0.
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itmodstruve0 -- Integral of the modified Struve function of order 0.
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Raw statistical functions
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-------------------------
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.. seealso:: :mod:`scipy.stats`: Friendly versions of these functions.
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||
|
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Binomial distribution
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^^^^^^^^^^^^^^^^^^^^^
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||
|
|
|
||
|
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.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
bdtr -- Binomial distribution cumulative distribution function.
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bdtrc -- Binomial distribution survival function.
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||
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|
bdtri -- Inverse function to `bdtr` with respect to `p`.
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bdtrik -- Inverse function to `bdtr` with respect to `k`.
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|
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bdtrin -- Inverse function to `bdtr` with respect to `n`.
|
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|
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|
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Beta distribution
|
||
|
|
^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
btdtr -- Cumulative distribution function of the beta distribution.
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||
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btdtri -- The `p`-th quantile of the beta distribution.
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btdtria -- Inverse of `btdtr` with respect to `a`.
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||
|
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btdtrib -- btdtria(a, p, x).
|
||
|
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F distribution
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^^^^^^^^^^^^^^
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||
|
|
|
||
|
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.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
fdtr -- F cumulative distribution function.
|
||
|
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fdtrc -- F survival function.
|
||
|
|
fdtri -- The `p`-th quantile of the F-distribution.
|
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|
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fdtridfd -- Inverse to `fdtr` vs dfd.
|
||
|
|
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||
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Gamma distribution
|
||
|
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^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
gdtr -- Gamma distribution cumulative distribution function.
|
||
|
|
gdtrc -- Gamma distribution survival function.
|
||
|
|
gdtria -- Inverse of `gdtr` vs a.
|
||
|
|
gdtrib -- Inverse of `gdtr` vs b.
|
||
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|
gdtrix -- Inverse of `gdtr` vs x.
|
||
|
|
|
||
|
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Negative binomial distribution
|
||
|
|
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
nbdtr -- Negative binomial cumulative distribution function.
|
||
|
|
nbdtrc -- Negative binomial survival function.
|
||
|
|
nbdtri -- Inverse of `nbdtr` vs `p`.
|
||
|
|
nbdtrik -- Inverse of `nbdtr` vs `k`.
|
||
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|
nbdtrin -- Inverse of `nbdtr` vs `n`.
|
||
|
|
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||
|
|
Noncentral F distribution
|
||
|
|
^^^^^^^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
ncfdtr -- Cumulative distribution function of the non-central F distribution.
|
||
|
|
ncfdtridfd -- Calculate degrees of freedom (denominator) for the noncentral F-distribution.
|
||
|
|
ncfdtridfn -- Calculate degrees of freedom (numerator) for the noncentral F-distribution.
|
||
|
|
ncfdtri -- Inverse cumulative distribution function of the non-central F distribution.
|
||
|
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ncfdtrinc -- Calculate non-centrality parameter for non-central F distribution.
|
||
|
|
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|
Noncentral t distribution
|
||
|
|
^^^^^^^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
nctdtr -- Cumulative distribution function of the non-central `t` distribution.
|
||
|
|
nctdtridf -- Calculate degrees of freedom for non-central t distribution.
|
||
|
|
nctdtrit -- Inverse cumulative distribution function of the non-central t distribution.
|
||
|
|
nctdtrinc -- Calculate non-centrality parameter for non-central t distribution.
|
||
|
|
|
||
|
|
Normal distribution
|
||
|
|
^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
nrdtrimn -- Calculate mean of normal distribution given other params.
|
||
|
|
nrdtrisd -- Calculate standard deviation of normal distribution given other params.
|
||
|
|
ndtr -- Normal cumulative distribution function.
|
||
|
|
log_ndtr -- Logarithm of normal cumulative distribution function.
|
||
|
|
ndtri -- Inverse of `ndtr` vs x.
|
||
|
|
ndtri_exp -- Inverse of `log_ndtr` vs x.
|
||
|
|
|
||
|
|
Poisson distribution
|
||
|
|
^^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
pdtr -- Poisson cumulative distribution function.
|
||
|
|
pdtrc -- Poisson survival function.
|
||
|
|
pdtri -- Inverse to `pdtr` vs m.
|
||
|
|
pdtrik -- Inverse to `pdtr` vs k.
|
||
|
|
|
||
|
|
Student t distribution
|
||
|
|
^^^^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
stdtr -- Student t distribution cumulative distribution function.
|
||
|
|
stdtridf -- Inverse of `stdtr` vs df.
|
||
|
|
stdtrit -- Inverse of `stdtr` vs `t`.
|
||
|
|
|
||
|
|
Chi square distribution
|
||
|
|
^^^^^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
chdtr -- Chi square cumulative distribution function.
|
||
|
|
chdtrc -- Chi square survival function.
|
||
|
|
chdtri -- Inverse to `chdtrc`.
|
||
|
|
chdtriv -- Inverse to `chdtr` vs `v`.
|
||
|
|
|
||
|
|
Non-central chi square distribution
|
||
|
|
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
chndtr -- Non-central chi square cumulative distribution function.
|
||
|
|
chndtridf -- Inverse to `chndtr` vs `df`.
|
||
|
|
chndtrinc -- Inverse to `chndtr` vs `nc`.
|
||
|
|
chndtrix -- Inverse to `chndtr` vs `x`.
|
||
|
|
|
||
|
|
Kolmogorov distribution
|
||
|
|
^^^^^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
smirnov -- Kolmogorov-Smirnov complementary cumulative distribution function.
|
||
|
|
smirnovi -- Inverse to `smirnov`.
|
||
|
|
kolmogorov -- Complementary cumulative distribution function of Kolmogorov distribution.
|
||
|
|
kolmogi -- Inverse function to `kolmogorov`.
|
||
|
|
|
||
|
|
Box-Cox transformation
|
||
|
|
^^^^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
boxcox -- Compute the Box-Cox transformation.
|
||
|
|
boxcox1p -- Compute the Box-Cox transformation of 1 + `x`.
|
||
|
|
inv_boxcox -- Compute the inverse of the Box-Cox transformation.
|
||
|
|
inv_boxcox1p -- Compute the inverse of the Box-Cox transformation.
|
||
|
|
|
||
|
|
|
||
|
|
Sigmoidal functions
|
||
|
|
^^^^^^^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
logit -- Logit ufunc for ndarrays.
|
||
|
|
expit -- Logistic sigmoid function.
|
||
|
|
log_expit -- Logarithm of the logistic sigmoid function.
|
||
|
|
|
||
|
|
Miscellaneous
|
||
|
|
^^^^^^^^^^^^^
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
tklmbda -- Tukey-Lambda cumulative distribution function.
|
||
|
|
owens_t -- Owen's T Function.
|
||
|
|
|
||
|
|
|
||
|
|
Information Theory functions
|
||
|
|
----------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
entr -- Elementwise function for computing entropy.
|
||
|
|
rel_entr -- Elementwise function for computing relative entropy.
|
||
|
|
kl_div -- Elementwise function for computing Kullback-Leibler divergence.
|
||
|
|
huber -- Huber loss function.
|
||
|
|
pseudo_huber -- Pseudo-Huber loss function.
|
||
|
|
|
||
|
|
|
||
|
|
Gamma and related functions
|
||
|
|
---------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
gamma -- Gamma function.
|
||
|
|
gammaln -- Logarithm of the absolute value of the Gamma function for real inputs.
|
||
|
|
loggamma -- Principal branch of the logarithm of the Gamma function.
|
||
|
|
gammasgn -- Sign of the gamma function.
|
||
|
|
gammainc -- Regularized lower incomplete gamma function.
|
||
|
|
gammaincinv -- Inverse to `gammainc`.
|
||
|
|
gammaincc -- Regularized upper incomplete gamma function.
|
||
|
|
gammainccinv -- Inverse to `gammaincc`.
|
||
|
|
beta -- Beta function.
|
||
|
|
betaln -- Natural logarithm of absolute value of beta function.
|
||
|
|
betainc -- Incomplete beta integral.
|
||
|
|
betaincc -- Complemented incomplete beta integral.
|
||
|
|
betaincinv -- Inverse function to beta integral.
|
||
|
|
betainccinv -- Inverse of the complemented incomplete beta integral.
|
||
|
|
psi -- The digamma function.
|
||
|
|
rgamma -- Gamma function inverted.
|
||
|
|
polygamma -- Polygamma function n.
|
||
|
|
multigammaln -- Returns the log of multivariate gamma, also sometimes called the generalized gamma.
|
||
|
|
digamma -- psi(x[, out]).
|
||
|
|
poch -- Rising factorial (z)_m.
|
||
|
|
|
||
|
|
|
||
|
|
Error function and Fresnel integrals
|
||
|
|
------------------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
erf -- Returns the error function of complex argument.
|
||
|
|
erfc -- Complementary error function, ``1 - erf(x)``.
|
||
|
|
erfcx -- Scaled complementary error function, ``exp(x**2) * erfc(x)``.
|
||
|
|
erfi -- Imaginary error function, ``-i erf(i z)``.
|
||
|
|
erfinv -- Inverse function for erf.
|
||
|
|
erfcinv -- Inverse function for erfc.
|
||
|
|
wofz -- Faddeeva function.
|
||
|
|
dawsn -- Dawson's integral.
|
||
|
|
fresnel -- Fresnel sin and cos integrals.
|
||
|
|
fresnel_zeros -- Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z).
|
||
|
|
modfresnelp -- Modified Fresnel positive integrals.
|
||
|
|
modfresnelm -- Modified Fresnel negative integrals.
|
||
|
|
voigt_profile -- Voigt profile.
|
||
|
|
|
||
|
|
The following functions do not accept NumPy arrays (they are not
|
||
|
|
universal functions):
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
erf_zeros -- Compute nt complex zeros of error function erf(z).
|
||
|
|
fresnelc_zeros -- Compute nt complex zeros of cosine Fresnel integral C(z).
|
||
|
|
fresnels_zeros -- Compute nt complex zeros of sine Fresnel integral S(z).
|
||
|
|
|
||
|
|
Legendre functions
|
||
|
|
------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
lpmv -- Associated Legendre function of integer order and real degree.
|
||
|
|
sph_harm -- Compute spherical harmonics.
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
clpmn -- Associated Legendre function of the first kind for complex arguments.
|
||
|
|
lpn -- Legendre function of the first kind.
|
||
|
|
lqn -- Legendre function of the second kind.
|
||
|
|
lpmn -- Sequence of associated Legendre functions of the first kind.
|
||
|
|
lqmn -- Sequence of associated Legendre functions of the second kind.
|
||
|
|
|
||
|
|
Ellipsoidal harmonics
|
||
|
|
---------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
ellip_harm -- Ellipsoidal harmonic functions E^p_n(l).
|
||
|
|
ellip_harm_2 -- Ellipsoidal harmonic functions F^p_n(l).
|
||
|
|
ellip_normal -- Ellipsoidal harmonic normalization constants gamma^p_n.
|
||
|
|
|
||
|
|
Orthogonal polynomials
|
||
|
|
----------------------
|
||
|
|
|
||
|
|
The following functions evaluate values of orthogonal polynomials:
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
assoc_laguerre -- Compute the generalized (associated) Laguerre polynomial of degree n and order k.
|
||
|
|
eval_legendre -- Evaluate Legendre polynomial at a point.
|
||
|
|
eval_chebyt -- Evaluate Chebyshev polynomial of the first kind at a point.
|
||
|
|
eval_chebyu -- Evaluate Chebyshev polynomial of the second kind at a point.
|
||
|
|
eval_chebyc -- Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a point.
|
||
|
|
eval_chebys -- Evaluate Chebyshev polynomial of the second kind on [-2, 2] at a point.
|
||
|
|
eval_jacobi -- Evaluate Jacobi polynomial at a point.
|
||
|
|
eval_laguerre -- Evaluate Laguerre polynomial at a point.
|
||
|
|
eval_genlaguerre -- Evaluate generalized Laguerre polynomial at a point.
|
||
|
|
eval_hermite -- Evaluate physicist's Hermite polynomial at a point.
|
||
|
|
eval_hermitenorm -- Evaluate probabilist's (normalized) Hermite polynomial at a point.
|
||
|
|
eval_gegenbauer -- Evaluate Gegenbauer polynomial at a point.
|
||
|
|
eval_sh_legendre -- Evaluate shifted Legendre polynomial at a point.
|
||
|
|
eval_sh_chebyt -- Evaluate shifted Chebyshev polynomial of the first kind at a point.
|
||
|
|
eval_sh_chebyu -- Evaluate shifted Chebyshev polynomial of the second kind at a point.
|
||
|
|
eval_sh_jacobi -- Evaluate shifted Jacobi polynomial at a point.
|
||
|
|
|
||
|
|
The following functions compute roots and quadrature weights for
|
||
|
|
orthogonal polynomials:
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
roots_legendre -- Gauss-Legendre quadrature.
|
||
|
|
roots_chebyt -- Gauss-Chebyshev (first kind) quadrature.
|
||
|
|
roots_chebyu -- Gauss-Chebyshev (second kind) quadrature.
|
||
|
|
roots_chebyc -- Gauss-Chebyshev (first kind) quadrature.
|
||
|
|
roots_chebys -- Gauss-Chebyshev (second kind) quadrature.
|
||
|
|
roots_jacobi -- Gauss-Jacobi quadrature.
|
||
|
|
roots_laguerre -- Gauss-Laguerre quadrature.
|
||
|
|
roots_genlaguerre -- Gauss-generalized Laguerre quadrature.
|
||
|
|
roots_hermite -- Gauss-Hermite (physicst's) quadrature.
|
||
|
|
roots_hermitenorm -- Gauss-Hermite (statistician's) quadrature.
|
||
|
|
roots_gegenbauer -- Gauss-Gegenbauer quadrature.
|
||
|
|
roots_sh_legendre -- Gauss-Legendre (shifted) quadrature.
|
||
|
|
roots_sh_chebyt -- Gauss-Chebyshev (first kind, shifted) quadrature.
|
||
|
|
roots_sh_chebyu -- Gauss-Chebyshev (second kind, shifted) quadrature.
|
||
|
|
roots_sh_jacobi -- Gauss-Jacobi (shifted) quadrature.
|
||
|
|
|
||
|
|
The functions below, in turn, return the polynomial coefficients in
|
||
|
|
``orthopoly1d`` objects, which function similarly as `numpy.poly1d`.
|
||
|
|
The ``orthopoly1d`` class also has an attribute ``weights``, which returns
|
||
|
|
the roots, weights, and total weights for the appropriate form of Gaussian
|
||
|
|
quadrature. These are returned in an ``n x 3`` array with roots in the first
|
||
|
|
column, weights in the second column, and total weights in the final column.
|
||
|
|
Note that ``orthopoly1d`` objects are converted to `~numpy.poly1d` when doing
|
||
|
|
arithmetic, and lose information of the original orthogonal polynomial.
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
legendre -- Legendre polynomial.
|
||
|
|
chebyt -- Chebyshev polynomial of the first kind.
|
||
|
|
chebyu -- Chebyshev polynomial of the second kind.
|
||
|
|
chebyc -- Chebyshev polynomial of the first kind on :math:`[-2, 2]`.
|
||
|
|
chebys -- Chebyshev polynomial of the second kind on :math:`[-2, 2]`.
|
||
|
|
jacobi -- Jacobi polynomial.
|
||
|
|
laguerre -- Laguerre polynomial.
|
||
|
|
genlaguerre -- Generalized (associated) Laguerre polynomial.
|
||
|
|
hermite -- Physicist's Hermite polynomial.
|
||
|
|
hermitenorm -- Normalized (probabilist's) Hermite polynomial.
|
||
|
|
gegenbauer -- Gegenbauer (ultraspherical) polynomial.
|
||
|
|
sh_legendre -- Shifted Legendre polynomial.
|
||
|
|
sh_chebyt -- Shifted Chebyshev polynomial of the first kind.
|
||
|
|
sh_chebyu -- Shifted Chebyshev polynomial of the second kind.
|
||
|
|
sh_jacobi -- Shifted Jacobi polynomial.
|
||
|
|
|
||
|
|
.. warning::
|
||
|
|
|
||
|
|
Computing values of high-order polynomials (around ``order > 20``) using
|
||
|
|
polynomial coefficients is numerically unstable. To evaluate polynomial
|
||
|
|
values, the ``eval_*`` functions should be used instead.
|
||
|
|
|
||
|
|
|
||
|
|
Hypergeometric functions
|
||
|
|
------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
hyp2f1 -- Gauss hypergeometric function 2F1(a, b; c; z).
|
||
|
|
hyp1f1 -- Confluent hypergeometric function 1F1(a, b; x).
|
||
|
|
hyperu -- Confluent hypergeometric function U(a, b, x) of the second kind.
|
||
|
|
hyp0f1 -- Confluent hypergeometric limit function 0F1.
|
||
|
|
|
||
|
|
|
||
|
|
Parabolic cylinder functions
|
||
|
|
----------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
pbdv -- Parabolic cylinder function D.
|
||
|
|
pbvv -- Parabolic cylinder function V.
|
||
|
|
pbwa -- Parabolic cylinder function W.
|
||
|
|
|
||
|
|
The following functions do not accept NumPy arrays (they are not
|
||
|
|
universal functions):
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
pbdv_seq -- Parabolic cylinder functions Dv(x) and derivatives.
|
||
|
|
pbvv_seq -- Parabolic cylinder functions Vv(x) and derivatives.
|
||
|
|
pbdn_seq -- Parabolic cylinder functions Dn(z) and derivatives.
|
||
|
|
|
||
|
|
Mathieu and related functions
|
||
|
|
-----------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
mathieu_a -- Characteristic value of even Mathieu functions.
|
||
|
|
mathieu_b -- Characteristic value of odd Mathieu functions.
|
||
|
|
|
||
|
|
The following functions do not accept NumPy arrays (they are not
|
||
|
|
universal functions):
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
mathieu_even_coef -- Fourier coefficients for even Mathieu and modified Mathieu functions.
|
||
|
|
mathieu_odd_coef -- Fourier coefficients for even Mathieu and modified Mathieu functions.
|
||
|
|
|
||
|
|
The following return both function and first derivative:
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
mathieu_cem -- Even Mathieu function and its derivative.
|
||
|
|
mathieu_sem -- Odd Mathieu function and its derivative.
|
||
|
|
mathieu_modcem1 -- Even modified Mathieu function of the first kind and its derivative.
|
||
|
|
mathieu_modcem2 -- Even modified Mathieu function of the second kind and its derivative.
|
||
|
|
mathieu_modsem1 -- Odd modified Mathieu function of the first kind and its derivative.
|
||
|
|
mathieu_modsem2 -- Odd modified Mathieu function of the second kind and its derivative.
|
||
|
|
|
||
|
|
Spheroidal wave functions
|
||
|
|
-------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
pro_ang1 -- Prolate spheroidal angular function of the first kind and its derivative.
|
||
|
|
pro_rad1 -- Prolate spheroidal radial function of the first kind and its derivative.
|
||
|
|
pro_rad2 -- Prolate spheroidal radial function of the second kind and its derivative.
|
||
|
|
obl_ang1 -- Oblate spheroidal angular function of the first kind and its derivative.
|
||
|
|
obl_rad1 -- Oblate spheroidal radial function of the first kind and its derivative.
|
||
|
|
obl_rad2 -- Oblate spheroidal radial function of the second kind and its derivative.
|
||
|
|
pro_cv -- Characteristic value of prolate spheroidal function.
|
||
|
|
obl_cv -- Characteristic value of oblate spheroidal function.
|
||
|
|
pro_cv_seq -- Characteristic values for prolate spheroidal wave functions.
|
||
|
|
obl_cv_seq -- Characteristic values for oblate spheroidal wave functions.
|
||
|
|
|
||
|
|
The following functions require pre-computed characteristic value:
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
pro_ang1_cv -- Prolate spheroidal angular function pro_ang1 for precomputed characteristic value.
|
||
|
|
pro_rad1_cv -- Prolate spheroidal radial function pro_rad1 for precomputed characteristic value.
|
||
|
|
pro_rad2_cv -- Prolate spheroidal radial function pro_rad2 for precomputed characteristic value.
|
||
|
|
obl_ang1_cv -- Oblate spheroidal angular function obl_ang1 for precomputed characteristic value.
|
||
|
|
obl_rad1_cv -- Oblate spheroidal radial function obl_rad1 for precomputed characteristic value.
|
||
|
|
obl_rad2_cv -- Oblate spheroidal radial function obl_rad2 for precomputed characteristic value.
|
||
|
|
|
||
|
|
Kelvin functions
|
||
|
|
----------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
kelvin -- Kelvin functions as complex numbers.
|
||
|
|
kelvin_zeros -- Compute nt zeros of all Kelvin functions.
|
||
|
|
ber -- Kelvin function ber.
|
||
|
|
bei -- Kelvin function bei
|
||
|
|
berp -- Derivative of the Kelvin function `ber`.
|
||
|
|
beip -- Derivative of the Kelvin function `bei`.
|
||
|
|
ker -- Kelvin function ker.
|
||
|
|
kei -- Kelvin function ker.
|
||
|
|
kerp -- Derivative of the Kelvin function ker.
|
||
|
|
keip -- Derivative of the Kelvin function kei.
|
||
|
|
|
||
|
|
The following functions do not accept NumPy arrays (they are not
|
||
|
|
universal functions):
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
ber_zeros -- Compute nt zeros of the Kelvin function ber(x).
|
||
|
|
bei_zeros -- Compute nt zeros of the Kelvin function bei(x).
|
||
|
|
berp_zeros -- Compute nt zeros of the Kelvin function ber'(x).
|
||
|
|
beip_zeros -- Compute nt zeros of the Kelvin function bei'(x).
|
||
|
|
ker_zeros -- Compute nt zeros of the Kelvin function ker(x).
|
||
|
|
kei_zeros -- Compute nt zeros of the Kelvin function kei(x).
|
||
|
|
kerp_zeros -- Compute nt zeros of the Kelvin function ker'(x).
|
||
|
|
keip_zeros -- Compute nt zeros of the Kelvin function kei'(x).
|
||
|
|
|
||
|
|
Combinatorics
|
||
|
|
-------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
comb -- The number of combinations of N things taken k at a time.
|
||
|
|
perm -- Permutations of N things taken k at a time, i.e., k-permutations of N.
|
||
|
|
stirling2 -- Stirling numbers of the second kind.
|
||
|
|
|
||
|
|
Lambert W and related functions
|
||
|
|
-------------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
lambertw -- Lambert W function.
|
||
|
|
wrightomega -- Wright Omega function.
|
||
|
|
|
||
|
|
Other special functions
|
||
|
|
-----------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
agm -- Arithmetic, Geometric Mean.
|
||
|
|
bernoulli -- Bernoulli numbers B0..Bn (inclusive).
|
||
|
|
binom -- Binomial coefficient
|
||
|
|
diric -- Periodic sinc function, also called the Dirichlet function.
|
||
|
|
euler -- Euler numbers E0..En (inclusive).
|
||
|
|
expn -- Exponential integral E_n.
|
||
|
|
exp1 -- Exponential integral E_1 of complex argument z.
|
||
|
|
expi -- Exponential integral Ei.
|
||
|
|
factorial -- The factorial of a number or array of numbers.
|
||
|
|
factorial2 -- Double factorial.
|
||
|
|
factorialk -- Multifactorial of n of order k, n(!!...!).
|
||
|
|
shichi -- Hyperbolic sine and cosine integrals.
|
||
|
|
sici -- Sine and cosine integrals.
|
||
|
|
softmax -- Softmax function.
|
||
|
|
log_softmax -- Logarithm of softmax function.
|
||
|
|
spence -- Spence's function, also known as the dilogarithm.
|
||
|
|
zeta -- Riemann zeta function.
|
||
|
|
zetac -- Riemann zeta function minus 1.
|
||
|
|
|
||
|
|
Convenience functions
|
||
|
|
---------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
cbrt -- Cube root of `x`.
|
||
|
|
exp10 -- 10**x.
|
||
|
|
exp2 -- 2**x.
|
||
|
|
radian -- Convert from degrees to radians.
|
||
|
|
cosdg -- Cosine of the angle `x` given in degrees.
|
||
|
|
sindg -- Sine of angle given in degrees.
|
||
|
|
tandg -- Tangent of angle x given in degrees.
|
||
|
|
cotdg -- Cotangent of the angle `x` given in degrees.
|
||
|
|
log1p -- Calculates log(1+x) for use when `x` is near zero.
|
||
|
|
expm1 -- ``exp(x) - 1`` for use when `x` is near zero.
|
||
|
|
cosm1 -- ``cos(x) - 1`` for use when `x` is near zero.
|
||
|
|
powm1 -- ``x**y - 1`` for use when `y` is near zero or `x` is near 1.
|
||
|
|
round -- Round to nearest integer.
|
||
|
|
xlogy -- Compute ``x*log(y)`` so that the result is 0 if ``x = 0``.
|
||
|
|
xlog1py -- Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``.
|
||
|
|
logsumexp -- Compute the log of the sum of exponentials of input elements.
|
||
|
|
exprel -- Relative error exponential, (exp(x)-1)/x, for use when `x` is near zero.
|
||
|
|
sinc -- Return the sinc function.
|
||
|
|
|
||
|
|
""" # noqa: E501
|
||
|
|
|
||
|
|
import os
|
||
|
|
import warnings
|
||
|
|
|
||
|
|
|
||
|
|
def _load_libsf_error_state():
|
||
|
|
"""Load libsf_error_state.dll shared library on Windows
|
||
|
|
|
||
|
|
libsf_error_state manages shared state used by
|
||
|
|
``scipy.special.seterr`` and ``scipy.special.geterr`` so that these
|
||
|
|
can work consistently between special functions provided by different
|
||
|
|
extension modules. This shared library is installed in scipy/special
|
||
|
|
alongside this __init__.py file. Due to lack of rpath support, Windows
|
||
|
|
cannot find shared libraries installed within wheels. To circumvent this,
|
||
|
|
we pre-load ``lib_sf_error_state.dll`` when on Windows.
|
||
|
|
|
||
|
|
The logic for this function was borrowed from the function ``make_init``
|
||
|
|
in `scipy/tools/openblas_support.py`:
|
||
|
|
https://github.com/scipy/scipy/blob/bb92c8014e21052e7dde67a76b28214dd1dcb94a/tools/openblas_support.py#L239-L274
|
||
|
|
""" # noqa: E501
|
||
|
|
if os.name == "nt":
|
||
|
|
try:
|
||
|
|
from ctypes import WinDLL
|
||
|
|
basedir = os.path.dirname(__file__)
|
||
|
|
except: # noqa: E722
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
dll_path = os.path.join(basedir, "libsf_error_state.dll")
|
||
|
|
if os.path.exists(dll_path):
|
||
|
|
WinDLL(dll_path)
|
||
|
|
|
||
|
|
|
||
|
|
_load_libsf_error_state()
|
||
|
|
|
||
|
|
|
||
|
|
from ._sf_error import SpecialFunctionWarning, SpecialFunctionError
|
||
|
|
|
||
|
|
from . import _ufuncs
|
||
|
|
from ._ufuncs import *
|
||
|
|
|
||
|
|
# Replace some function definitions from _ufuncs to add Array API support
|
||
|
|
from ._support_alternative_backends import (
|
||
|
|
log_ndtr, ndtr, ndtri, erf, erfc, i0, i0e, i1, i1e, gammaln,
|
||
|
|
gammainc, gammaincc, logit, expit, entr, rel_entr, xlogy, chdtrc)
|
||
|
|
|
||
|
|
from . import _basic
|
||
|
|
from ._basic import *
|
||
|
|
|
||
|
|
from ._logsumexp import logsumexp, softmax, log_softmax
|
||
|
|
|
||
|
|
from . import _orthogonal
|
||
|
|
from ._orthogonal import *
|
||
|
|
|
||
|
|
from ._spfun_stats import multigammaln
|
||
|
|
from ._ellip_harm import (
|
||
|
|
ellip_harm,
|
||
|
|
ellip_harm_2,
|
||
|
|
ellip_normal
|
||
|
|
)
|
||
|
|
from ._lambertw import lambertw
|
||
|
|
from ._spherical_bessel import (
|
||
|
|
spherical_jn,
|
||
|
|
spherical_yn,
|
||
|
|
spherical_in,
|
||
|
|
spherical_kn
|
||
|
|
)
|
||
|
|
|
||
|
|
# Deprecated namespaces, to be removed in v2.0.0
|
||
|
|
from . import add_newdocs, basic, orthogonal, specfun, sf_error, spfun_stats
|
||
|
|
|
||
|
|
# We replace some function definitions from _ufuncs with those from
|
||
|
|
# _support_alternative_backends above, but those are all listed in _ufuncs.__all__,
|
||
|
|
# so there is no need to consider _support_alternative_backends.__all__ here.
|
||
|
|
__all__ = _ufuncs.__all__ + _basic.__all__ + _orthogonal.__all__
|
||
|
|
__all__ += [
|
||
|
|
'SpecialFunctionWarning',
|
||
|
|
'SpecialFunctionError',
|
||
|
|
'logsumexp',
|
||
|
|
'softmax',
|
||
|
|
'log_softmax',
|
||
|
|
'multigammaln',
|
||
|
|
'ellip_harm',
|
||
|
|
'ellip_harm_2',
|
||
|
|
'ellip_normal',
|
||
|
|
'lambertw',
|
||
|
|
'spherical_jn',
|
||
|
|
'spherical_yn',
|
||
|
|
'spherical_in',
|
||
|
|
'spherical_kn',
|
||
|
|
]
|
||
|
|
|
||
|
|
from scipy._lib._testutils import PytestTester
|
||
|
|
test = PytestTester(__name__)
|
||
|
|
del PytestTester
|
||
|
|
|
||
|
|
_depr_msg = ('\nThis function was deprecated in SciPy 1.12.0, and will be '
|
||
|
|
'removed in SciPy 1.14.0. Use scipy.special.{} instead.')
|
||
|
|
|
||
|
|
|
||
|
|
def btdtr(*args, **kwargs): # type: ignore [no-redef]
|
||
|
|
warnings.warn(_depr_msg.format('betainc'), category=DeprecationWarning,
|
||
|
|
stacklevel=2)
|
||
|
|
return _ufuncs.btdtr(*args, **kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
btdtr.__doc__ = _ufuncs.btdtr.__doc__ # type: ignore [misc]
|
||
|
|
|
||
|
|
|
||
|
|
def btdtri(*args, **kwargs): # type: ignore [no-redef]
|
||
|
|
warnings.warn(_depr_msg.format('betaincinv'), category=DeprecationWarning,
|
||
|
|
stacklevel=2)
|
||
|
|
return _ufuncs.btdtri(*args, **kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
btdtri.__doc__ = _ufuncs.btdtri.__doc__ # type: ignore [misc]
|
||
|
|
|
||
|
|
|
||
|
|
def _get_include():
|
||
|
|
"""This function is for development purposes only.
|
||
|
|
|
||
|
|
This function could disappear or its behavior could change at any time.
|
||
|
|
"""
|
||
|
|
import os
|
||
|
|
return os.path.dirname(__file__)
|
||
|
|
|