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# Docstrings for generated ufuncs
#
# The syntax is designed to look like the function add_newdoc is being
# called from numpy.lib, but in this file add_newdoc puts the
# docstrings in a dictionary. This dictionary is used in
# _generate_pyx.py to generate the docstrings for the ufuncs in
# scipy.special at the C level when the ufuncs are created at compile
# time.
docdict: dict[str, str] = {}
def get(name):
return docdict.get(name)
def add_newdoc(name, doc):
docdict[name] = doc
add_newdoc("_sf_error_test_function",
"""
Private function; do not use.
""")
add_newdoc("_cosine_cdf",
"""
_cosine_cdf(x)
Cumulative distribution function (CDF) of the cosine distribution::
{ 0, x < -pi
cdf(x) = { (pi + x + sin(x))/(2*pi), -pi <= x <= pi
{ 1, x > pi
Parameters
----------
x : array_like
`x` must contain real numbers.
Returns
-------
scalar or ndarray
The cosine distribution CDF evaluated at `x`.
""")
add_newdoc("_cosine_invcdf",
"""
_cosine_invcdf(p)
Inverse of the cumulative distribution function (CDF) of the cosine
distribution.
The CDF of the cosine distribution is::
cdf(x) = (pi + x + sin(x))/(2*pi)
This function computes the inverse of cdf(x).
Parameters
----------
p : array_like
`p` must contain real numbers in the interval ``0 <= p <= 1``.
`nan` is returned for values of `p` outside the interval [0, 1].
Returns
-------
scalar or ndarray
The inverse of the cosine distribution CDF evaluated at `p`.
""")
add_newdoc("sph_harm",
r"""
sph_harm(m, n, theta, phi, out=None)
Compute spherical harmonics.
The spherical harmonics are defined as
.. math::
Y^m_n(\theta,\phi) = \sqrt{\frac{2n+1}{4\pi} \frac{(n-m)!}{(n+m)!}}
e^{i m \theta} P^m_n(\cos(\phi))
where :math:`P_n^m` are the associated Legendre functions; see `lpmv`.
Parameters
----------
m : array_like
Order of the harmonic (int); must have ``|m| <= n``.
n : array_like
Degree of the harmonic (int); must have ``n >= 0``. This is
often denoted by ``l`` (lower case L) in descriptions of
spherical harmonics.
theta : array_like
Azimuthal (longitudinal) coordinate; must be in ``[0, 2*pi]``.
phi : array_like
Polar (colatitudinal) coordinate; must be in ``[0, pi]``.
out : ndarray, optional
Optional output array for the function values
Returns
-------
y_mn : complex scalar or ndarray
The harmonic :math:`Y^m_n` sampled at ``theta`` and ``phi``.
Notes
-----
There are different conventions for the meanings of the input
arguments ``theta`` and ``phi``. In SciPy ``theta`` is the
azimuthal angle and ``phi`` is the polar angle. It is common to
see the opposite convention, that is, ``theta`` as the polar angle
and ``phi`` as the azimuthal angle.
Note that SciPy's spherical harmonics include the Condon-Shortley
phase [2]_ because it is part of `lpmv`.
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\theta} \sin(\phi) \\
Y_1^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{\pi}}
\cos(\phi) \\
Y_1^1(\theta, \phi) &= -\frac{1}{2} \sqrt{\frac{3}{2\pi}}
e^{i\theta} \sin(\phi).
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
""")
add_newdoc("_ellip_harm",
"""
Internal function, use `ellip_harm` instead.
""")
add_newdoc("_ellip_norm",
"""
Internal function, use `ellip_norm` instead.
""")
add_newdoc("_lambertw",
"""
Internal function, use `lambertw` instead.
""")
add_newdoc("voigt_profile",
r"""
voigt_profile(x, sigma, gamma, out=None)
Voigt profile.
The Voigt profile is a convolution of a 1-D Normal distribution with
standard deviation ``sigma`` and a 1-D Cauchy distribution with half-width at
half-maximum ``gamma``.
If ``sigma = 0``, PDF of Cauchy distribution is returned.
Conversely, if ``gamma = 0``, PDF of Normal distribution is returned.
If ``sigma = gamma = 0``, the return value is ``Inf`` for ``x = 0``,
and ``0`` for all other ``x``.
Parameters
----------
x : array_like
Real argument
sigma : array_like
The standard deviation of the Normal distribution part
gamma : array_like
The half-width at half-maximum of the Cauchy distribution part
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
The Voigt profile at the given arguments
See Also
--------
wofz : Faddeeva function
Notes
-----
It can be expressed in terms of Faddeeva function
.. math:: V(x; \sigma, \gamma) = \frac{Re[w(z)]}{\sigma\sqrt{2\pi}},
.. math:: z = \frac{x + i\gamma}{\sqrt{2}\sigma}
where :math:`w(z)` is the Faddeeva function.
References
----------
.. [1] https://en.wikipedia.org/wiki/Voigt_profile
Examples
--------
Calculate the function at point 2 for ``sigma=1`` and ``gamma=1``.
>>> from scipy.special import voigt_profile
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> voigt_profile(2, 1., 1.)
0.09071519942627544
Calculate the function at several points by providing a NumPy array
for `x`.
>>> values = np.array([-2., 0., 5])
>>> voigt_profile(values, 1., 1.)
array([0.0907152 , 0.20870928, 0.01388492])
Plot the function for different parameter sets.
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> x = np.linspace(-10, 10, 500)
>>> parameters_list = [(1.5, 0., "solid"), (1.3, 0.5, "dashed"),
... (0., 1.8, "dotted"), (1., 1., "dashdot")]
>>> for params in parameters_list:
... sigma, gamma, linestyle = params
... voigt = voigt_profile(x, sigma, gamma)
... ax.plot(x, voigt, label=rf"$\sigma={sigma},\, \gamma={gamma}$",
... ls=linestyle)
>>> ax.legend()
>>> plt.show()
Verify visually that the Voigt profile indeed arises as the convolution
of a normal and a Cauchy distribution.
>>> from scipy.signal import convolve
>>> x, dx = np.linspace(-10, 10, 500, retstep=True)
>>> def gaussian(x, sigma):
... return np.exp(-0.5 * x**2/sigma**2)/(sigma * np.sqrt(2*np.pi))
>>> def cauchy(x, gamma):
... return gamma/(np.pi * (np.square(x)+gamma**2))
>>> sigma = 2
>>> gamma = 1
>>> gauss_profile = gaussian(x, sigma)
>>> cauchy_profile = cauchy(x, gamma)
>>> convolved = dx * convolve(cauchy_profile, gauss_profile, mode="same")
>>> voigt = voigt_profile(x, sigma, gamma)
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> ax.plot(x, gauss_profile, label="Gauss: $G$", c='b')
>>> ax.plot(x, cauchy_profile, label="Cauchy: $C$", c='y', ls="dashed")
>>> xx = 0.5*(x[1:] + x[:-1]) # midpoints
>>> ax.plot(xx, convolved[1:], label="Convolution: $G * C$", ls='dashdot',
... c='k')
>>> ax.plot(x, voigt, label="Voigt", ls='dotted', c='r')
>>> ax.legend()
>>> plt.show()
""")
add_newdoc("wrightomega",
r"""
wrightomega(z, out=None)
Wright Omega function.
Defined as the solution to
.. math::
\omega + \log(\omega) = z
where :math:`\log` is the principal branch of the complex logarithm.
Parameters
----------
z : array_like
Points at which to evaluate the Wright Omega function
out : ndarray, optional
Optional output array for the function values
Returns
-------
omega : scalar or ndarray
Values of the Wright Omega function
See Also
--------
lambertw : The Lambert W function
Notes
-----
.. versionadded:: 0.19.0
The function can also be defined as
.. math::
\omega(z) = W_{K(z)}(e^z)
where :math:`K(z) = \lceil (\Im(z) - \pi)/(2\pi) \rceil` is the
unwinding number and :math:`W` is the Lambert W function.
The implementation here is taken from [1]_.
References
----------
.. [1] Lawrence, Corless, and Jeffrey, "Algorithm 917: Complex
Double-Precision Evaluation of the Wright :math:`\omega`
Function." ACM Transactions on Mathematical Software,
2012. :doi:`10.1145/2168773.2168779`.
Examples
--------
>>> import numpy as np
>>> from scipy.special import wrightomega, lambertw
>>> wrightomega([-2, -1, 0, 1, 2])
array([0.12002824, 0.27846454, 0.56714329, 1. , 1.5571456 ])
Complex input:
>>> wrightomega(3 + 5j)
(1.5804428632097158+3.8213626783287937j)
Verify that ``wrightomega(z)`` satisfies ``w + log(w) = z``:
>>> w = -5 + 4j
>>> wrightomega(w + np.log(w))
(-5+4j)
Verify the connection to ``lambertw``:
>>> z = 0.5 + 3j
>>> wrightomega(z)
(0.0966015889280649+1.4937828458191993j)
>>> lambertw(np.exp(z))
(0.09660158892806493+1.4937828458191993j)
>>> z = 0.5 + 4j
>>> wrightomega(z)
(-0.3362123489037213+2.282986001579032j)
>>> lambertw(np.exp(z), k=1)
(-0.33621234890372115+2.282986001579032j)
""")
add_newdoc("agm",
"""
agm(a, b, out=None)
Compute the arithmetic-geometric mean of `a` and `b`.
Start with a_0 = a and b_0 = b and iteratively compute::
a_{n+1} = (a_n + b_n)/2
b_{n+1} = sqrt(a_n*b_n)
a_n and b_n converge to the same limit as n increases; their common
limit is agm(a, b).
Parameters
----------
a, b : array_like
Real values only. If the values are both negative, the result
is negative. If one value is negative and the other is positive,
`nan` is returned.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
The arithmetic-geometric mean of `a` and `b`.
Examples
--------
>>> import numpy as np
>>> from scipy.special import agm
>>> a, b = 24.0, 6.0
>>> agm(a, b)
13.458171481725614
Compare that result to the iteration:
>>> while a != b:
... a, b = (a + b)/2, np.sqrt(a*b)
... print("a = %19.16f b=%19.16f" % (a, b))
...
a = 15.0000000000000000 b=12.0000000000000000
a = 13.5000000000000000 b=13.4164078649987388
a = 13.4582039324993694 b=13.4581390309909850
a = 13.4581714817451772 b=13.4581714817060547
a = 13.4581714817256159 b=13.4581714817256159
When array-like arguments are given, broadcasting applies:
>>> a = np.array([[1.5], [3], [6]]) # a has shape (3, 1).
>>> b = np.array([6, 12, 24, 48]) # b has shape (4,).
>>> agm(a, b)
array([[ 3.36454287, 5.42363427, 9.05798751, 15.53650756],
[ 4.37037309, 6.72908574, 10.84726853, 18.11597502],
[ 6. , 8.74074619, 13.45817148, 21.69453707]])
""")
add_newdoc("airy",
r"""
airy(z, out=None)
Airy functions and their derivatives.
Parameters
----------
z : array_like
Real or complex argument.
out : tuple of ndarray, optional
Optional output arrays for the function values
Returns
-------
Ai, Aip, Bi, Bip : 4-tuple of scalar or ndarray
Airy functions Ai and Bi, and their derivatives Aip and Bip.
See Also
--------
airye : exponentially scaled Airy functions.
Notes
-----
The Airy functions Ai and Bi are two independent solutions of
.. math:: y''(x) = x y(x).
For real `z` in [-10, 10], the computation is carried out by calling
the Cephes [1]_ `airy` routine, which uses power series summation
for small `z` and rational minimax approximations for large `z`.
Outside this range, the AMOS [2]_ `zairy` and `zbiry` routines are
employed. They are computed using power series for :math:`|z| < 1` and
the following relations to modified Bessel functions for larger `z`
(where :math:`t \equiv 2 z^{3/2}/3`):
.. math::
Ai(z) = \frac{1}{\pi \sqrt{3}} K_{1/3}(t)
Ai'(z) = -\frac{z}{\pi \sqrt{3}} K_{2/3}(t)
Bi(z) = \sqrt{\frac{z}{3}} \left(I_{-1/3}(t) + I_{1/3}(t) \right)
Bi'(z) = \frac{z}{\sqrt{3}} \left(I_{-2/3}(t) + I_{2/3}(t)\right)
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
.. [2] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
Examples
--------
Compute the Airy functions on the interval [-15, 5].
>>> import numpy as np
>>> from scipy import special
>>> x = np.linspace(-15, 5, 201)
>>> ai, aip, bi, bip = special.airy(x)
Plot Ai(x) and Bi(x).
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, ai, 'r', label='Ai(x)')
>>> plt.plot(x, bi, 'b--', label='Bi(x)')
>>> plt.ylim(-0.5, 1.0)
>>> plt.grid()
>>> plt.legend(loc='upper left')
>>> plt.show()
""")
add_newdoc("airye",
"""
airye(z, out=None)
Exponentially scaled Airy functions and their derivatives.
Scaling::
eAi = Ai * exp(2.0/3.0*z*sqrt(z))
eAip = Aip * exp(2.0/3.0*z*sqrt(z))
eBi = Bi * exp(-abs(2.0/3.0*(z*sqrt(z)).real))
eBip = Bip * exp(-abs(2.0/3.0*(z*sqrt(z)).real))
Parameters
----------
z : array_like
Real or complex argument.
out : tuple of ndarray, optional
Optional output arrays for the function values
Returns
-------
eAi, eAip, eBi, eBip : 4-tuple of scalar or ndarray
Exponentially scaled Airy functions eAi and eBi, and their derivatives
eAip and eBip
See Also
--------
airy
Notes
-----
Wrapper for the AMOS [1]_ routines `zairy` and `zbiry`.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
Examples
--------
We can compute exponentially scaled Airy functions and their derivatives:
>>> import numpy as np
>>> from scipy.special import airye
>>> import matplotlib.pyplot as plt
>>> z = np.linspace(0, 50, 500)
>>> eAi, eAip, eBi, eBip = airye(z)
>>> f, ax = plt.subplots(2, 1, sharex=True)
>>> for ind, data in enumerate([[eAi, eAip, ["eAi", "eAip"]],
... [eBi, eBip, ["eBi", "eBip"]]]):
... ax[ind].plot(z, data[0], "-r", z, data[1], "-b")
... ax[ind].legend(data[2])
... ax[ind].grid(True)
>>> plt.show()
We can compute these using usual non-scaled Airy functions by:
>>> from scipy.special import airy
>>> Ai, Aip, Bi, Bip = airy(z)
>>> np.allclose(eAi, Ai * np.exp(2.0 / 3.0 * z * np.sqrt(z)))
True
>>> np.allclose(eAip, Aip * np.exp(2.0 / 3.0 * z * np.sqrt(z)))
True
>>> np.allclose(eBi, Bi * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z)))))
True
>>> np.allclose(eBip, Bip * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z)))))
True
Comparing non-scaled and exponentially scaled ones, the usual non-scaled
function quickly underflows for large values, whereas the exponentially
scaled function does not.
>>> airy(200)
(0.0, 0.0, nan, nan)
>>> airye(200)
(0.07501041684381093, -1.0609012305109042, 0.15003188417418148, 2.1215836725571093)
""")
add_newdoc("bdtr",
r"""
bdtr(k, n, p, out=None)
Binomial distribution cumulative distribution function.
Sum of the terms 0 through `floor(k)` of the Binomial probability density.
.. math::
\mathrm{bdtr}(k, n, p) =
\sum_{j=0}^{\lfloor k \rfloor} {{n}\choose{j}} p^j (1-p)^{n-j}
Parameters
----------
k : array_like
Number of successes (double), rounded down to the nearest integer.
n : array_like
Number of events (int).
p : array_like
Probability of success in a single event (float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
y : scalar or ndarray
Probability of `floor(k)` or fewer successes in `n` independent events with
success probabilities of `p`.
Notes
-----
The terms are not summed directly; instead the regularized incomplete beta
function is employed, according to the formula,
.. math::
\mathrm{bdtr}(k, n, p) =
I_{1 - p}(n - \lfloor k \rfloor, \lfloor k \rfloor + 1).
Wrapper for the Cephes [1]_ routine `bdtr`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
""")
add_newdoc("bdtrc",
r"""
bdtrc(k, n, p, out=None)
Binomial distribution survival function.
Sum of the terms `floor(k) + 1` through `n` of the binomial probability
density,
.. math::
\mathrm{bdtrc}(k, n, p) =
\sum_{j=\lfloor k \rfloor +1}^n {{n}\choose{j}} p^j (1-p)^{n-j}
Parameters
----------
k : array_like
Number of successes (double), rounded down to nearest integer.
n : array_like
Number of events (int)
p : array_like
Probability of success in a single event.
out : ndarray, optional
Optional output array for the function values
Returns
-------
y : scalar or ndarray
Probability of `floor(k) + 1` or more successes in `n` independent
events with success probabilities of `p`.
See Also
--------
bdtr
betainc
Notes
-----
The terms are not summed directly; instead the regularized incomplete beta
function is employed, according to the formula,
.. math::
\mathrm{bdtrc}(k, n, p) = I_{p}(\lfloor k \rfloor + 1, n - \lfloor k \rfloor).
Wrapper for the Cephes [1]_ routine `bdtrc`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
""")
add_newdoc("bdtri",
r"""
bdtri(k, n, y, out=None)
Inverse function to `bdtr` with respect to `p`.
Finds the event probability `p` such that the sum of the terms 0 through
`k` of the binomial probability density is equal to the given cumulative
probability `y`.
Parameters
----------
k : array_like
Number of successes (float), rounded down to the nearest integer.
n : array_like
Number of events (float)
y : array_like
Cumulative probability (probability of `k` or fewer successes in `n`
events).
out : ndarray, optional
Optional output array for the function values
Returns
-------
p : scalar or ndarray
The event probability such that `bdtr(\lfloor k \rfloor, n, p) = y`.
See Also
--------
bdtr
betaincinv
Notes
-----
The computation is carried out using the inverse beta integral function
and the relation,::
1 - p = betaincinv(n - k, k + 1, y).
Wrapper for the Cephes [1]_ routine `bdtri`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
""")
add_newdoc("bdtrik",
"""
bdtrik(y, n, p, out=None)
Inverse function to `bdtr` with respect to `k`.
Finds the number of successes `k` such that the sum of the terms 0 through
`k` of the Binomial probability density for `n` events with probability
`p` is equal to the given cumulative probability `y`.
Parameters
----------
y : array_like
Cumulative probability (probability of `k` or fewer successes in `n`
events).
n : array_like
Number of events (float).
p : array_like
Success probability (float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
k : scalar or ndarray
The number of successes `k` such that `bdtr(k, n, p) = y`.
See Also
--------
bdtr
Notes
-----
Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the
cumulative incomplete beta distribution.
Computation of `k` involves a search for a value that produces the desired
value of `y`. The search relies on the monotonicity of `y` with `k`.
Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`.
References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [2] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
""")
add_newdoc("bdtrin",
"""
bdtrin(k, y, p, out=None)
Inverse function to `bdtr` with respect to `n`.
Finds the number of events `n` such that the sum of the terms 0 through
`k` of the Binomial probability density for events with probability `p` is
equal to the given cumulative probability `y`.
Parameters
----------
k : array_like
Number of successes (float).
y : array_like
Cumulative probability (probability of `k` or fewer successes in `n`
events).
p : array_like
Success probability (float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
n : scalar or ndarray
The number of events `n` such that `bdtr(k, n, p) = y`.
See Also
--------
bdtr
Notes
-----
Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the
cumulative incomplete beta distribution.
Computation of `n` involves a search for a value that produces the desired
value of `y`. The search relies on the monotonicity of `y` with `n`.
Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`.
References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [2] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
""")
add_newdoc(
"binom",
r"""
binom(x, y, out=None)
Binomial coefficient considered as a function of two real variables.
For real arguments, the binomial coefficient is defined as
.. math::
\binom{x}{y} = \frac{\Gamma(x + 1)}{\Gamma(y + 1)\Gamma(x - y + 1)} =
\frac{1}{(x + 1)\mathrm{B}(x - y + 1, y + 1)}
Where :math:`\Gamma` is the Gamma function (`gamma`) and :math:`\mathrm{B}`
is the Beta function (`beta`) [1]_.
Parameters
----------
x, y: array_like
Real arguments to :math:`\binom{x}{y}`.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Value of binomial coefficient.
See Also
--------
comb : The number of combinations of N things taken k at a time.
Notes
-----
The Gamma function has poles at non-positive integers and tends to either
positive or negative infinity depending on the direction on the real line
from which a pole is approached. When considered as a function of two real
variables, :math:`\binom{x}{y}` is thus undefined when `x` is a negative
integer. `binom` returns ``nan`` when ``x`` is a negative integer. This
is the case even when ``x`` is a negative integer and ``y`` an integer,
contrary to the usual convention for defining :math:`\binom{n}{k}` when it
is considered as a function of two integer variables.
References
----------
.. [1] https://en.wikipedia.org/wiki/Binomial_coefficient
Examples
--------
The following examples illustrate the ways in which `binom` differs from
the function `comb`.
>>> from scipy.special import binom, comb
When ``exact=False`` and ``x`` and ``y`` are both positive, `comb` calls
`binom` internally.
>>> x, y = 3, 2
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(3.0, 3.0, 3)
For larger values, `comb` with ``exact=True`` no longer agrees
with `binom`.
>>> x, y = 43, 23
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(960566918219.9999, 960566918219.9999, 960566918220)
`binom` returns ``nan`` when ``x`` is a negative integer, but is otherwise
defined for negative arguments. `comb` returns 0 whenever one of ``x`` or
``y`` is negative or ``x`` is less than ``y``.
>>> x, y = -3, 2
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(nan, 0.0, 0)
>>> x, y = -3.1, 2.2
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(18.714147876804432, 0.0, 0)
>>> x, y = 2.2, 3.1
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(0.037399983365134115, 0.0, 0)
"""
)
add_newdoc("btdtria",
r"""
btdtria(p, b, x, out=None)
Inverse of `btdtr` with respect to `a`.
This is the inverse of the beta cumulative distribution function, `btdtr`,
considered as a function of `a`, returning the value of `a` for which
`btdtr(a, b, x) = p`, or
.. math::
p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt
Parameters
----------
p : array_like
Cumulative probability, in [0, 1].
b : array_like
Shape parameter (`b` > 0).
x : array_like
The quantile, in [0, 1].
out : ndarray, optional
Optional output array for the function values
Returns
-------
a : scalar or ndarray
The value of the shape parameter `a` such that `btdtr(a, b, x) = p`.
See Also
--------
btdtr : Cumulative distribution function of the beta distribution.
btdtri : Inverse with respect to `x`.
btdtrib : Inverse with respect to `b`.
Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`.
The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `a` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `a`.
References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
Algorithm 708: Significant Digit Computation of the Incomplete Beta
Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373.
""")
add_newdoc("btdtrib",
r"""
btdtria(a, p, x, out=None)
Inverse of `btdtr` with respect to `b`.
This is the inverse of the beta cumulative distribution function, `btdtr`,
considered as a function of `b`, returning the value of `b` for which
`btdtr(a, b, x) = p`, or
.. math::
p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt
Parameters
----------
a : array_like
Shape parameter (`a` > 0).
p : array_like
Cumulative probability, in [0, 1].
x : array_like
The quantile, in [0, 1].
out : ndarray, optional
Optional output array for the function values
Returns
-------
b : scalar or ndarray
The value of the shape parameter `b` such that `btdtr(a, b, x) = p`.
See Also
--------
btdtr : Cumulative distribution function of the beta distribution.
btdtri : Inverse with respect to `x`.
btdtria : Inverse with respect to `a`.
Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`.
The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `b` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `b`.
References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
Algorithm 708: Significant Digit Computation of the Incomplete Beta
Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373.
""")
add_newdoc("bei",
r"""
bei(x, out=None)
Kelvin function bei.
Defined as
.. math::
\mathrm{bei}(x) = \Im[J_0(x e^{3 \pi i / 4})]
where :math:`J_0` is the Bessel function of the first kind of
order zero (see `jv`). See [dlmf]_ for more details.
Parameters
----------
x : array_like
Real argument.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of the Kelvin function.
See Also
--------
ber : the corresponding real part
beip : the derivative of bei
jv : Bessel function of the first kind
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/10.61
Examples
--------
It can be expressed using Bessel functions.
>>> import numpy as np
>>> import scipy.special as sc
>>> x = np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.jv(0, x * np.exp(3 * np.pi * 1j / 4)).imag
array([0.24956604, 0.97229163, 1.93758679, 2.29269032])
>>> sc.bei(x)
array([0.24956604, 0.97229163, 1.93758679, 2.29269032])
""")
add_newdoc("beip",
r"""
beip(x, out=None)
Derivative of the Kelvin function bei.
Parameters
----------
x : array_like
Real argument.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
The values of the derivative of bei.
See Also
--------
bei
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/10#PT5
""")
add_newdoc("ber",
r"""
ber(x, out=None)
Kelvin function ber.
Defined as
.. math::
\mathrm{ber}(x) = \Re[J_0(x e^{3 \pi i / 4})]
where :math:`J_0` is the Bessel function of the first kind of
order zero (see `jv`). See [dlmf]_ for more details.
Parameters
----------
x : array_like
Real argument.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of the Kelvin function.
See Also
--------
bei : the corresponding real part
berp : the derivative of bei
jv : Bessel function of the first kind
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/10.61
Examples
--------
It can be expressed using Bessel functions.
>>> import numpy as np
>>> import scipy.special as sc
>>> x = np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.jv(0, x * np.exp(3 * np.pi * 1j / 4)).real
array([ 0.98438178, 0.75173418, -0.22138025, -2.56341656])
>>> sc.ber(x)
array([ 0.98438178, 0.75173418, -0.22138025, -2.56341656])
""")
add_newdoc("berp",
r"""
berp(x, out=None)
Derivative of the Kelvin function ber.
Parameters
----------
x : array_like
Real argument.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
The values of the derivative of ber.
See Also
--------
ber
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/10#PT5
""")
add_newdoc("besselpoly",
r"""
besselpoly(a, lmb, nu, out=None)
Weighted integral of the Bessel function of the first kind.
Computes
.. math::
\int_0^1 x^\lambda J_\nu(2 a x) \, dx
where :math:`J_\nu` is a Bessel function and :math:`\lambda=lmb`,
:math:`\nu=nu`.
Parameters
----------
a : array_like
Scale factor inside the Bessel function.
lmb : array_like
Power of `x`
nu : array_like
Order of the Bessel function.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Value of the integral.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Evaluate the function for one parameter set.
>>> from scipy.special import besselpoly
>>> besselpoly(1, 1, 1)
0.24449718372863877
Evaluate the function for different scale factors.
>>> import numpy as np
>>> factors = np.array([0., 3., 6.])
>>> besselpoly(factors, 1, 1)
array([ 0. , -0.00549029, 0.00140174])
Plot the function for varying powers, orders and scales.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> powers = np.linspace(0, 10, 100)
>>> orders = [1, 2, 3]
>>> scales = [1, 2]
>>> all_combinations = [(order, scale) for order in orders
... for scale in scales]
>>> for order, scale in all_combinations:
... ax.plot(powers, besselpoly(scale, powers, order),
... label=rf"$\nu={order}, a={scale}$")
>>> ax.legend()
>>> ax.set_xlabel(r"$\lambda$")
>>> ax.set_ylabel(r"$\int_0^1 x^{\lambda} J_{\nu}(2ax)\,dx$")
>>> plt.show()
""")
add_newdoc("beta",
r"""
beta(a, b, out=None)
Beta function.
This function is defined in [1]_ as
.. math::
B(a, b) = \int_0^1 t^{a-1}(1-t)^{b-1}dt
= \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},
where :math:`\Gamma` is the gamma function.
Parameters
----------
a, b : array_like
Real-valued arguments
out : ndarray, optional
Optional output array for the function result
Returns
-------
scalar or ndarray
Value of the beta function
See Also
--------
gamma : the gamma function
betainc : the regularized incomplete beta function
betaln : the natural logarithm of the absolute
value of the beta function
References
----------
.. [1] NIST Digital Library of Mathematical Functions,
Eq. 5.12.1. https://dlmf.nist.gov/5.12
Examples
--------
>>> import scipy.special as sc
The beta function relates to the gamma function by the
definition given above:
>>> sc.beta(2, 3)
0.08333333333333333
>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)
0.08333333333333333
As this relationship demonstrates, the beta function
is symmetric:
>>> sc.beta(1.7, 2.4)
0.16567527689031739
>>> sc.beta(2.4, 1.7)
0.16567527689031739
This function satisfies :math:`B(1, b) = 1/b`:
>>> sc.beta(1, 4)
0.25
""")
add_newdoc(
"betainc",
r"""
betainc(a, b, x, out=None)
Regularized incomplete beta function.
Computes the regularized incomplete beta function, defined as [1]_:
.. math::
I_x(a, b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^x
t^{a-1}(1-t)^{b-1}dt,
for :math:`0 \leq x \leq 1`.
This function is the cumulative distribution function for the beta
distribution; its range is [0, 1].
Parameters
----------
a, b : array_like
Positive, real-valued parameters
x : array_like
Real-valued such that :math:`0 \leq x \leq 1`,
the upper limit of integration
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Value of the regularized incomplete beta function
See Also
--------
beta : beta function
betaincinv : inverse of the regularized incomplete beta function
betaincc : complement of the regularized incomplete beta function
scipy.stats.beta : beta distribution
Notes
-----
The term *regularized* in the name of this function refers to the
scaling of the function by the gamma function terms shown in the
formula. When not qualified as *regularized*, the name *incomplete
beta function* often refers to just the integral expression,
without the gamma terms. One can use the function `beta` from
`scipy.special` to get this "nonregularized" incomplete beta
function by multiplying the result of ``betainc(a, b, x)`` by
``beta(a, b)``.
References
----------
.. [1] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/8.17
Examples
--------
Let :math:`B(a, b)` be the `beta` function.
>>> import scipy.special as sc
The coefficient in terms of `gamma` is equal to
:math:`1/B(a, b)`. Also, when :math:`x=1`
the integral is equal to :math:`B(a, b)`.
Therefore, :math:`I_{x=1}(a, b) = 1` for any :math:`a, b`.
>>> sc.betainc(0.2, 3.5, 1.0)
1.0
It satisfies
:math:`I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b))`,
where :math:`F` is the hypergeometric function `hyp2f1`:
>>> a, b, x = 1.4, 3.1, 0.5
>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))
0.8148904036225295
>>> sc.betainc(a, b, x)
0.8148904036225296
This functions satisfies the relationship
:math:`I_x(a, b) = 1 - I_{1-x}(b, a)`:
>>> sc.betainc(2.2, 3.1, 0.4)
0.49339638807619446
>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)
0.49339638807619446
""")
add_newdoc(
"betaincc",
r"""
betaincc(a, b, x, out=None)
Complement of the regularized incomplete beta function.
Computes the complement of the regularized incomplete beta function,
defined as [1]_:
.. math::
\bar{I}_x(a, b) = 1 - I_x(a, b)
= 1 - \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^x
t^{a-1}(1-t)^{b-1}dt,
for :math:`0 \leq x \leq 1`.
Parameters
----------
a, b : array_like
Positive, real-valued parameters
x : array_like
Real-valued such that :math:`0 \leq x \leq 1`,
the upper limit of integration
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Value of the regularized incomplete beta function
See Also
--------
betainc : regularized incomplete beta function
betaincinv : inverse of the regularized incomplete beta function
betainccinv :
inverse of the complement of the regularized incomplete beta function
beta : beta function
scipy.stats.beta : beta distribution
Notes
-----
.. versionadded:: 1.11.0
References
----------
.. [1] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/8.17
Examples
--------
>>> from scipy.special import betaincc, betainc
The naive calculation ``1 - betainc(a, b, x)`` loses precision when
the values of ``betainc(a, b, x)`` are close to 1:
>>> 1 - betainc(0.5, 8, [0.9, 0.99, 0.999])
array([2.0574632e-09, 0.0000000e+00, 0.0000000e+00])
By using ``betaincc``, we get the correct values:
>>> betaincc(0.5, 8, [0.9, 0.99, 0.999])
array([2.05746321e-09, 1.97259354e-17, 1.96467954e-25])
""")
add_newdoc(
"betaincinv",
r"""
betaincinv(a, b, y, out=None)
Inverse of the regularized incomplete beta function.
Computes :math:`x` such that:
.. math::
y = I_x(a, b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}
\int_0^x t^{a-1}(1-t)^{b-1}dt,
where :math:`I_x` is the normalized incomplete beta function `betainc`
and :math:`\Gamma` is the `gamma` function [1]_.
Parameters
----------
a, b : array_like
Positive, real-valued parameters
y : array_like
Real-valued input
out : ndarray, optional
Optional output array for function values
Returns
-------
scalar or ndarray
Value of the inverse of the regularized incomplete beta function
See Also
--------
betainc : regularized incomplete beta function
gamma : gamma function
References
----------
.. [1] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/8.17
Examples
--------
>>> import scipy.special as sc
This function is the inverse of `betainc` for fixed
values of :math:`a` and :math:`b`.
>>> a, b = 1.2, 3.1
>>> y = sc.betainc(a, b, 0.2)
>>> sc.betaincinv(a, b, y)
0.2
>>>
>>> a, b = 7.5, 0.4
>>> x = sc.betaincinv(a, b, 0.5)
>>> sc.betainc(a, b, x)
0.5
""")
add_newdoc(
"betainccinv",
r"""
betainccinv(a, b, y, out=None)
Inverse of the complemented regularized incomplete beta function.
Computes :math:`x` such that:
.. math::
y = 1 - I_x(a, b) = 1 - \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}
\int_0^x t^{a-1}(1-t)^{b-1}dt,
where :math:`I_x` is the normalized incomplete beta function `betainc`
and :math:`\Gamma` is the `gamma` function [1]_.
Parameters
----------
a, b : array_like
Positive, real-valued parameters
y : array_like
Real-valued input
out : ndarray, optional
Optional output array for function values
Returns
-------
scalar or ndarray
Value of the inverse of the regularized incomplete beta function
See Also
--------
betainc : regularized incomplete beta function
betaincc : complement of the regularized incomplete beta function
Notes
-----
.. versionadded:: 1.11.0
References
----------
.. [1] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/8.17
Examples
--------
>>> from scipy.special import betainccinv, betaincc
This function is the inverse of `betaincc` for fixed
values of :math:`a` and :math:`b`.
>>> a, b = 1.2, 3.1
>>> y = betaincc(a, b, 0.2)
>>> betainccinv(a, b, y)
0.2
>>> a, b = 7, 2.5
>>> x = betainccinv(a, b, 0.875)
>>> betaincc(a, b, x)
0.875
""")
add_newdoc("betaln",
"""
betaln(a, b, out=None)
Natural logarithm of absolute value of beta function.
Computes ``ln(abs(beta(a, b)))``.
Parameters
----------
a, b : array_like
Positive, real-valued parameters
out : ndarray, optional
Optional output array for function values
Returns
-------
scalar or ndarray
Value of the betaln function
See Also
--------
gamma : the gamma function
betainc : the regularized incomplete beta function
beta : the beta function
Examples
--------
>>> import numpy as np
>>> from scipy.special import betaln, beta
Verify that, for moderate values of ``a`` and ``b``, ``betaln(a, b)``
is the same as ``log(beta(a, b))``:
>>> betaln(3, 4)
-4.0943445622221
>>> np.log(beta(3, 4))
-4.0943445622221
In the following ``beta(a, b)`` underflows to 0, so we can't compute
the logarithm of the actual value.
>>> a = 400
>>> b = 900
>>> beta(a, b)
0.0
We can compute the logarithm of ``beta(a, b)`` by using `betaln`:
>>> betaln(a, b)
-804.3069951764146
""")
add_newdoc("boxcox",
"""
boxcox(x, lmbda, out=None)
Compute the Box-Cox transformation.
The Box-Cox transformation is::
y = (x**lmbda - 1) / lmbda if lmbda != 0
log(x) if lmbda == 0
Returns `nan` if ``x < 0``.
Returns `-inf` if ``x == 0`` and ``lmbda < 0``.
Parameters
----------
x : array_like
Data to be transformed.
lmbda : array_like
Power parameter of the Box-Cox transform.
out : ndarray, optional
Optional output array for the function values
Returns
-------
y : scalar or ndarray
Transformed data.
Notes
-----
.. versionadded:: 0.14.0
Examples
--------
>>> from scipy.special import boxcox
>>> boxcox([1, 4, 10], 2.5)
array([ 0. , 12.4 , 126.09110641])
>>> boxcox(2, [0, 1, 2])
array([ 0.69314718, 1. , 1.5 ])
""")
add_newdoc("boxcox1p",
"""
boxcox1p(x, lmbda, out=None)
Compute the Box-Cox transformation of 1 + `x`.
The Box-Cox transformation computed by `boxcox1p` is::
y = ((1+x)**lmbda - 1) / lmbda if lmbda != 0
log(1+x) if lmbda == 0
Returns `nan` if ``x < -1``.
Returns `-inf` if ``x == -1`` and ``lmbda < 0``.
Parameters
----------
x : array_like
Data to be transformed.
lmbda : array_like
Power parameter of the Box-Cox transform.
out : ndarray, optional
Optional output array for the function values
Returns
-------
y : scalar or ndarray
Transformed data.
Notes
-----
.. versionadded:: 0.14.0
Examples
--------
>>> from scipy.special import boxcox1p
>>> boxcox1p(1e-4, [0, 0.5, 1])
array([ 9.99950003e-05, 9.99975001e-05, 1.00000000e-04])
>>> boxcox1p([0.01, 0.1], 0.25)
array([ 0.00996272, 0.09645476])
""")
add_newdoc("inv_boxcox",
"""
inv_boxcox(y, lmbda, out=None)
Compute the inverse of the Box-Cox transformation.
Find ``x`` such that::
y = (x**lmbda - 1) / lmbda if lmbda != 0
log(x) if lmbda == 0
Parameters
----------
y : array_like
Data to be transformed.
lmbda : array_like
Power parameter of the Box-Cox transform.
out : ndarray, optional
Optional output array for the function values
Returns
-------
x : scalar or ndarray
Transformed data.
Notes
-----
.. versionadded:: 0.16.0
Examples
--------
>>> from scipy.special import boxcox, inv_boxcox
>>> y = boxcox([1, 4, 10], 2.5)
>>> inv_boxcox(y, 2.5)
array([1., 4., 10.])
""")
add_newdoc("inv_boxcox1p",
"""
inv_boxcox1p(y, lmbda, out=None)
Compute the inverse of the Box-Cox transformation.
Find ``x`` such that::
y = ((1+x)**lmbda - 1) / lmbda if lmbda != 0
log(1+x) if lmbda == 0
Parameters
----------
y : array_like
Data to be transformed.
lmbda : array_like
Power parameter of the Box-Cox transform.
out : ndarray, optional
Optional output array for the function values
Returns
-------
x : scalar or ndarray
Transformed data.
Notes
-----
.. versionadded:: 0.16.0
Examples
--------
>>> from scipy.special import boxcox1p, inv_boxcox1p
>>> y = boxcox1p([1, 4, 10], 2.5)
>>> inv_boxcox1p(y, 2.5)
array([1., 4., 10.])
""")
add_newdoc("btdtr",
r"""
btdtr(a, b, x, out=None)
Cumulative distribution function of the beta distribution.
Returns the integral from zero to `x` of the beta probability density
function,
.. math::
I = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt
where :math:`\Gamma` is the gamma function.
.. deprecated:: 1.12.0
This function is deprecated and will be removed from SciPy 1.14.0.
Use `scipy.special.betainc` instead.
Parameters
----------
a : array_like
Shape parameter (a > 0).
b : array_like
Shape parameter (b > 0).
x : array_like
Upper limit of integration, in [0, 1].
out : ndarray, optional
Optional output array for the function values
Returns
-------
I : scalar or ndarray
Cumulative distribution function of the beta distribution with
parameters `a` and `b` at `x`.
See Also
--------
betainc
Notes
-----
This function is identical to the incomplete beta integral function
`betainc`.
Wrapper for the Cephes [1]_ routine `btdtr`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
""")
add_newdoc("btdtri",
r"""
btdtri(a, b, p, out=None)
The `p`-th quantile of the beta distribution.
This function is the inverse of the beta cumulative distribution function,
`btdtr`, returning the value of `x` for which `btdtr(a, b, x) = p`, or
.. math::
p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt
.. deprecated:: 1.12.0
This function is deprecated and will be removed from SciPy 1.14.0.
Use `scipy.special.betaincinv` instead.
Parameters
----------
a : array_like
Shape parameter (`a` > 0).
b : array_like
Shape parameter (`b` > 0).
p : array_like
Cumulative probability, in [0, 1].
out : ndarray, optional
Optional output array for the function values
Returns
-------
x : scalar or ndarray
The quantile corresponding to `p`.
See Also
--------
betaincinv
btdtr
Notes
-----
The value of `x` is found by interval halving or Newton iterations.
Wrapper for the Cephes [1]_ routine `incbi`, which solves the equivalent
problem of finding the inverse of the incomplete beta integral.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
""")
add_newdoc("cbrt",
"""
cbrt(x, out=None)
Element-wise cube root of `x`.
Parameters
----------
x : array_like
`x` must contain real numbers.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
The cube root of each value in `x`.
Examples
--------
>>> from scipy.special import cbrt
>>> cbrt(8)
2.0
>>> cbrt([-8, -3, 0.125, 1.331])
array([-2. , -1.44224957, 0.5 , 1.1 ])
""")
add_newdoc("chdtr",
r"""
chdtr(v, x, out=None)
Chi square cumulative distribution function.
Returns the area under the left tail (from 0 to `x`) of the Chi
square probability density function with `v` degrees of freedom:
.. math::
\frac{1}{2^{v/2} \Gamma(v/2)} \int_0^x t^{v/2 - 1} e^{-t/2} dt
Here :math:`\Gamma` is the Gamma function; see `gamma`. This
integral can be expressed in terms of the regularized lower
incomplete gamma function `gammainc` as
``gammainc(v / 2, x / 2)``. [1]_
Parameters
----------
v : array_like
Degrees of freedom.
x : array_like
Upper bound of the integral.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of the cumulative distribution function.
See Also
--------
chdtrc, chdtri, chdtriv, gammainc
References
----------
.. [1] Chi-Square distribution,
https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It can be expressed in terms of the regularized lower incomplete
gamma function.
>>> v = 1
>>> x = np.arange(4)
>>> sc.chdtr(v, x)
array([0. , 0.68268949, 0.84270079, 0.91673548])
>>> sc.gammainc(v / 2, x / 2)
array([0. , 0.68268949, 0.84270079, 0.91673548])
""")
add_newdoc("chdtrc",
r"""
chdtrc(v, x, out=None)
Chi square survival function.
Returns the area under the right hand tail (from `x` to infinity)
of the Chi square probability density function with `v` degrees of
freedom:
.. math::
\frac{1}{2^{v/2} \Gamma(v/2)} \int_x^\infty t^{v/2 - 1} e^{-t/2} dt
Here :math:`\Gamma` is the Gamma function; see `gamma`. This
integral can be expressed in terms of the regularized upper
incomplete gamma function `gammaincc` as
``gammaincc(v / 2, x / 2)``. [1]_
Parameters
----------
v : array_like
Degrees of freedom.
x : array_like
Lower bound of the integral.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of the survival function.
See Also
--------
chdtr, chdtri, chdtriv, gammaincc
References
----------
.. [1] Chi-Square distribution,
https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It can be expressed in terms of the regularized upper incomplete
gamma function.
>>> v = 1
>>> x = np.arange(4)
>>> sc.chdtrc(v, x)
array([1. , 0.31731051, 0.15729921, 0.08326452])
>>> sc.gammaincc(v / 2, x / 2)
array([1. , 0.31731051, 0.15729921, 0.08326452])
""")
add_newdoc("chdtri",
"""
chdtri(v, p, out=None)
Inverse to `chdtrc` with respect to `x`.
Returns `x` such that ``chdtrc(v, x) == p``.
Parameters
----------
v : array_like
Degrees of freedom.
p : array_like
Probability.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
x : scalar or ndarray
Value so that the probability a Chi square random variable
with `v` degrees of freedom is greater than `x` equals `p`.
See Also
--------
chdtrc, chdtr, chdtriv
References
----------
.. [1] Chi-Square distribution,
https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
Examples
--------
>>> import scipy.special as sc
It inverts `chdtrc`.
>>> v, p = 1, 0.3
>>> sc.chdtrc(v, sc.chdtri(v, p))
0.3
>>> x = 1
>>> sc.chdtri(v, sc.chdtrc(v, x))
1.0
""")
add_newdoc("chdtriv",
"""
chdtriv(p, x, out=None)
Inverse to `chdtr` with respect to `v`.
Returns `v` such that ``chdtr(v, x) == p``.
Parameters
----------
p : array_like
Probability that the Chi square random variable is less than
or equal to `x`.
x : array_like
Nonnegative input.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Degrees of freedom.
See Also
--------
chdtr, chdtrc, chdtri
References
----------
.. [1] Chi-Square distribution,
https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
Examples
--------
>>> import scipy.special as sc
It inverts `chdtr`.
>>> p, x = 0.5, 1
>>> sc.chdtr(sc.chdtriv(p, x), x)
0.5000000000202172
>>> v = 1
>>> sc.chdtriv(sc.chdtr(v, x), v)
1.0000000000000013
""")
add_newdoc("chndtr",
r"""
chndtr(x, df, nc, out=None)
Non-central chi square cumulative distribution function
The cumulative distribution function is given by:
.. math::
P(\chi^{\prime 2} \vert \nu, \lambda) =\sum_{j=0}^{\infty}
e^{-\lambda /2}
\frac{(\lambda /2)^j}{j!} P(\chi^{\prime 2} \vert \nu + 2j),
where :math:`\nu > 0` is the degrees of freedom (``df``) and
:math:`\lambda \geq 0` is the non-centrality parameter (``nc``).
Parameters
----------
x : array_like
Upper bound of the integral; must satisfy ``x >= 0``
df : array_like
Degrees of freedom; must satisfy ``df > 0``
nc : array_like
Non-centrality parameter; must satisfy ``nc >= 0``
out : ndarray, optional
Optional output array for the function results
Returns
-------
x : scalar or ndarray
Value of the non-central chi square cumulative distribution function.
See Also
--------
chndtrix, chndtridf, chndtrinc
""")
add_newdoc("chndtrix",
"""
chndtrix(p, df, nc, out=None)
Inverse to `chndtr` vs `x`
Calculated using a search to find a value for `x` that produces the
desired value of `p`.
Parameters
----------
p : array_like
Probability; must satisfy ``0 <= p < 1``
df : array_like
Degrees of freedom; must satisfy ``df > 0``
nc : array_like
Non-centrality parameter; must satisfy ``nc >= 0``
out : ndarray, optional
Optional output array for the function results
Returns
-------
x : scalar or ndarray
Value so that the probability a non-central Chi square random variable
with `df` degrees of freedom and non-centrality, `nc`, is greater than
`x` equals `p`.
See Also
--------
chndtr, chndtridf, chndtrinc
""")
add_newdoc("chndtridf",
"""
chndtridf(x, p, nc, out=None)
Inverse to `chndtr` vs `df`
Calculated using a search to find a value for `df` that produces the
desired value of `p`.
Parameters
----------
x : array_like
Upper bound of the integral; must satisfy ``x >= 0``
p : array_like
Probability; must satisfy ``0 <= p < 1``
nc : array_like
Non-centrality parameter; must satisfy ``nc >= 0``
out : ndarray, optional
Optional output array for the function results
Returns
-------
df : scalar or ndarray
Degrees of freedom
See Also
--------
chndtr, chndtrix, chndtrinc
""")
add_newdoc("chndtrinc",
"""
chndtrinc(x, df, p, out=None)
Inverse to `chndtr` vs `nc`
Calculated using a search to find a value for `df` that produces the
desired value of `p`.
Parameters
----------
x : array_like
Upper bound of the integral; must satisfy ``x >= 0``
df : array_like
Degrees of freedom; must satisfy ``df > 0``
p : array_like
Probability; must satisfy ``0 <= p < 1``
out : ndarray, optional
Optional output array for the function results
Returns
-------
nc : scalar or ndarray
Non-centrality
See Also
--------
chndtr, chndtrix, chndtrinc
""")
add_newdoc("cosdg",
"""
cosdg(x, out=None)
Cosine of the angle `x` given in degrees.
Parameters
----------
x : array_like
Angle, given in degrees.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Cosine of the input.
See Also
--------
sindg, tandg, cotdg
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is more accurate than using cosine directly.
>>> x = 90 + 180 * np.arange(3)
>>> sc.cosdg(x)
array([-0., 0., -0.])
>>> np.cos(x * np.pi / 180)
array([ 6.1232340e-17, -1.8369702e-16, 3.0616170e-16])
""")
add_newdoc("cosm1",
"""
cosm1(x, out=None)
cos(x) - 1 for use when `x` is near zero.
Parameters
----------
x : array_like
Real valued argument.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of ``cos(x) - 1``.
See Also
--------
expm1, log1p
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is more accurate than computing ``cos(x) - 1`` directly for
``x`` around 0.
>>> x = 1e-30
>>> np.cos(x) - 1
0.0
>>> sc.cosm1(x)
-5.0000000000000005e-61
""")
add_newdoc("cotdg",
"""
cotdg(x, out=None)
Cotangent of the angle `x` given in degrees.
Parameters
----------
x : array_like
Angle, given in degrees.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Cotangent at the input.
See Also
--------
sindg, cosdg, tandg
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is more accurate than using cotangent directly.
>>> x = 90 + 180 * np.arange(3)
>>> sc.cotdg(x)
array([0., 0., 0.])
>>> 1 / np.tan(x * np.pi / 180)
array([6.1232340e-17, 1.8369702e-16, 3.0616170e-16])
""")
add_newdoc("dawsn",
"""
dawsn(x, out=None)
Dawson's integral.
Computes::
exp(-x**2) * integral(exp(t**2), t=0..x).
Parameters
----------
x : array_like
Function parameter.
out : ndarray, optional
Optional output array for the function values
Returns
-------
y : scalar or ndarray
Value of the integral.
See Also
--------
wofz, erf, erfc, erfcx, erfi
References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
http://ab-initio.mit.edu/Faddeeva
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-15, 15, num=1000)
>>> plt.plot(x, special.dawsn(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$dawsn(x)$')
>>> plt.show()
""")
add_newdoc("ellipe",
r"""
ellipe(m, out=None)
Complete elliptic integral of the second kind
This function is defined as
.. math:: E(m) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{1/2} dt
Parameters
----------
m : array_like
Defines the parameter of the elliptic integral.
out : ndarray, optional
Optional output array for the function values
Returns
-------
E : scalar or ndarray
Value of the elliptic integral.
See Also
--------
ellipkm1 : Complete elliptic integral of the first kind, near `m` = 1
ellipk : Complete elliptic integral of the first kind
ellipkinc : Incomplete elliptic integral of the first kind
ellipeinc : Incomplete elliptic integral of the second kind
elliprd : Symmetric elliptic integral of the second kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
Notes
-----
Wrapper for the Cephes [1]_ routine `ellpe`.
For `m > 0` the computation uses the approximation,
.. math:: E(m) \approx P(1-m) - (1-m) \log(1-m) Q(1-m),
where :math:`P` and :math:`Q` are tenth-order polynomials. For
`m < 0`, the relation
.. math:: E(m) = E(m/(m - 1)) \sqrt(1-m)
is used.
The parameterization in terms of :math:`m` follows that of section
17.2 in [2]_. Other parameterizations in terms of the
complementary parameter :math:`1 - m`, modular angle
:math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also
used, so be careful that you choose the correct parameter.
The Legendre E integral is related to Carlson's symmetric R_D or R_G
functions in multiple ways [3]_. For example,
.. math:: E(m) = 2 R_G(0, 1-k^2, 1) .
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [3] NIST Digital Library of Mathematical
Functions. http://dlmf.nist.gov/, Release 1.0.28 of
2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i
Examples
--------
This function is used in finding the circumference of an
ellipse with semi-major axis `a` and semi-minor axis `b`.
>>> import numpy as np
>>> from scipy import special
>>> a = 3.5
>>> b = 2.1
>>> e_sq = 1.0 - b**2/a**2 # eccentricity squared
Then the circumference is found using the following:
>>> C = 4*a*special.ellipe(e_sq) # circumference formula
>>> C
17.868899204378693
When `a` and `b` are the same (meaning eccentricity is 0),
this reduces to the circumference of a circle.
>>> 4*a*special.ellipe(0.0) # formula for ellipse with a = b
21.991148575128552
>>> 2*np.pi*a # formula for circle of radius a
21.991148575128552
""")
add_newdoc("ellipeinc",
r"""
ellipeinc(phi, m, out=None)
Incomplete elliptic integral of the second kind
This function is defined as
.. math:: E(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{1/2} dt
Parameters
----------
phi : array_like
amplitude of the elliptic integral.
m : array_like
parameter of the elliptic integral.
out : ndarray, optional
Optional output array for the function values
Returns
-------
E : scalar or ndarray
Value of the elliptic integral.
See Also
--------
ellipkm1 : Complete elliptic integral of the first kind, near `m` = 1
ellipk : Complete elliptic integral of the first kind
ellipkinc : Incomplete elliptic integral of the first kind
ellipe : Complete elliptic integral of the second kind
elliprd : Symmetric elliptic integral of the second kind.
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
Notes
-----
Wrapper for the Cephes [1]_ routine `ellie`.
Computation uses arithmetic-geometric means algorithm.
The parameterization in terms of :math:`m` follows that of section
17.2 in [2]_. Other parameterizations in terms of the
complementary parameter :math:`1 - m`, modular angle
:math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also
used, so be careful that you choose the correct parameter.
The Legendre E incomplete integral can be related to combinations
of Carlson's symmetric integrals R_D, R_F, and R_G in multiple
ways [3]_. For example, with :math:`c = \csc^2\phi`,
.. math::
E(\phi, m) = R_F(c-1, c-k^2, c)
- \frac{1}{3} k^2 R_D(c-1, c-k^2, c) .
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [3] NIST Digital Library of Mathematical
Functions. http://dlmf.nist.gov/, Release 1.0.28 of
2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i
""")
add_newdoc("ellipj",
"""
ellipj(u, m, out=None)
Jacobian elliptic functions
Calculates the Jacobian elliptic functions of parameter `m` between
0 and 1, and real argument `u`.
Parameters
----------
m : array_like
Parameter.
u : array_like
Argument.
out : tuple of ndarray, optional
Optional output arrays for the function values
Returns
-------
sn, cn, dn, ph : 4-tuple of scalar or ndarray
The returned functions::
sn(u|m), cn(u|m), dn(u|m)
The value `ph` is such that if `u = ellipkinc(ph, m)`,
then `sn(u|m) = sin(ph)` and `cn(u|m) = cos(ph)`.
See Also
--------
ellipk : Complete elliptic integral of the first kind
ellipkinc : Incomplete elliptic integral of the first kind
Notes
-----
Wrapper for the Cephes [1]_ routine `ellpj`.
These functions are periodic, with quarter-period on the real axis
equal to the complete elliptic integral `ellipk(m)`.
Relation to incomplete elliptic integral: If `u = ellipkinc(phi,m)`, then
`sn(u|m) = sin(phi)`, and `cn(u|m) = cos(phi)`. The `phi` is called
the amplitude of `u`.
Computation is by means of the arithmetic-geometric mean algorithm,
except when `m` is within 1e-9 of 0 or 1. In the latter case with `m`
close to 1, the approximation applies only for `phi < pi/2`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
""")
add_newdoc("ellipkm1",
"""
ellipkm1(p, out=None)
Complete elliptic integral of the first kind around `m` = 1
This function is defined as
.. math:: K(p) = \\int_0^{\\pi/2} [1 - m \\sin(t)^2]^{-1/2} dt
where `m = 1 - p`.
Parameters
----------
p : array_like
Defines the parameter of the elliptic integral as `m = 1 - p`.
out : ndarray, optional
Optional output array for the function values
Returns
-------
K : scalar or ndarray
Value of the elliptic integral.
See Also
--------
ellipk : Complete elliptic integral of the first kind
ellipkinc : Incomplete elliptic integral of the first kind
ellipe : Complete elliptic integral of the second kind
ellipeinc : Incomplete elliptic integral of the second kind
elliprf : Completely-symmetric elliptic integral of the first kind.
Notes
-----
Wrapper for the Cephes [1]_ routine `ellpk`.
For `p <= 1`, computation uses the approximation,
.. math:: K(p) \\approx P(p) - \\log(p) Q(p),
where :math:`P` and :math:`Q` are tenth-order polynomials. The
argument `p` is used internally rather than `m` so that the logarithmic
singularity at `m = 1` will be shifted to the origin; this preserves
maximum accuracy. For `p > 1`, the identity
.. math:: K(p) = K(1/p)/\\sqrt(p)
is used.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
""")
add_newdoc("ellipk",
r"""
ellipk(m, out=None)
Complete elliptic integral of the first kind.
This function is defined as
.. math:: K(m) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{-1/2} dt
Parameters
----------
m : array_like
The parameter of the elliptic integral.
out : ndarray, optional
Optional output array for the function values
Returns
-------
K : scalar or ndarray
Value of the elliptic integral.
See Also
--------
ellipkm1 : Complete elliptic integral of the first kind around m = 1
ellipkinc : Incomplete elliptic integral of the first kind
ellipe : Complete elliptic integral of the second kind
ellipeinc : Incomplete elliptic integral of the second kind
elliprf : Completely-symmetric elliptic integral of the first kind.
Notes
-----
For more precision around point m = 1, use `ellipkm1`, which this
function calls.
The parameterization in terms of :math:`m` follows that of section
17.2 in [1]_. Other parameterizations in terms of the
complementary parameter :math:`1 - m`, modular angle
:math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also
used, so be careful that you choose the correct parameter.
The Legendre K integral is related to Carlson's symmetric R_F
function by [2]_:
.. math:: K(m) = R_F(0, 1-k^2, 1) .
References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [2] NIST Digital Library of Mathematical
Functions. http://dlmf.nist.gov/, Release 1.0.28 of
2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i
""")
add_newdoc("ellipkinc",
r"""
ellipkinc(phi, m, out=None)
Incomplete elliptic integral of the first kind
This function is defined as
.. math:: K(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{-1/2} dt
This function is also called :math:`F(\phi, m)`.
Parameters
----------
phi : array_like
amplitude of the elliptic integral
m : array_like
parameter of the elliptic integral
out : ndarray, optional
Optional output array for the function values
Returns
-------
K : scalar or ndarray
Value of the elliptic integral
See Also
--------
ellipkm1 : Complete elliptic integral of the first kind, near `m` = 1
ellipk : Complete elliptic integral of the first kind
ellipe : Complete elliptic integral of the second kind
ellipeinc : Incomplete elliptic integral of the second kind
elliprf : Completely-symmetric elliptic integral of the first kind.
Notes
-----
Wrapper for the Cephes [1]_ routine `ellik`. The computation is
carried out using the arithmetic-geometric mean algorithm.
The parameterization in terms of :math:`m` follows that of section
17.2 in [2]_. Other parameterizations in terms of the
complementary parameter :math:`1 - m`, modular angle
:math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also
used, so be careful that you choose the correct parameter.
The Legendre K incomplete integral (or F integral) is related to
Carlson's symmetric R_F function [3]_.
Setting :math:`c = \csc^2\phi`,
.. math:: F(\phi, m) = R_F(c-1, c-k^2, c) .
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [3] NIST Digital Library of Mathematical
Functions. http://dlmf.nist.gov/, Release 1.0.28 of
2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i
""")
add_newdoc(
"elliprc",
r"""
elliprc(x, y, out=None)
Degenerate symmetric elliptic integral.
The function RC is defined as [1]_
.. math::
R_{\mathrm{C}}(x, y) =
\frac{1}{2} \int_0^{+\infty} (t + x)^{-1/2} (t + y)^{-1} dt
= R_{\mathrm{F}}(x, y, y)
Parameters
----------
x, y : array_like
Real or complex input parameters. `x` can be any number in the
complex plane cut along the negative real axis. `y` must be non-zero.
out : ndarray, optional
Optional output array for the function values
Returns
-------
R : scalar or ndarray
Value of the integral. If `y` is real and negative, the Cauchy
principal value is returned. If both of `x` and `y` are real, the
return value is real. Otherwise, the return value is complex.
See Also
--------
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprd : Symmetric elliptic integral of the second kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
elliprj : Symmetric elliptic integral of the third kind.
Notes
-----
RC is a degenerate case of the symmetric integral RF: ``elliprc(x, y) ==
elliprf(x, y, y)``. It is an elementary function rather than an elliptic
integral.
The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order. [2]_
.. versionadded:: 1.8.0
References
----------
.. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
Functions," NIST, US Dept. of Commerce.
https://dlmf.nist.gov/19.16.E6
.. [2] B. C. Carlson, "Numerical computation of real or complex elliptic
integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
https://arxiv.org/abs/math/9409227
https://doi.org/10.1007/BF02198293
Examples
--------
Basic homogeneity property:
>>> import numpy as np
>>> from scipy.special import elliprc
>>> x = 1.2 + 3.4j
>>> y = 5.
>>> scale = 0.3 + 0.4j
>>> elliprc(scale*x, scale*y)
(0.5484493976710874-0.4169557678995833j)
>>> elliprc(x, y)/np.sqrt(scale)
(0.5484493976710874-0.41695576789958333j)
When the two arguments coincide, the integral is particularly
simple:
>>> x = 1.2 + 3.4j
>>> elliprc(x, x)
(0.4299173120614631-0.3041729818745595j)
>>> 1/np.sqrt(x)
(0.4299173120614631-0.30417298187455954j)
Another simple case: the first argument vanishes:
>>> y = 1.2 + 3.4j
>>> elliprc(0, y)
(0.6753125346116815-0.47779380263880866j)
>>> np.pi/2/np.sqrt(y)
(0.6753125346116815-0.4777938026388088j)
When `x` and `y` are both positive, we can express
:math:`R_C(x,y)` in terms of more elementary functions. For the
case :math:`0 \le x < y`,
>>> x = 3.2
>>> y = 6.
>>> elliprc(x, y)
0.44942991498453444
>>> np.arctan(np.sqrt((y-x)/x))/np.sqrt(y-x)
0.44942991498453433
And for the case :math:`0 \le y < x`,
>>> x = 6.
>>> y = 3.2
>>> elliprc(x,y)
0.4989837501576147
>>> np.log((np.sqrt(x)+np.sqrt(x-y))/np.sqrt(y))/np.sqrt(x-y)
0.49898375015761476
""")
add_newdoc(
"elliprd",
r"""
elliprd(x, y, z, out=None)
Symmetric elliptic integral of the second kind.
The function RD is defined as [1]_
.. math::
R_{\mathrm{D}}(x, y, z) =
\frac{3}{2} \int_0^{+\infty} [(t + x) (t + y)]^{-1/2} (t + z)^{-3/2}
dt
Parameters
----------
x, y, z : array_like
Real or complex input parameters. `x` or `y` can be any number in the
complex plane cut along the negative real axis, but at most one of them
can be zero, while `z` must be non-zero.
out : ndarray, optional
Optional output array for the function values
Returns
-------
R : scalar or ndarray
Value of the integral. If all of `x`, `y`, and `z` are real, the
return value is real. Otherwise, the return value is complex.
See Also
--------
elliprc : Degenerate symmetric elliptic integral.
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
elliprj : Symmetric elliptic integral of the third kind.
Notes
-----
RD is a degenerate case of the elliptic integral RJ: ``elliprd(x, y, z) ==
elliprj(x, y, z, z)``.
The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order. [2]_
.. versionadded:: 1.8.0
References
----------
.. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
Functions," NIST, US Dept. of Commerce.
https://dlmf.nist.gov/19.16.E5
.. [2] B. C. Carlson, "Numerical computation of real or complex elliptic
integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
https://arxiv.org/abs/math/9409227
https://doi.org/10.1007/BF02198293
Examples
--------
Basic homogeneity property:
>>> import numpy as np
>>> from scipy.special import elliprd
>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprd(scale*x, scale*y, scale*z)
(-0.03703043835680379-0.24500934665683802j)
>>> elliprd(x, y, z)*np.power(scale, -1.5)
(-0.0370304383568038-0.24500934665683805j)
All three arguments coincide:
>>> x = 1.2 + 3.4j
>>> elliprd(x, x, x)
(-0.03986825876151896-0.14051741840449586j)
>>> np.power(x, -1.5)
(-0.03986825876151894-0.14051741840449583j)
The so-called "second lemniscate constant":
>>> elliprd(0, 2, 1)/3
0.5990701173677961
>>> from scipy.special import gamma
>>> gamma(0.75)**2/np.sqrt(2*np.pi)
0.5990701173677959
""")
add_newdoc(
"elliprf",
r"""
elliprf(x, y, z, out=None)
Completely-symmetric elliptic integral of the first kind.
The function RF is defined as [1]_
.. math::
R_{\mathrm{F}}(x, y, z) =
\frac{1}{2} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} dt
Parameters
----------
x, y, z : array_like
Real or complex input parameters. `x`, `y`, or `z` can be any number in
the complex plane cut along the negative real axis, but at most one of
them can be zero.
out : ndarray, optional
Optional output array for the function values
Returns
-------
R : scalar or ndarray
Value of the integral. If all of `x`, `y`, and `z` are real, the return
value is real. Otherwise, the return value is complex.
See Also
--------
elliprc : Degenerate symmetric integral.
elliprd : Symmetric elliptic integral of the second kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
elliprj : Symmetric elliptic integral of the third kind.
Notes
-----
The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order (cf.:
https://dlmf.nist.gov/19.36.i) and the AGM algorithm for the complete
integral. [2]_
.. versionadded:: 1.8.0
References
----------
.. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
Functions," NIST, US Dept. of Commerce.
https://dlmf.nist.gov/19.16.E1
.. [2] B. C. Carlson, "Numerical computation of real or complex elliptic
integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
https://arxiv.org/abs/math/9409227
https://doi.org/10.1007/BF02198293
Examples
--------
Basic homogeneity property:
>>> import numpy as np
>>> from scipy.special import elliprf
>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprf(scale*x, scale*y, scale*z)
(0.5328051227278146-0.4008623567957094j)
>>> elliprf(x, y, z)/np.sqrt(scale)
(0.5328051227278147-0.4008623567957095j)
All three arguments coincide:
>>> x = 1.2 + 3.4j
>>> elliprf(x, x, x)
(0.42991731206146316-0.30417298187455954j)
>>> 1/np.sqrt(x)
(0.4299173120614631-0.30417298187455954j)
The so-called "first lemniscate constant":
>>> elliprf(0, 1, 2)
1.3110287771460598
>>> from scipy.special import gamma
>>> gamma(0.25)**2/(4*np.sqrt(2*np.pi))
1.3110287771460598
""")
add_newdoc(
"elliprg",
r"""
elliprg(x, y, z, out=None)
Completely-symmetric elliptic integral of the second kind.
The function RG is defined as [1]_
.. math::
R_{\mathrm{G}}(x, y, z) =
\frac{1}{4} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2}
\left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) t
dt
Parameters
----------
x, y, z : array_like
Real or complex input parameters. `x`, `y`, or `z` can be any number in
the complex plane cut along the negative real axis.
out : ndarray, optional
Optional output array for the function values
Returns
-------
R : scalar or ndarray
Value of the integral. If all of `x`, `y`, and `z` are real, the return
value is real. Otherwise, the return value is complex.
See Also
--------
elliprc : Degenerate symmetric integral.
elliprd : Symmetric elliptic integral of the second kind.
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprj : Symmetric elliptic integral of the third kind.
Notes
-----
The implementation uses the relation [1]_
.. math::
2 R_{\mathrm{G}}(x, y, z) =
z R_{\mathrm{F}}(x, y, z) -
\frac{1}{3} (x - z) (y - z) R_{\mathrm{D}}(x, y, z) +
\sqrt{\frac{x y}{z}}
and the symmetry of `x`, `y`, `z` when at least one non-zero parameter can
be chosen as the pivot. When one of the arguments is close to zero, the AGM
method is applied instead. Other special cases are computed following Ref.
[2]_
.. versionadded:: 1.8.0
References
----------
.. [1] B. C. Carlson, "Numerical computation of real or complex elliptic
integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
https://arxiv.org/abs/math/9409227
https://doi.org/10.1007/BF02198293
.. [2] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
Functions," NIST, US Dept. of Commerce.
https://dlmf.nist.gov/19.16.E1
https://dlmf.nist.gov/19.20.ii
Examples
--------
Basic homogeneity property:
>>> import numpy as np
>>> from scipy.special import elliprg
>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprg(scale*x, scale*y, scale*z)
(1.195936862005246+0.8470988320464167j)
>>> elliprg(x, y, z)*np.sqrt(scale)
(1.195936862005246+0.8470988320464165j)
Simplifications:
>>> elliprg(0, y, y)
1.756203682760182
>>> 0.25*np.pi*np.sqrt(y)
1.7562036827601817
>>> elliprg(0, 0, z)
1.224744871391589
>>> 0.5*np.sqrt(z)
1.224744871391589
The surface area of a triaxial ellipsoid with semiaxes ``a``, ``b``, and
``c`` is given by
.. math::
S = 4 \pi a b c R_{\mathrm{G}}(1 / a^2, 1 / b^2, 1 / c^2).
>>> def ellipsoid_area(a, b, c):
... r = 4.0 * np.pi * a * b * c
... return r * elliprg(1.0 / (a * a), 1.0 / (b * b), 1.0 / (c * c))
>>> print(ellipsoid_area(1, 3, 5))
108.62688289491807
""")
add_newdoc(
"elliprj",
r"""
elliprj(x, y, z, p, out=None)
Symmetric elliptic integral of the third kind.
The function RJ is defined as [1]_
.. math::
R_{\mathrm{J}}(x, y, z, p) =
\frac{3}{2} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2}
(t + p)^{-1} dt
.. warning::
This function should be considered experimental when the inputs are
unbalanced. Check correctness with another independent implementation.
Parameters
----------
x, y, z, p : array_like
Real or complex input parameters. `x`, `y`, or `z` are numbers in
the complex plane cut along the negative real axis (subject to further
constraints, see Notes), and at most one of them can be zero. `p` must
be non-zero.
out : ndarray, optional
Optional output array for the function values
Returns
-------
R : scalar or ndarray
Value of the integral. If all of `x`, `y`, `z`, and `p` are real, the
return value is real. Otherwise, the return value is complex.
If `p` is real and negative, while `x`, `y`, and `z` are real,
non-negative, and at most one of them is zero, the Cauchy principal
value is returned. [1]_ [2]_
See Also
--------
elliprc : Degenerate symmetric integral.
elliprd : Symmetric elliptic integral of the second kind.
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
Notes
-----
The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order. [3]_ The algorithm is slightly
different from its earlier incarnation as it appears in [1]_, in that the
call to `elliprc` (or ``atan``/``atanh``, see [4]_) is no longer needed in
the inner loop. Asymptotic approximations are used where arguments differ
widely in the order of magnitude. [5]_
The input values are subject to certain sufficient but not necessary
constraints when input arguments are complex. Notably, ``x``, ``y``, and
``z`` must have non-negative real parts, unless two of them are
non-negative and complex-conjugates to each other while the other is a real
non-negative number. [1]_ If the inputs do not satisfy the sufficient
condition described in Ref. [1]_ they are rejected outright with the output
set to NaN.
In the case where one of ``x``, ``y``, and ``z`` is equal to ``p``, the
function ``elliprd`` should be preferred because of its less restrictive
domain.
.. versionadded:: 1.8.0
References
----------
.. [1] B. C. Carlson, "Numerical computation of real or complex elliptic
integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
https://arxiv.org/abs/math/9409227
https://doi.org/10.1007/BF02198293
.. [2] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
Functions," NIST, US Dept. of Commerce.
https://dlmf.nist.gov/19.20.iii
.. [3] B. C. Carlson, J. FitzSimmons, "Reduction Theorems for Elliptic
Integrands with the Square Root of Two Quadratic Factors," J.
Comput. Appl. Math., vol. 118, nos. 1-2, pp. 71-85, 2000.
https://doi.org/10.1016/S0377-0427(00)00282-X
.. [4] F. Johansson, "Numerical Evaluation of Elliptic Functions, Elliptic
Integrals and Modular Forms," in J. Blumlein, C. Schneider, P.
Paule, eds., "Elliptic Integrals, Elliptic Functions and Modular
Forms in Quantum Field Theory," pp. 269-293, 2019 (Cham,
Switzerland: Springer Nature Switzerland)
https://arxiv.org/abs/1806.06725
https://doi.org/10.1007/978-3-030-04480-0
.. [5] B. C. Carlson, J. L. Gustafson, "Asymptotic Approximations for
Symmetric Elliptic Integrals," SIAM J. Math. Anls., vol. 25, no. 2,
pp. 288-303, 1994.
https://arxiv.org/abs/math/9310223
https://doi.org/10.1137/S0036141092228477
Examples
--------
Basic homogeneity property:
>>> import numpy as np
>>> from scipy.special import elliprj
>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> p = 7.
>>> scale = 0.3 - 0.4j
>>> elliprj(scale*x, scale*y, scale*z, scale*p)
(0.10834905565679157+0.19694950747103812j)
>>> elliprj(x, y, z, p)*np.power(scale, -1.5)
(0.10834905565679556+0.19694950747103854j)
Reduction to simpler elliptic integral:
>>> elliprj(x, y, z, z)
(0.08288462362195129-0.028376809745123258j)
>>> from scipy.special import elliprd
>>> elliprd(x, y, z)
(0.08288462362195136-0.028376809745123296j)
All arguments coincide:
>>> elliprj(x, x, x, x)
(-0.03986825876151896-0.14051741840449586j)
>>> np.power(x, -1.5)
(-0.03986825876151894-0.14051741840449583j)
""")
add_newdoc("entr",
r"""
entr(x, out=None)
Elementwise function for computing entropy.
.. math:: \text{entr}(x) = \begin{cases} - x \log(x) & x > 0 \\ 0 & x = 0
\\ -\infty & \text{otherwise} \end{cases}
Parameters
----------
x : ndarray
Input array.
out : ndarray, optional
Optional output array for the function values
Returns
-------
res : scalar or ndarray
The value of the elementwise entropy function at the given points `x`.
See Also
--------
kl_div, rel_entr, scipy.stats.entropy
Notes
-----
.. versionadded:: 0.15.0
This function is concave.
The origin of this function is in convex programming; see [1]_.
Given a probability distribution :math:`p_1, \ldots, p_n`,
the definition of entropy in the context of *information theory* is
.. math::
\sum_{i = 1}^n \mathrm{entr}(p_i).
To compute the latter quantity, use `scipy.stats.entropy`.
References
----------
.. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*.
Cambridge University Press, 2004.
:doi:`https://doi.org/10.1017/CBO9780511804441`
""")
add_newdoc("erf",
"""
erf(z, out=None)
Returns the error function of complex argument.
It is defined as ``2/sqrt(pi)*integral(exp(-t**2), t=0..z)``.
Parameters
----------
x : ndarray
Input array.
out : ndarray, optional
Optional output array for the function values
Returns
-------
res : scalar or ndarray
The values of the error function at the given points `x`.
See Also
--------
erfc, erfinv, erfcinv, wofz, erfcx, erfi
Notes
-----
The cumulative of the unit normal distribution is given by
``Phi(z) = 1/2[1 + erf(z/sqrt(2))]``.
References
----------
.. [1] https://en.wikipedia.org/wiki/Error_function
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover,
1972. http://www.math.sfu.ca/~cbm/aands/page_297.htm
.. [3] Steven G. Johnson, Faddeeva W function implementation.
http://ab-initio.mit.edu/Faddeeva
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-3, 3)
>>> plt.plot(x, special.erf(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$erf(x)$')
>>> plt.show()
""")
add_newdoc("erfc",
"""
erfc(x, out=None)
Complementary error function, ``1 - erf(x)``.
Parameters
----------
x : array_like
Real or complex valued argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the complementary error function
See Also
--------
erf, erfi, erfcx, dawsn, wofz
References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
http://ab-initio.mit.edu/Faddeeva
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-3, 3)
>>> plt.plot(x, special.erfc(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$erfc(x)$')
>>> plt.show()
""")
add_newdoc("erfi",
"""
erfi(z, out=None)
Imaginary error function, ``-i erf(i z)``.
Parameters
----------
z : array_like
Real or complex valued argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the imaginary error function
See Also
--------
erf, erfc, erfcx, dawsn, wofz
Notes
-----
.. versionadded:: 0.12.0
References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
http://ab-initio.mit.edu/Faddeeva
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-3, 3)
>>> plt.plot(x, special.erfi(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$erfi(x)$')
>>> plt.show()
""")
add_newdoc("erfcx",
"""
erfcx(x, out=None)
Scaled complementary error function, ``exp(x**2) * erfc(x)``.
Parameters
----------
x : array_like
Real or complex valued argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the scaled complementary error function
See Also
--------
erf, erfc, erfi, dawsn, wofz
Notes
-----
.. versionadded:: 0.12.0
References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
http://ab-initio.mit.edu/Faddeeva
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-3, 3)
>>> plt.plot(x, special.erfcx(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$erfcx(x)$')
>>> plt.show()
""")
add_newdoc(
"erfinv",
"""
erfinv(y, out=None)
Inverse of the error function.
Computes the inverse of the error function.
In the complex domain, there is no unique complex number w satisfying
erf(w)=z. This indicates a true inverse function would be multivalued.
When the domain restricts to the real, -1 < x < 1, there is a unique real
number satisfying erf(erfinv(x)) = x.
Parameters
----------
y : ndarray
Argument at which to evaluate. Domain: [-1, 1]
out : ndarray, optional
Optional output array for the function values
Returns
-------
erfinv : scalar or ndarray
The inverse of erf of y, element-wise
See Also
--------
erf : Error function of a complex argument
erfc : Complementary error function, ``1 - erf(x)``
erfcinv : Inverse of the complementary error function
Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import erfinv, erf
>>> erfinv(0.5)
0.4769362762044699
>>> y = np.linspace(-1.0, 1.0, num=9)
>>> x = erfinv(y)
>>> x
array([ -inf, -0.81341985, -0.47693628, -0.22531206, 0. ,
0.22531206, 0.47693628, 0.81341985, inf])
Verify that ``erf(erfinv(y))`` is ``y``.
>>> erf(x)
array([-1. , -0.75, -0.5 , -0.25, 0. , 0.25, 0.5 , 0.75, 1. ])
Plot the function:
>>> y = np.linspace(-1, 1, 200)
>>> fig, ax = plt.subplots()
>>> ax.plot(y, erfinv(y))
>>> ax.grid(True)
>>> ax.set_xlabel('y')
>>> ax.set_title('erfinv(y)')
>>> plt.show()
""")
add_newdoc(
"erfcinv",
"""
erfcinv(y, out=None)
Inverse of the complementary error function.
Computes the inverse of the complementary error function.
In the complex domain, there is no unique complex number w satisfying
erfc(w)=z. This indicates a true inverse function would be multivalued.
When the domain restricts to the real, 0 < x < 2, there is a unique real
number satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).
It is related to inverse of the error function by erfcinv(1-x) = erfinv(x)
Parameters
----------
y : ndarray
Argument at which to evaluate. Domain: [0, 2]
out : ndarray, optional
Optional output array for the function values
Returns
-------
erfcinv : scalar or ndarray
The inverse of erfc of y, element-wise
See Also
--------
erf : Error function of a complex argument
erfc : Complementary error function, ``1 - erf(x)``
erfinv : Inverse of the error function
Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import erfcinv
>>> erfcinv(0.5)
0.4769362762044699
>>> y = np.linspace(0.0, 2.0, num=11)
>>> erfcinv(y)
array([ inf, 0.9061938 , 0.59511608, 0.37080716, 0.17914345,
-0. , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,
-inf])
Plot the function:
>>> y = np.linspace(0, 2, 200)
>>> fig, ax = plt.subplots()
>>> ax.plot(y, erfcinv(y))
>>> ax.grid(True)
>>> ax.set_xlabel('y')
>>> ax.set_title('erfcinv(y)')
>>> plt.show()
""")
add_newdoc("eval_jacobi",
r"""
eval_jacobi(n, alpha, beta, x, out=None)
Evaluate Jacobi polynomial at a point.
The Jacobi polynomials can be defined via the Gauss hypergeometric
function :math:`{}_2F_1` as
.. math::
P_n^{(\alpha, \beta)}(x) = \frac{(\alpha + 1)_n}{\Gamma(n + 1)}
{}_2F_1(-n, 1 + \alpha + \beta + n; \alpha + 1; (1 - z)/2)
where :math:`(\cdot)_n` is the Pochhammer symbol; see `poch`. When
:math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.42 in [AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer the result is
determined via the relation to the Gauss hypergeometric
function.
alpha : array_like
Parameter
beta : array_like
Parameter
x : array_like
Points at which to evaluate the polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
P : scalar or ndarray
Values of the Jacobi polynomial
See Also
--------
roots_jacobi : roots and quadrature weights of Jacobi polynomials
jacobi : Jacobi polynomial object
hyp2f1 : Gauss hypergeometric function
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_sh_jacobi",
r"""
eval_sh_jacobi(n, p, q, x, out=None)
Evaluate shifted Jacobi polynomial at a point.
Defined by
.. math::
G_n^{(p, q)}(x)
= \binom{2n + p - 1}{n}^{-1} P_n^{(p - q, q - 1)}(2x - 1),
where :math:`P_n^{(\cdot, \cdot)}` is the n-th Jacobi
polynomial. See 22.5.2 in [AS]_ for details.
Parameters
----------
n : int
Degree of the polynomial. If not an integer, the result is
determined via the relation to `binom` and `eval_jacobi`.
p : float
Parameter
q : float
Parameter
out : ndarray, optional
Optional output array for the function values
Returns
-------
G : scalar or ndarray
Values of the shifted Jacobi polynomial.
See Also
--------
roots_sh_jacobi : roots and quadrature weights of shifted Jacobi
polynomials
sh_jacobi : shifted Jacobi polynomial object
eval_jacobi : evaluate Jacobi polynomials
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_gegenbauer",
r"""
eval_gegenbauer(n, alpha, x, out=None)
Evaluate Gegenbauer polynomial at a point.
The Gegenbauer polynomials can be defined via the Gauss
hypergeometric function :math:`{}_2F_1` as
.. math::
C_n^{(\alpha)} = \frac{(2\alpha)_n}{\Gamma(n + 1)}
{}_2F_1(-n, 2\alpha + n; \alpha + 1/2; (1 - z)/2).
When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.46 in [AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the result is
determined via the relation to the Gauss hypergeometric
function.
alpha : array_like
Parameter
x : array_like
Points at which to evaluate the Gegenbauer polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
C : scalar or ndarray
Values of the Gegenbauer polynomial
See Also
--------
roots_gegenbauer : roots and quadrature weights of Gegenbauer
polynomials
gegenbauer : Gegenbauer polynomial object
hyp2f1 : Gauss hypergeometric function
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_chebyt",
r"""
eval_chebyt(n, x, out=None)
Evaluate Chebyshev polynomial of the first kind at a point.
The Chebyshev polynomials of the first kind can be defined via the
Gauss hypergeometric function :math:`{}_2F_1` as
.. math::
T_n(x) = {}_2F_1(n, -n; 1/2; (1 - x)/2).
When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.47 in [AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the result is
determined via the relation to the Gauss hypergeometric
function.
x : array_like
Points at which to evaluate the Chebyshev polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
T : scalar or ndarray
Values of the Chebyshev polynomial
See Also
--------
roots_chebyt : roots and quadrature weights of Chebyshev
polynomials of the first kind
chebyu : Chebychev polynomial object
eval_chebyu : evaluate Chebyshev polynomials of the second kind
hyp2f1 : Gauss hypergeometric function
numpy.polynomial.chebyshev.Chebyshev : Chebyshev series
Notes
-----
This routine is numerically stable for `x` in ``[-1, 1]`` at least
up to order ``10000``.
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_chebyu",
r"""
eval_chebyu(n, x, out=None)
Evaluate Chebyshev polynomial of the second kind at a point.
The Chebyshev polynomials of the second kind can be defined via
the Gauss hypergeometric function :math:`{}_2F_1` as
.. math::
U_n(x) = (n + 1) {}_2F_1(-n, n + 2; 3/2; (1 - x)/2).
When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.48 in [AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the result is
determined via the relation to the Gauss hypergeometric
function.
x : array_like
Points at which to evaluate the Chebyshev polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
U : scalar or ndarray
Values of the Chebyshev polynomial
See Also
--------
roots_chebyu : roots and quadrature weights of Chebyshev
polynomials of the second kind
chebyu : Chebyshev polynomial object
eval_chebyt : evaluate Chebyshev polynomials of the first kind
hyp2f1 : Gauss hypergeometric function
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_chebys",
r"""
eval_chebys(n, x, out=None)
Evaluate Chebyshev polynomial of the second kind on [-2, 2] at a
point.
These polynomials are defined as
.. math::
S_n(x) = U_n(x/2)
where :math:`U_n` is a Chebyshev polynomial of the second
kind. See 22.5.13 in [AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the result is
determined via the relation to `eval_chebyu`.
x : array_like
Points at which to evaluate the Chebyshev polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
S : scalar or ndarray
Values of the Chebyshev polynomial
See Also
--------
roots_chebys : roots and quadrature weights of Chebyshev
polynomials of the second kind on [-2, 2]
chebys : Chebyshev polynomial object
eval_chebyu : evaluate Chebyshev polynomials of the second kind
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
They are a scaled version of the Chebyshev polynomials of the
second kind.
>>> x = np.linspace(-2, 2, 6)
>>> sc.eval_chebys(3, x)
array([-4. , 0.672, 0.736, -0.736, -0.672, 4. ])
>>> sc.eval_chebyu(3, x / 2)
array([-4. , 0.672, 0.736, -0.736, -0.672, 4. ])
""")
add_newdoc("eval_chebyc",
r"""
eval_chebyc(n, x, out=None)
Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a
point.
These polynomials are defined as
.. math::
C_n(x) = 2 T_n(x/2)
where :math:`T_n` is a Chebyshev polynomial of the first kind. See
22.5.11 in [AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the result is
determined via the relation to `eval_chebyt`.
x : array_like
Points at which to evaluate the Chebyshev polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
C : scalar or ndarray
Values of the Chebyshev polynomial
See Also
--------
roots_chebyc : roots and quadrature weights of Chebyshev
polynomials of the first kind on [-2, 2]
chebyc : Chebyshev polynomial object
numpy.polynomial.chebyshev.Chebyshev : Chebyshev series
eval_chebyt : evaluate Chebycshev polynomials of the first kind
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
They are a scaled version of the Chebyshev polynomials of the
first kind.
>>> x = np.linspace(-2, 2, 6)
>>> sc.eval_chebyc(3, x)
array([-2. , 1.872, 1.136, -1.136, -1.872, 2. ])
>>> 2 * sc.eval_chebyt(3, x / 2)
array([-2. , 1.872, 1.136, -1.136, -1.872, 2. ])
""")
add_newdoc("eval_sh_chebyt",
r"""
eval_sh_chebyt(n, x, out=None)
Evaluate shifted Chebyshev polynomial of the first kind at a
point.
These polynomials are defined as
.. math::
T_n^*(x) = T_n(2x - 1)
where :math:`T_n` is a Chebyshev polynomial of the first kind. See
22.5.14 in [AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the result is
determined via the relation to `eval_chebyt`.
x : array_like
Points at which to evaluate the shifted Chebyshev polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
T : scalar or ndarray
Values of the shifted Chebyshev polynomial
See Also
--------
roots_sh_chebyt : roots and quadrature weights of shifted
Chebyshev polynomials of the first kind
sh_chebyt : shifted Chebyshev polynomial object
eval_chebyt : evaluate Chebyshev polynomials of the first kind
numpy.polynomial.chebyshev.Chebyshev : Chebyshev series
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_sh_chebyu",
r"""
eval_sh_chebyu(n, x, out=None)
Evaluate shifted Chebyshev polynomial of the second kind at a
point.
These polynomials are defined as
.. math::
U_n^*(x) = U_n(2x - 1)
where :math:`U_n` is a Chebyshev polynomial of the first kind. See
22.5.15 in [AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the result is
determined via the relation to `eval_chebyu`.
x : array_like
Points at which to evaluate the shifted Chebyshev polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
U : scalar or ndarray
Values of the shifted Chebyshev polynomial
See Also
--------
roots_sh_chebyu : roots and quadrature weights of shifted
Chebychev polynomials of the second kind
sh_chebyu : shifted Chebyshev polynomial object
eval_chebyu : evaluate Chebyshev polynomials of the second kind
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_legendre",
r"""
eval_legendre(n, x, out=None)
Evaluate Legendre polynomial at a point.
The Legendre polynomials can be defined via the Gauss
hypergeometric function :math:`{}_2F_1` as
.. math::
P_n(x) = {}_2F_1(-n, n + 1; 1; (1 - x)/2).
When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.49 in [AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the result is
determined via the relation to the Gauss hypergeometric
function.
x : array_like
Points at which to evaluate the Legendre polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
P : scalar or ndarray
Values of the Legendre polynomial
See Also
--------
roots_legendre : roots and quadrature weights of Legendre
polynomials
legendre : Legendre polynomial object
hyp2f1 : Gauss hypergeometric function
numpy.polynomial.legendre.Legendre : Legendre series
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
Examples
--------
>>> import numpy as np
>>> from scipy.special import eval_legendre
Evaluate the zero-order Legendre polynomial at x = 0
>>> eval_legendre(0, 0)
1.0
Evaluate the first-order Legendre polynomial between -1 and 1
>>> X = np.linspace(-1, 1, 5) # Domain of Legendre polynomials
>>> eval_legendre(1, X)
array([-1. , -0.5, 0. , 0.5, 1. ])
Evaluate Legendre polynomials of order 0 through 4 at x = 0
>>> N = range(0, 5)
>>> eval_legendre(N, 0)
array([ 1. , 0. , -0.5 , 0. , 0.375])
Plot Legendre polynomials of order 0 through 4
>>> X = np.linspace(-1, 1)
>>> import matplotlib.pyplot as plt
>>> for n in range(0, 5):
... y = eval_legendre(n, X)
... plt.plot(X, y, label=r'$P_{}(x)$'.format(n))
>>> plt.title("Legendre Polynomials")
>>> plt.xlabel("x")
>>> plt.ylabel(r'$P_n(x)$')
>>> plt.legend(loc='lower right')
>>> plt.show()
""")
add_newdoc("eval_sh_legendre",
r"""
eval_sh_legendre(n, x, out=None)
Evaluate shifted Legendre polynomial at a point.
These polynomials are defined as
.. math::
P_n^*(x) = P_n(2x - 1)
where :math:`P_n` is a Legendre polynomial. See 2.2.11 in [AS]_
for details.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the value is
determined via the relation to `eval_legendre`.
x : array_like
Points at which to evaluate the shifted Legendre polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
P : scalar or ndarray
Values of the shifted Legendre polynomial
See Also
--------
roots_sh_legendre : roots and quadrature weights of shifted
Legendre polynomials
sh_legendre : shifted Legendre polynomial object
eval_legendre : evaluate Legendre polynomials
numpy.polynomial.legendre.Legendre : Legendre series
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_genlaguerre",
r"""
eval_genlaguerre(n, alpha, x, out=None)
Evaluate generalized Laguerre polynomial at a point.
The generalized Laguerre polynomials can be defined via the
confluent hypergeometric function :math:`{}_1F_1` as
.. math::
L_n^{(\alpha)}(x) = \binom{n + \alpha}{n}
{}_1F_1(-n, \alpha + 1, x).
When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.54 in [AS]_ for details. The Laguerre
polynomials are the special case where :math:`\alpha = 0`.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer, the result is
determined via the relation to the confluent hypergeometric
function.
alpha : array_like
Parameter; must have ``alpha > -1``
x : array_like
Points at which to evaluate the generalized Laguerre
polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
L : scalar or ndarray
Values of the generalized Laguerre polynomial
See Also
--------
roots_genlaguerre : roots and quadrature weights of generalized
Laguerre polynomials
genlaguerre : generalized Laguerre polynomial object
hyp1f1 : confluent hypergeometric function
eval_laguerre : evaluate Laguerre polynomials
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_laguerre",
r"""
eval_laguerre(n, x, out=None)
Evaluate Laguerre polynomial at a point.
The Laguerre polynomials can be defined via the confluent
hypergeometric function :math:`{}_1F_1` as
.. math::
L_n(x) = {}_1F_1(-n, 1, x).
See 22.5.16 and 22.5.54 in [AS]_ for details. When :math:`n` is an
integer the result is a polynomial of degree :math:`n`.
Parameters
----------
n : array_like
Degree of the polynomial. If not an integer the result is
determined via the relation to the confluent hypergeometric
function.
x : array_like
Points at which to evaluate the Laguerre polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
L : scalar or ndarray
Values of the Laguerre polynomial
See Also
--------
roots_laguerre : roots and quadrature weights of Laguerre
polynomials
laguerre : Laguerre polynomial object
numpy.polynomial.laguerre.Laguerre : Laguerre series
eval_genlaguerre : evaluate generalized Laguerre polynomials
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_hermite",
r"""
eval_hermite(n, x, out=None)
Evaluate physicist's Hermite polynomial at a point.
Defined by
.. math::
H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2};
:math:`H_n` is a polynomial of degree :math:`n`. See 22.11.7 in
[AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial
x : array_like
Points at which to evaluate the Hermite polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
H : scalar or ndarray
Values of the Hermite polynomial
See Also
--------
roots_hermite : roots and quadrature weights of physicist's
Hermite polynomials
hermite : physicist's Hermite polynomial object
numpy.polynomial.hermite.Hermite : Physicist's Hermite series
eval_hermitenorm : evaluate Probabilist's Hermite polynomials
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("eval_hermitenorm",
r"""
eval_hermitenorm(n, x, out=None)
Evaluate probabilist's (normalized) Hermite polynomial at a
point.
Defined by
.. math::
He_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2};
:math:`He_n` is a polynomial of degree :math:`n`. See 22.11.8 in
[AS]_ for details.
Parameters
----------
n : array_like
Degree of the polynomial
x : array_like
Points at which to evaluate the Hermite polynomial
out : ndarray, optional
Optional output array for the function values
Returns
-------
He : scalar or ndarray
Values of the Hermite polynomial
See Also
--------
roots_hermitenorm : roots and quadrature weights of probabilist's
Hermite polynomials
hermitenorm : probabilist's Hermite polynomial object
numpy.polynomial.hermite_e.HermiteE : Probabilist's Hermite series
eval_hermite : evaluate physicist's Hermite polynomials
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
""")
add_newdoc("exp1",
r"""
exp1(z, out=None)
Exponential integral E1.
For complex :math:`z \ne 0` the exponential integral can be defined as
[1]_
.. math::
E_1(z) = \int_z^\infty \frac{e^{-t}}{t} dt,
where the path of the integral does not cross the negative real
axis or pass through the origin.
Parameters
----------
z: array_like
Real or complex argument.
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the exponential integral E1
See Also
--------
expi : exponential integral :math:`Ei`
expn : generalization of :math:`E_1`
Notes
-----
For :math:`x > 0` it is related to the exponential integral
:math:`Ei` (see `expi`) via the relation
.. math::
E_1(x) = -Ei(-x).
References
----------
.. [1] Digital Library of Mathematical Functions, 6.2.1
https://dlmf.nist.gov/6.2#E1
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It has a pole at 0.
>>> sc.exp1(0)
inf
It has a branch cut on the negative real axis.
>>> sc.exp1(-1)
nan
>>> sc.exp1(complex(-1, 0))
(-1.8951178163559368-3.141592653589793j)
>>> sc.exp1(complex(-1, -0.0))
(-1.8951178163559368+3.141592653589793j)
It approaches 0 along the positive real axis.
>>> sc.exp1([1, 10, 100, 1000])
array([2.19383934e-01, 4.15696893e-06, 3.68359776e-46, 0.00000000e+00])
It is related to `expi`.
>>> x = np.array([1, 2, 3, 4])
>>> sc.exp1(x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
>>> -sc.expi(-x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
""")
add_newdoc(
"_scaled_exp1",
"""
_scaled_exp1(x, out=None):
Compute the scaled exponential integral.
This is a private function, subject to change or removal with no
deprecation.
This function computes F(x), where F is the factor remaining in E_1(x)
when exp(-x)/x is factored out. That is,::
E_1(x) = exp(-x)/x * F(x)
or
F(x) = x * exp(x) * E_1(x)
The function is defined for real x >= 0. For x < 0, nan is returned.
F has the properties:
* F(0) = 0
* F(x) is increasing on [0, inf).
* The limit as x goes to infinity of F(x) is 1.
Parameters
----------
x: array_like
The input values. Must be real. The implementation is limited to
double precision floating point, so other types will be cast to
to double precision.
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the scaled exponential integral.
See Also
--------
exp1 : exponential integral E_1
Examples
--------
>>> from scipy.special import _scaled_exp1
>>> _scaled_exp1([0, 0.1, 1, 10, 100])
"""
)
add_newdoc("exp10",
"""
exp10(x, out=None)
Compute ``10**x`` element-wise.
Parameters
----------
x : array_like
`x` must contain real numbers.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
``10**x``, computed element-wise.
Examples
--------
>>> import numpy as np
>>> from scipy.special import exp10
>>> exp10(3)
1000.0
>>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]])
>>> exp10(x)
array([[ 0.1 , 0.31622777, 1. ],
[ 3.16227766, 10. , 31.6227766 ]])
""")
add_newdoc("exp2",
"""
exp2(x, out=None)
Compute ``2**x`` element-wise.
Parameters
----------
x : array_like
`x` must contain real numbers.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
``2**x``, computed element-wise.
Examples
--------
>>> import numpy as np
>>> from scipy.special import exp2
>>> exp2(3)
8.0
>>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]])
>>> exp2(x)
array([[ 0.5 , 0.70710678, 1. ],
[ 1.41421356, 2. , 2.82842712]])
""")
add_newdoc("expi",
r"""
expi(x, out=None)
Exponential integral Ei.
For real :math:`x`, the exponential integral is defined as [1]_
.. math::
Ei(x) = \int_{-\infty}^x \frac{e^t}{t} dt.
For :math:`x > 0` the integral is understood as a Cauchy principal
value.
It is extended to the complex plane by analytic continuation of
the function on the interval :math:`(0, \infty)`. The complex
variant has a branch cut on the negative real axis.
Parameters
----------
x : array_like
Real or complex valued argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the exponential integral
See Also
--------
exp1 : Exponential integral :math:`E_1`
expn : Generalized exponential integral :math:`E_n`
Notes
-----
The exponential integrals :math:`E_1` and :math:`Ei` satisfy the
relation
.. math::
E_1(x) = -Ei(-x)
for :math:`x > 0`.
References
----------
.. [1] Digital Library of Mathematical Functions, 6.2.5
https://dlmf.nist.gov/6.2#E5
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is related to `exp1`.
>>> x = np.array([1, 2, 3, 4])
>>> -sc.expi(-x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
>>> sc.exp1(x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
The complex variant has a branch cut on the negative real axis.
>>> sc.expi(-1 + 1e-12j)
(-0.21938393439552062+3.1415926535894254j)
>>> sc.expi(-1 - 1e-12j)
(-0.21938393439552062-3.1415926535894254j)
As the complex variant approaches the branch cut, the real parts
approach the value of the real variant.
>>> sc.expi(-1)
-0.21938393439552062
The SciPy implementation returns the real variant for complex
values on the branch cut.
>>> sc.expi(complex(-1, 0.0))
(-0.21938393439552062-0j)
>>> sc.expi(complex(-1, -0.0))
(-0.21938393439552062-0j)
""")
add_newdoc('expit',
"""
expit(x, out=None)
Expit (a.k.a. logistic sigmoid) ufunc for ndarrays.
The expit function, also known as the logistic sigmoid function, is
defined as ``expit(x) = 1/(1+exp(-x))``. It is the inverse of the
logit function.
Parameters
----------
x : ndarray
The ndarray to apply expit to element-wise.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
An ndarray of the same shape as x. Its entries
are `expit` of the corresponding entry of x.
See Also
--------
logit
Notes
-----
As a ufunc expit takes a number of optional
keyword arguments. For more information
see `ufuncs <https://docs.scipy.org/doc/numpy/reference/ufuncs.html>`_
.. versionadded:: 0.10.0
Examples
--------
>>> import numpy as np
>>> from scipy.special import expit, logit
>>> expit([-np.inf, -1.5, 0, 1.5, np.inf])
array([ 0. , 0.18242552, 0.5 , 0.81757448, 1. ])
`logit` is the inverse of `expit`:
>>> logit(expit([-2.5, 0, 3.1, 5.0]))
array([-2.5, 0. , 3.1, 5. ])
Plot expit(x) for x in [-6, 6]:
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-6, 6, 121)
>>> y = expit(x)
>>> plt.plot(x, y)
>>> plt.grid()
>>> plt.xlim(-6, 6)
>>> plt.xlabel('x')
>>> plt.title('expit(x)')
>>> plt.show()
""")
add_newdoc("expm1",
"""
expm1(x, out=None)
Compute ``exp(x) - 1``.
When `x` is near zero, ``exp(x)`` is near 1, so the numerical calculation
of ``exp(x) - 1`` can suffer from catastrophic loss of precision.
``expm1(x)`` is implemented to avoid the loss of precision that occurs when
`x` is near zero.
Parameters
----------
x : array_like
`x` must contain real numbers.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
``exp(x) - 1`` computed element-wise.
Examples
--------
>>> import numpy as np
>>> from scipy.special import expm1
>>> expm1(1.0)
1.7182818284590451
>>> expm1([-0.2, -0.1, 0, 0.1, 0.2])
array([-0.18126925, -0.09516258, 0. , 0.10517092, 0.22140276])
The exact value of ``exp(7.5e-13) - 1`` is::
7.5000000000028125000000007031250000001318...*10**-13.
Here is what ``expm1(7.5e-13)`` gives:
>>> expm1(7.5e-13)
7.5000000000028135e-13
Compare that to ``exp(7.5e-13) - 1``, where the subtraction results in
a "catastrophic" loss of precision:
>>> np.exp(7.5e-13) - 1
7.5006667543675576e-13
""")
add_newdoc("expn",
r"""
expn(n, x, out=None)
Generalized exponential integral En.
For integer :math:`n \geq 0` and real :math:`x \geq 0` the
generalized exponential integral is defined as [dlmf]_
.. math::
E_n(x) = x^{n - 1} \int_x^\infty \frac{e^{-t}}{t^n} dt.
Parameters
----------
n : array_like
Non-negative integers
x : array_like
Real argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the generalized exponential integral
See Also
--------
exp1 : special case of :math:`E_n` for :math:`n = 1`
expi : related to :math:`E_n` when :math:`n = 1`
References
----------
.. [dlmf] Digital Library of Mathematical Functions, 8.19.2
https://dlmf.nist.gov/8.19#E2
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
Its domain is nonnegative n and x.
>>> sc.expn(-1, 1.0), sc.expn(1, -1.0)
(nan, nan)
It has a pole at ``x = 0`` for ``n = 1, 2``; for larger ``n`` it
is equal to ``1 / (n - 1)``.
>>> sc.expn([0, 1, 2, 3, 4], 0)
array([ inf, inf, 1. , 0.5 , 0.33333333])
For n equal to 0 it reduces to ``exp(-x) / x``.
>>> x = np.array([1, 2, 3, 4])
>>> sc.expn(0, x)
array([0.36787944, 0.06766764, 0.01659569, 0.00457891])
>>> np.exp(-x) / x
array([0.36787944, 0.06766764, 0.01659569, 0.00457891])
For n equal to 1 it reduces to `exp1`.
>>> sc.expn(1, x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
>>> sc.exp1(x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
""")
add_newdoc("exprel",
r"""
exprel(x, out=None)
Relative error exponential, ``(exp(x) - 1)/x``.
When `x` is near zero, ``exp(x)`` is near 1, so the numerical calculation
of ``exp(x) - 1`` can suffer from catastrophic loss of precision.
``exprel(x)`` is implemented to avoid the loss of precision that occurs when
`x` is near zero.
Parameters
----------
x : ndarray
Input array. `x` must contain real numbers.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
``(exp(x) - 1)/x``, computed element-wise.
See Also
--------
expm1
Notes
-----
.. versionadded:: 0.17.0
Examples
--------
>>> import numpy as np
>>> from scipy.special import exprel
>>> exprel(0.01)
1.0050167084168056
>>> exprel([-0.25, -0.1, 0, 0.1, 0.25])
array([ 0.88479687, 0.95162582, 1. , 1.05170918, 1.13610167])
Compare ``exprel(5e-9)`` to the naive calculation. The exact value
is ``1.00000000250000000416...``.
>>> exprel(5e-9)
1.0000000025
>>> (np.exp(5e-9) - 1)/5e-9
0.99999999392252903
""")
add_newdoc("fdtr",
r"""
fdtr(dfn, dfd, x, out=None)
F cumulative distribution function.
Returns the value of the cumulative distribution function of the
F-distribution, also known as Snedecor's F-distribution or the
Fisher-Snedecor distribution.
The F-distribution with parameters :math:`d_n` and :math:`d_d` is the
distribution of the random variable,
.. math::
X = \frac{U_n/d_n}{U_d/d_d},
where :math:`U_n` and :math:`U_d` are random variables distributed
:math:`\chi^2`, with :math:`d_n` and :math:`d_d` degrees of freedom,
respectively.
Parameters
----------
dfn : array_like
First parameter (positive float).
dfd : array_like
Second parameter (positive float).
x : array_like
Argument (nonnegative float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
y : scalar or ndarray
The CDF of the F-distribution with parameters `dfn` and `dfd` at `x`.
See Also
--------
fdtrc : F distribution survival function
fdtri : F distribution inverse cumulative distribution
scipy.stats.f : F distribution
Notes
-----
The regularized incomplete beta function is used, according to the
formula,
.. math::
F(d_n, d_d; x) = I_{xd_n/(d_d + xd_n)}(d_n/2, d_d/2).
Wrapper for the Cephes [1]_ routine `fdtr`. The F distribution is also
available as `scipy.stats.f`. Calling `fdtr` directly can improve
performance compared to the ``cdf`` method of `scipy.stats.f` (see last
example below).
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``.
>>> import numpy as np
>>> from scipy.special import fdtr
>>> fdtr(1, 2, 1)
0.5773502691896258
Calculate the function at several points by providing a NumPy array for
`x`.
>>> x = np.array([0.5, 2., 3.])
>>> fdtr(1, 2, x)
array([0.4472136 , 0.70710678, 0.77459667])
Plot the function for several parameter sets.
>>> import matplotlib.pyplot as plt
>>> dfn_parameters = [1, 5, 10, 50]
>>> dfd_parameters = [1, 1, 2, 3]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(dfn_parameters, dfd_parameters,
... linestyles))
>>> x = np.linspace(0, 30, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
... dfn, dfd, style = parameter_set
... fdtr_vals = fdtr(dfn, dfd, x)
... ax.plot(x, fdtr_vals, label=rf"$d_n={dfn},\, d_d={dfd}$",
... ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("F distribution cumulative distribution function")
>>> plt.show()
The F distribution is also available as `scipy.stats.f`. Using `fdtr`
directly can be much faster than calling the ``cdf`` method of
`scipy.stats.f`, especially for small arrays or individual values.
To get the same results one must use the following parametrization:
``stats.f(dfn, dfd).cdf(x)=fdtr(dfn, dfd, x)``.
>>> from scipy.stats import f
>>> dfn, dfd = 1, 2
>>> x = 1
>>> fdtr_res = fdtr(dfn, dfd, x) # this will often be faster than below
>>> f_dist_res = f(dfn, dfd).cdf(x)
>>> fdtr_res == f_dist_res # test that results are equal
True
""")
add_newdoc("fdtrc",
r"""
fdtrc(dfn, dfd, x, out=None)
F survival function.
Returns the complemented F-distribution function (the integral of the
density from `x` to infinity).
Parameters
----------
dfn : array_like
First parameter (positive float).
dfd : array_like
Second parameter (positive float).
x : array_like
Argument (nonnegative float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
y : scalar or ndarray
The complemented F-distribution function with parameters `dfn` and
`dfd` at `x`.
See Also
--------
fdtr : F distribution cumulative distribution function
fdtri : F distribution inverse cumulative distribution function
scipy.stats.f : F distribution
Notes
-----
The regularized incomplete beta function is used, according to the
formula,
.. math::
F(d_n, d_d; x) = I_{d_d/(d_d + xd_n)}(d_d/2, d_n/2).
Wrapper for the Cephes [1]_ routine `fdtrc`. The F distribution is also
available as `scipy.stats.f`. Calling `fdtrc` directly can improve
performance compared to the ``sf`` method of `scipy.stats.f` (see last
example below).
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``.
>>> import numpy as np
>>> from scipy.special import fdtrc
>>> fdtrc(1, 2, 1)
0.42264973081037427
Calculate the function at several points by providing a NumPy array for
`x`.
>>> x = np.array([0.5, 2., 3.])
>>> fdtrc(1, 2, x)
array([0.5527864 , 0.29289322, 0.22540333])
Plot the function for several parameter sets.
>>> import matplotlib.pyplot as plt
>>> dfn_parameters = [1, 5, 10, 50]
>>> dfd_parameters = [1, 1, 2, 3]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(dfn_parameters, dfd_parameters,
... linestyles))
>>> x = np.linspace(0, 30, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
... dfn, dfd, style = parameter_set
... fdtrc_vals = fdtrc(dfn, dfd, x)
... ax.plot(x, fdtrc_vals, label=rf"$d_n={dfn},\, d_d={dfd}$",
... ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("F distribution survival function")
>>> plt.show()
The F distribution is also available as `scipy.stats.f`. Using `fdtrc`
directly can be much faster than calling the ``sf`` method of
`scipy.stats.f`, especially for small arrays or individual values.
To get the same results one must use the following parametrization:
``stats.f(dfn, dfd).sf(x)=fdtrc(dfn, dfd, x)``.
>>> from scipy.stats import f
>>> dfn, dfd = 1, 2
>>> x = 1
>>> fdtrc_res = fdtrc(dfn, dfd, x) # this will often be faster than below
>>> f_dist_res = f(dfn, dfd).sf(x)
>>> f_dist_res == fdtrc_res # test that results are equal
True
""")
add_newdoc("fdtri",
r"""
fdtri(dfn, dfd, p, out=None)
The `p`-th quantile of the F-distribution.
This function is the inverse of the F-distribution CDF, `fdtr`, returning
the `x` such that `fdtr(dfn, dfd, x) = p`.
Parameters
----------
dfn : array_like
First parameter (positive float).
dfd : array_like
Second parameter (positive float).
p : array_like
Cumulative probability, in [0, 1].
out : ndarray, optional
Optional output array for the function values
Returns
-------
x : scalar or ndarray
The quantile corresponding to `p`.
See Also
--------
fdtr : F distribution cumulative distribution function
fdtrc : F distribution survival function
scipy.stats.f : F distribution
Notes
-----
The computation is carried out using the relation to the inverse
regularized beta function, :math:`I^{-1}_x(a, b)`. Let
:math:`z = I^{-1}_p(d_d/2, d_n/2).` Then,
.. math::
x = \frac{d_d (1 - z)}{d_n z}.
If `p` is such that :math:`x < 0.5`, the following relation is used
instead for improved stability: let
:math:`z' = I^{-1}_{1 - p}(d_n/2, d_d/2).` Then,
.. math::
x = \frac{d_d z'}{d_n (1 - z')}.
Wrapper for the Cephes [1]_ routine `fdtri`.
The F distribution is also available as `scipy.stats.f`. Calling
`fdtri` directly can improve performance compared to the ``ppf``
method of `scipy.stats.f` (see last example below).
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
`fdtri` represents the inverse of the F distribution CDF which is
available as `fdtr`. Here, we calculate the CDF for ``df1=1``, ``df2=2``
at ``x=3``. `fdtri` then returns ``3`` given the same values for `df1`,
`df2` and the computed CDF value.
>>> import numpy as np
>>> from scipy.special import fdtri, fdtr
>>> df1, df2 = 1, 2
>>> x = 3
>>> cdf_value = fdtr(df1, df2, x)
>>> fdtri(df1, df2, cdf_value)
3.000000000000006
Calculate the function at several points by providing a NumPy array for
`x`.
>>> x = np.array([0.1, 0.4, 0.7])
>>> fdtri(1, 2, x)
array([0.02020202, 0.38095238, 1.92156863])
Plot the function for several parameter sets.
>>> import matplotlib.pyplot as plt
>>> dfn_parameters = [50, 10, 1, 50]
>>> dfd_parameters = [0.5, 1, 1, 5]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(dfn_parameters, dfd_parameters,
... linestyles))
>>> x = np.linspace(0, 1, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
... dfn, dfd, style = parameter_set
... fdtri_vals = fdtri(dfn, dfd, x)
... ax.plot(x, fdtri_vals, label=rf"$d_n={dfn},\, d_d={dfd}$",
... ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> title = "F distribution inverse cumulative distribution function"
>>> ax.set_title(title)
>>> ax.set_ylim(0, 30)
>>> plt.show()
The F distribution is also available as `scipy.stats.f`. Using `fdtri`
directly can be much faster than calling the ``ppf`` method of
`scipy.stats.f`, especially for small arrays or individual values.
To get the same results one must use the following parametrization:
``stats.f(dfn, dfd).ppf(x)=fdtri(dfn, dfd, x)``.
>>> from scipy.stats import f
>>> dfn, dfd = 1, 2
>>> x = 0.7
>>> fdtri_res = fdtri(dfn, dfd, x) # this will often be faster than below
>>> f_dist_res = f(dfn, dfd).ppf(x)
>>> f_dist_res == fdtri_res # test that results are equal
True
""")
add_newdoc("fdtridfd",
"""
fdtridfd(dfn, p, x, out=None)
Inverse to `fdtr` vs dfd
Finds the F density argument dfd such that ``fdtr(dfn, dfd, x) == p``.
Parameters
----------
dfn : array_like
First parameter (positive float).
p : array_like
Cumulative probability, in [0, 1].
x : array_like
Argument (nonnegative float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
dfd : scalar or ndarray
`dfd` such that ``fdtr(dfn, dfd, x) == p``.
See Also
--------
fdtr : F distribution cumulative distribution function
fdtrc : F distribution survival function
fdtri : F distribution quantile function
scipy.stats.f : F distribution
Examples
--------
Compute the F distribution cumulative distribution function for one
parameter set.
>>> from scipy.special import fdtridfd, fdtr
>>> dfn, dfd, x = 10, 5, 2
>>> cdf_value = fdtr(dfn, dfd, x)
>>> cdf_value
0.7700248806501017
Verify that `fdtridfd` recovers the original value for `dfd`:
>>> fdtridfd(dfn, cdf_value, x)
5.0
""")
'''
commented out as fdtridfn seems to have bugs and is not in functions.json
see: https://github.com/scipy/scipy/pull/15622#discussion_r811440983
add_newdoc(
"fdtridfn",
"""
fdtridfn(p, dfd, x, out=None)
Inverse to `fdtr` vs dfn
finds the F density argument dfn such that ``fdtr(dfn, dfd, x) == p``.
Parameters
----------
p : array_like
Cumulative probability, in [0, 1].
dfd : array_like
Second parameter (positive float).
x : array_like
Argument (nonnegative float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
dfn : scalar or ndarray
`dfn` such that ``fdtr(dfn, dfd, x) == p``.
See Also
--------
fdtr, fdtrc, fdtri, fdtridfd
""")
'''
add_newdoc("fresnel",
r"""
fresnel(z, out=None)
Fresnel integrals.
The Fresnel integrals are defined as
.. math::
S(z) &= \int_0^z \sin(\pi t^2 /2) dt \\
C(z) &= \int_0^z \cos(\pi t^2 /2) dt.
See [dlmf]_ for details.
Parameters
----------
z : array_like
Real or complex valued argument
out : 2-tuple of ndarrays, optional
Optional output arrays for the function results
Returns
-------
S, C : 2-tuple of scalar or ndarray
Values of the Fresnel integrals
See Also
--------
fresnel_zeros : zeros of the Fresnel integrals
References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/7.2#iii
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
As z goes to infinity along the real axis, S and C converge to 0.5.
>>> S, C = sc.fresnel([0.1, 1, 10, 100, np.inf])
>>> S
array([0.00052359, 0.43825915, 0.46816998, 0.4968169 , 0.5 ])
>>> C
array([0.09999753, 0.7798934 , 0.49989869, 0.4999999 , 0.5 ])
They are related to the error function `erf`.
>>> z = np.array([1, 2, 3, 4])
>>> zeta = 0.5 * np.sqrt(np.pi) * (1 - 1j) * z
>>> S, C = sc.fresnel(z)
>>> C + 1j*S
array([0.7798934 +0.43825915j, 0.48825341+0.34341568j,
0.60572079+0.496313j , 0.49842603+0.42051575j])
>>> 0.5 * (1 + 1j) * sc.erf(zeta)
array([0.7798934 +0.43825915j, 0.48825341+0.34341568j,
0.60572079+0.496313j , 0.49842603+0.42051575j])
""")
add_newdoc("gamma",
r"""
gamma(z, out=None)
gamma function.
The gamma function is defined as
.. math::
\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt
for :math:`\Re(z) > 0` and is extended to the rest of the complex
plane by analytic continuation. See [dlmf]_ for more details.
Parameters
----------
z : array_like
Real or complex valued argument
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the gamma function
Notes
-----
The gamma function is often referred to as the generalized
factorial since :math:`\Gamma(n + 1) = n!` for natural numbers
:math:`n`. More generally it satisfies the recurrence relation
:math:`\Gamma(z + 1) = z \cdot \Gamma(z)` for complex :math:`z`,
which, combined with the fact that :math:`\Gamma(1) = 1`, implies
the above identity for :math:`z = n`.
References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/5.2#E1
Examples
--------
>>> import numpy as np
>>> from scipy.special import gamma, factorial
>>> gamma([0, 0.5, 1, 5])
array([ inf, 1.77245385, 1. , 24. ])
>>> z = 2.5 + 1j
>>> gamma(z)
(0.77476210455108352+0.70763120437959293j)
>>> gamma(z+1), z*gamma(z) # Recurrence property
((1.2292740569981171+2.5438401155000685j),
(1.2292740569981158+2.5438401155000658j))
>>> gamma(0.5)**2 # gamma(0.5) = sqrt(pi)
3.1415926535897927
Plot gamma(x) for real x
>>> x = np.linspace(-3.5, 5.5, 2251)
>>> y = gamma(x)
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')
>>> k = np.arange(1, 7)
>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,
... label='(x-1)!, x = 1, 2, ...')
>>> plt.xlim(-3.5, 5.5)
>>> plt.ylim(-10, 25)
>>> plt.grid()
>>> plt.xlabel('x')
>>> plt.legend(loc='lower right')
>>> plt.show()
""")
add_newdoc("gammainc",
r"""
gammainc(a, x, out=None)
Regularized lower incomplete gamma function.
It is defined as
.. math::
P(a, x) = \frac{1}{\Gamma(a)} \int_0^x t^{a - 1}e^{-t} dt
for :math:`a > 0` and :math:`x \geq 0`. See [dlmf]_ for details.
Parameters
----------
a : array_like
Positive parameter
x : array_like
Nonnegative argument
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the lower incomplete gamma function
See Also
--------
gammaincc : regularized upper incomplete gamma function
gammaincinv : inverse of the regularized lower incomplete gamma function
gammainccinv : inverse of the regularized upper incomplete gamma function
Notes
-----
The function satisfies the relation ``gammainc(a, x) +
gammaincc(a, x) = 1`` where `gammaincc` is the regularized upper
incomplete gamma function.
The implementation largely follows that of [boost]_.
References
----------
.. [dlmf] NIST Digital Library of Mathematical functions
https://dlmf.nist.gov/8.2#E4
.. [boost] Maddock et. al., "Incomplete Gamma Functions",
https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
Examples
--------
>>> import scipy.special as sc
It is the CDF of the gamma distribution, so it starts at 0 and
monotonically increases to 1.
>>> sc.gammainc(0.5, [0, 1, 10, 100])
array([0. , 0.84270079, 0.99999226, 1. ])
It is equal to one minus the upper incomplete gamma function.
>>> a, x = 0.5, 0.4
>>> sc.gammainc(a, x)
0.6289066304773024
>>> 1 - sc.gammaincc(a, x)
0.6289066304773024
""")
add_newdoc("gammaincc",
r"""
gammaincc(a, x, out=None)
Regularized upper incomplete gamma function.
It is defined as
.. math::
Q(a, x) = \frac{1}{\Gamma(a)} \int_x^\infty t^{a - 1}e^{-t} dt
for :math:`a > 0` and :math:`x \geq 0`. See [dlmf]_ for details.
Parameters
----------
a : array_like
Positive parameter
x : array_like
Nonnegative argument
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the upper incomplete gamma function
See Also
--------
gammainc : regularized lower incomplete gamma function
gammaincinv : inverse of the regularized lower incomplete gamma function
gammainccinv : inverse of the regularized upper incomplete gamma function
Notes
-----
The function satisfies the relation ``gammainc(a, x) +
gammaincc(a, x) = 1`` where `gammainc` is the regularized lower
incomplete gamma function.
The implementation largely follows that of [boost]_.
References
----------
.. [dlmf] NIST Digital Library of Mathematical functions
https://dlmf.nist.gov/8.2#E4
.. [boost] Maddock et. al., "Incomplete Gamma Functions",
https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
Examples
--------
>>> import scipy.special as sc
It is the survival function of the gamma distribution, so it
starts at 1 and monotonically decreases to 0.
>>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])
array([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,
0.00000000e+00])
It is equal to one minus the lower incomplete gamma function.
>>> a, x = 0.5, 0.4
>>> sc.gammaincc(a, x)
0.37109336952269756
>>> 1 - sc.gammainc(a, x)
0.37109336952269756
""")
add_newdoc("gammainccinv",
"""
gammainccinv(a, y, out=None)
Inverse of the regularized upper incomplete gamma function.
Given an input :math:`y` between 0 and 1, returns :math:`x` such
that :math:`y = Q(a, x)`. Here :math:`Q` is the regularized upper
incomplete gamma function; see `gammaincc`. This is well-defined
because the upper incomplete gamma function is monotonic as can
be seen from its definition in [dlmf]_.
Parameters
----------
a : array_like
Positive parameter
y : array_like
Argument between 0 and 1, inclusive
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the inverse of the upper incomplete gamma function
See Also
--------
gammaincc : regularized upper incomplete gamma function
gammainc : regularized lower incomplete gamma function
gammaincinv : inverse of the regularized lower incomplete gamma function
References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/8.2#E4
Examples
--------
>>> import scipy.special as sc
It starts at infinity and monotonically decreases to 0.
>>> sc.gammainccinv(0.5, [0, 0.1, 0.5, 1])
array([ inf, 1.35277173, 0.22746821, 0. ])
It inverts the upper incomplete gamma function.
>>> a, x = 0.5, [0, 0.1, 0.5, 1]
>>> sc.gammaincc(a, sc.gammainccinv(a, x))
array([0. , 0.1, 0.5, 1. ])
>>> a, x = 0.5, [0, 10, 50]
>>> sc.gammainccinv(a, sc.gammaincc(a, x))
array([ 0., 10., 50.])
""")
add_newdoc("gammaincinv",
"""
gammaincinv(a, y, out=None)
Inverse to the regularized lower incomplete gamma function.
Given an input :math:`y` between 0 and 1, returns :math:`x` such
that :math:`y = P(a, x)`. Here :math:`P` is the regularized lower
incomplete gamma function; see `gammainc`. This is well-defined
because the lower incomplete gamma function is monotonic as can be
seen from its definition in [dlmf]_.
Parameters
----------
a : array_like
Positive parameter
y : array_like
Parameter between 0 and 1, inclusive
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the inverse of the lower incomplete gamma function
See Also
--------
gammainc : regularized lower incomplete gamma function
gammaincc : regularized upper incomplete gamma function
gammainccinv : inverse of the regularized upper incomplete gamma function
References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/8.2#E4
Examples
--------
>>> import scipy.special as sc
It starts at 0 and monotonically increases to infinity.
>>> sc.gammaincinv(0.5, [0, 0.1 ,0.5, 1])
array([0. , 0.00789539, 0.22746821, inf])
It inverts the lower incomplete gamma function.
>>> a, x = 0.5, [0, 0.1, 0.5, 1]
>>> sc.gammainc(a, sc.gammaincinv(a, x))
array([0. , 0.1, 0.5, 1. ])
>>> a, x = 0.5, [0, 10, 25]
>>> sc.gammaincinv(a, sc.gammainc(a, x))
array([ 0. , 10. , 25.00001465])
""")
add_newdoc("gammaln",
r"""
gammaln(x, out=None)
Logarithm of the absolute value of the gamma function.
Defined as
.. math::
\ln(\lvert\Gamma(x)\rvert)
where :math:`\Gamma` is the gamma function. For more details on
the gamma function, see [dlmf]_.
Parameters
----------
x : array_like
Real argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the log of the absolute value of gamma
See Also
--------
gammasgn : sign of the gamma function
loggamma : principal branch of the logarithm of the gamma function
Notes
-----
It is the same function as the Python standard library function
:func:`math.lgamma`.
When used in conjunction with `gammasgn`, this function is useful
for working in logspace on the real axis without having to deal
with complex numbers via the relation ``exp(gammaln(x)) =
gammasgn(x) * gamma(x)``.
For complex-valued log-gamma, use `loggamma` instead of `gammaln`.
References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/5
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It has two positive zeros.
>>> sc.gammaln([1, 2])
array([0., 0.])
It has poles at nonpositive integers.
>>> sc.gammaln([0, -1, -2, -3, -4])
array([inf, inf, inf, inf, inf])
It asymptotically approaches ``x * log(x)`` (Stirling's formula).
>>> x = np.array([1e10, 1e20, 1e40, 1e80])
>>> sc.gammaln(x)
array([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])
>>> x * np.log(x)
array([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])
""")
add_newdoc("gammasgn",
r"""
gammasgn(x, out=None)
Sign of the gamma function.
It is defined as
.. math::
\text{gammasgn}(x) =
\begin{cases}
+1 & \Gamma(x) > 0 \\
-1 & \Gamma(x) < 0
\end{cases}
where :math:`\Gamma` is the gamma function; see `gamma`. This
definition is complete since the gamma function is never zero;
see the discussion after [dlmf]_.
Parameters
----------
x : array_like
Real argument
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Sign of the gamma function
See Also
--------
gamma : the gamma function
gammaln : log of the absolute value of the gamma function
loggamma : analytic continuation of the log of the gamma function
Notes
-----
The gamma function can be computed as ``gammasgn(x) *
np.exp(gammaln(x))``.
References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/5.2#E1
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is 1 for `x > 0`.
>>> sc.gammasgn([1, 2, 3, 4])
array([1., 1., 1., 1.])
It alternates between -1 and 1 for negative integers.
>>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])
array([-1., 1., -1., 1.])
It can be used to compute the gamma function.
>>> x = [1.5, 0.5, -0.5, -1.5]
>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))
array([ 0.88622693, 1.77245385, -3.5449077 , 2.3632718 ])
>>> sc.gamma(x)
array([ 0.88622693, 1.77245385, -3.5449077 , 2.3632718 ])
""")
add_newdoc("gdtr",
r"""
gdtr(a, b, x, out=None)
Gamma distribution cumulative distribution function.
Returns the integral from zero to `x` of the gamma probability density
function,
.. math::
F = \int_0^x \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt,
where :math:`\Gamma` is the gamma function.
Parameters
----------
a : array_like
The rate parameter of the gamma distribution, sometimes denoted
:math:`\beta` (float). It is also the reciprocal of the scale
parameter :math:`\theta`.
b : array_like
The shape parameter of the gamma distribution, sometimes denoted
:math:`\alpha` (float).
x : array_like
The quantile (upper limit of integration; float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
F : scalar or ndarray
The CDF of the gamma distribution with parameters `a` and `b`
evaluated at `x`.
See Also
--------
gdtrc : 1 - CDF of the gamma distribution.
scipy.stats.gamma: Gamma distribution
Notes
-----
The evaluation is carried out using the relation to the incomplete gamma
integral (regularized gamma function).
Wrapper for the Cephes [1]_ routine `gdtr`. Calling `gdtr` directly can
improve performance compared to the ``cdf`` method of `scipy.stats.gamma`
(see last example below).
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Compute the function for ``a=1``, ``b=2`` at ``x=5``.
>>> import numpy as np
>>> from scipy.special import gdtr
>>> import matplotlib.pyplot as plt
>>> gdtr(1., 2., 5.)
0.9595723180054873
Compute the function for ``a=1`` and ``b=2`` at several points by
providing a NumPy array for `x`.
>>> xvalues = np.array([1., 2., 3., 4])
>>> gdtr(1., 1., xvalues)
array([0.63212056, 0.86466472, 0.95021293, 0.98168436])
`gdtr` can evaluate different parameter sets by providing arrays with
broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the
function for three different `a` at four positions `x` and ``b=3``,
resulting in a 3x4 array.
>>> a = np.array([[0.5], [1.5], [2.5]])
>>> x = np.array([1., 2., 3., 4])
>>> a.shape, x.shape
((3, 1), (4,))
>>> gdtr(a, 3., x)
array([[0.01438768, 0.0803014 , 0.19115317, 0.32332358],
[0.19115317, 0.57680992, 0.82642193, 0.9380312 ],
[0.45618688, 0.87534798, 0.97974328, 0.9972306 ]])
Plot the function for four different parameter sets.
>>> a_parameters = [0.3, 1, 2, 6]
>>> b_parameters = [2, 10, 15, 20]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(a_parameters, b_parameters, linestyles))
>>> x = np.linspace(0, 30, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
... a, b, style = parameter_set
... gdtr_vals = gdtr(a, b, x)
... ax.plot(x, gdtr_vals, label=fr"$a= {a},\, b={b}$", ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("Gamma distribution cumulative distribution function")
>>> plt.show()
The gamma distribution is also available as `scipy.stats.gamma`. Using
`gdtr` directly can be much faster than calling the ``cdf`` method of
`scipy.stats.gamma`, especially for small arrays or individual values.
To get the same results one must use the following parametrization:
``stats.gamma(b, scale=1/a).cdf(x)=gdtr(a, b, x)``.
>>> from scipy.stats import gamma
>>> a = 2.
>>> b = 3
>>> x = 1.
>>> gdtr_result = gdtr(a, b, x) # this will often be faster than below
>>> gamma_dist_result = gamma(b, scale=1/a).cdf(x)
>>> gdtr_result == gamma_dist_result # test that results are equal
True
""")
add_newdoc("gdtrc",
r"""
gdtrc(a, b, x, out=None)
Gamma distribution survival function.
Integral from `x` to infinity of the gamma probability density function,
.. math::
F = \int_x^\infty \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt,
where :math:`\Gamma` is the gamma function.
Parameters
----------
a : array_like
The rate parameter of the gamma distribution, sometimes denoted
:math:`\beta` (float). It is also the reciprocal of the scale
parameter :math:`\theta`.
b : array_like
The shape parameter of the gamma distribution, sometimes denoted
:math:`\alpha` (float).
x : array_like
The quantile (lower limit of integration; float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
F : scalar or ndarray
The survival function of the gamma distribution with parameters `a`
and `b` evaluated at `x`.
See Also
--------
gdtr: Gamma distribution cumulative distribution function
scipy.stats.gamma: Gamma distribution
gdtrix
Notes
-----
The evaluation is carried out using the relation to the incomplete gamma
integral (regularized gamma function).
Wrapper for the Cephes [1]_ routine `gdtrc`. Calling `gdtrc` directly can
improve performance compared to the ``sf`` method of `scipy.stats.gamma`
(see last example below).
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Compute the function for ``a=1`` and ``b=2`` at ``x=5``.
>>> import numpy as np
>>> from scipy.special import gdtrc
>>> import matplotlib.pyplot as plt
>>> gdtrc(1., 2., 5.)
0.04042768199451279
Compute the function for ``a=1``, ``b=2`` at several points by providing
a NumPy array for `x`.
>>> xvalues = np.array([1., 2., 3., 4])
>>> gdtrc(1., 1., xvalues)
array([0.36787944, 0.13533528, 0.04978707, 0.01831564])
`gdtrc` can evaluate different parameter sets by providing arrays with
broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the
function for three different `a` at four positions `x` and ``b=3``,
resulting in a 3x4 array.
>>> a = np.array([[0.5], [1.5], [2.5]])
>>> x = np.array([1., 2., 3., 4])
>>> a.shape, x.shape
((3, 1), (4,))
>>> gdtrc(a, 3., x)
array([[0.98561232, 0.9196986 , 0.80884683, 0.67667642],
[0.80884683, 0.42319008, 0.17357807, 0.0619688 ],
[0.54381312, 0.12465202, 0.02025672, 0.0027694 ]])
Plot the function for four different parameter sets.
>>> a_parameters = [0.3, 1, 2, 6]
>>> b_parameters = [2, 10, 15, 20]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(a_parameters, b_parameters, linestyles))
>>> x = np.linspace(0, 30, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
... a, b, style = parameter_set
... gdtrc_vals = gdtrc(a, b, x)
... ax.plot(x, gdtrc_vals, label=fr"$a= {a},\, b={b}$", ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("Gamma distribution survival function")
>>> plt.show()
The gamma distribution is also available as `scipy.stats.gamma`.
Using `gdtrc` directly can be much faster than calling the ``sf`` method
of `scipy.stats.gamma`, especially for small arrays or individual
values. To get the same results one must use the following parametrization:
``stats.gamma(b, scale=1/a).sf(x)=gdtrc(a, b, x)``.
>>> from scipy.stats import gamma
>>> a = 2
>>> b = 3
>>> x = 1.
>>> gdtrc_result = gdtrc(a, b, x) # this will often be faster than below
>>> gamma_dist_result = gamma(b, scale=1/a).sf(x)
>>> gdtrc_result == gamma_dist_result # test that results are equal
True
""")
add_newdoc("gdtria",
"""
gdtria(p, b, x, out=None)
Inverse of `gdtr` vs a.
Returns the inverse with respect to the parameter `a` of ``p =
gdtr(a, b, x)``, the cumulative distribution function of the gamma
distribution.
Parameters
----------
p : array_like
Probability values.
b : array_like
`b` parameter values of `gdtr(a, b, x)`. `b` is the "shape" parameter
of the gamma distribution.
x : array_like
Nonnegative real values, from the domain of the gamma distribution.
out : ndarray, optional
If a fourth argument is given, it must be a numpy.ndarray whose size
matches the broadcast result of `a`, `b` and `x`. `out` is then the
array returned by the function.
Returns
-------
a : scalar or ndarray
Values of the `a` parameter such that `p = gdtr(a, b, x)`. `1/a`
is the "scale" parameter of the gamma distribution.
See Also
--------
gdtr : CDF of the gamma distribution.
gdtrib : Inverse with respect to `b` of `gdtr(a, b, x)`.
gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`.
Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`.
The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `a` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `a`.
References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
Computation of the incomplete gamma function ratios and their
inverse. ACM Trans. Math. Softw. 12 (1986), 377-393.
Examples
--------
First evaluate `gdtr`.
>>> from scipy.special import gdtr, gdtria
>>> p = gdtr(1.2, 3.4, 5.6)
>>> print(p)
0.94378087442
Verify the inverse.
>>> gdtria(p, 3.4, 5.6)
1.2
""")
add_newdoc("gdtrib",
"""
gdtrib(a, p, x, out=None)
Inverse of `gdtr` vs b.
Returns the inverse with respect to the parameter `b` of ``p =
gdtr(a, b, x)``, the cumulative distribution function of the gamma
distribution.
Parameters
----------
a : array_like
`a` parameter values of `gdtr(a, b, x)`. `1/a` is the "scale"
parameter of the gamma distribution.
p : array_like
Probability values.
x : array_like
Nonnegative real values, from the domain of the gamma distribution.
out : ndarray, optional
If a fourth argument is given, it must be a numpy.ndarray whose size
matches the broadcast result of `a`, `b` and `x`. `out` is then the
array returned by the function.
Returns
-------
b : scalar or ndarray
Values of the `b` parameter such that `p = gdtr(a, b, x)`. `b` is
the "shape" parameter of the gamma distribution.
See Also
--------
gdtr : CDF of the gamma distribution.
gdtria : Inverse with respect to `a` of `gdtr(a, b, x)`.
gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`.
Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`.
The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `b` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `b`.
References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
Computation of the incomplete gamma function ratios and their
inverse. ACM Trans. Math. Softw. 12 (1986), 377-393.
Examples
--------
First evaluate `gdtr`.
>>> from scipy.special import gdtr, gdtrib
>>> p = gdtr(1.2, 3.4, 5.6)
>>> print(p)
0.94378087442
Verify the inverse.
>>> gdtrib(1.2, p, 5.6)
3.3999999999723882
""")
add_newdoc("gdtrix",
"""
gdtrix(a, b, p, out=None)
Inverse of `gdtr` vs x.
Returns the inverse with respect to the parameter `x` of ``p =
gdtr(a, b, x)``, the cumulative distribution function of the gamma
distribution. This is also known as the pth quantile of the
distribution.
Parameters
----------
a : array_like
`a` parameter values of `gdtr(a, b, x)`. `1/a` is the "scale"
parameter of the gamma distribution.
b : array_like
`b` parameter values of `gdtr(a, b, x)`. `b` is the "shape" parameter
of the gamma distribution.
p : array_like
Probability values.
out : ndarray, optional
If a fourth argument is given, it must be a numpy.ndarray whose size
matches the broadcast result of `a`, `b` and `x`. `out` is then the
array returned by the function.
Returns
-------
x : scalar or ndarray
Values of the `x` parameter such that `p = gdtr(a, b, x)`.
See Also
--------
gdtr : CDF of the gamma distribution.
gdtria : Inverse with respect to `a` of `gdtr(a, b, x)`.
gdtrib : Inverse with respect to `b` of `gdtr(a, b, x)`.
Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`.
The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `x` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `x`.
References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
Computation of the incomplete gamma function ratios and their
inverse. ACM Trans. Math. Softw. 12 (1986), 377-393.
Examples
--------
First evaluate `gdtr`.
>>> from scipy.special import gdtr, gdtrix
>>> p = gdtr(1.2, 3.4, 5.6)
>>> print(p)
0.94378087442
Verify the inverse.
>>> gdtrix(1.2, 3.4, p)
5.5999999999999996
""")
add_newdoc("hankel1",
r"""
hankel1(v, z, out=None)
Hankel function of the first kind
Parameters
----------
v : array_like
Order (float).
z : array_like
Argument (float or complex).
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the Hankel function of the first kind.
See Also
--------
hankel1e : ndarray
This function with leading exponential behavior stripped off.
Notes
-----
A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the
computation using the relation,
.. math:: H^{(1)}_v(z) =
\frac{2}{\imath\pi} \exp(-\imath \pi v/2) K_v(z \exp(-\imath\pi/2))
where :math:`K_v` is the modified Bessel function of the second kind.
For negative orders, the relation
.. math:: H^{(1)}_{-v}(z) = H^{(1)}_v(z) \exp(\imath\pi v)
is used.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
""")
add_newdoc("hankel1e",
r"""
hankel1e(v, z, out=None)
Exponentially scaled Hankel function of the first kind
Defined as::
hankel1e(v, z) = hankel1(v, z) * exp(-1j * z)
Parameters
----------
v : array_like
Order (float).
z : array_like
Argument (float or complex).
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the exponentially scaled Hankel function.
Notes
-----
A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the
computation using the relation,
.. math:: H^{(1)}_v(z) =
\frac{2}{\imath\pi} \exp(-\imath \pi v/2) K_v(z \exp(-\imath\pi/2))
where :math:`K_v` is the modified Bessel function of the second kind.
For negative orders, the relation
.. math:: H^{(1)}_{-v}(z) = H^{(1)}_v(z) \exp(\imath\pi v)
is used.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
""")
add_newdoc("hankel2",
r"""
hankel2(v, z, out=None)
Hankel function of the second kind
Parameters
----------
v : array_like
Order (float).
z : array_like
Argument (float or complex).
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the Hankel function of the second kind.
See Also
--------
hankel2e : this function with leading exponential behavior stripped off.
Notes
-----
A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the
computation using the relation,
.. math:: H^{(2)}_v(z) =
-\frac{2}{\imath\pi} \exp(\imath \pi v/2) K_v(z \exp(\imath\pi/2))
where :math:`K_v` is the modified Bessel function of the second kind.
For negative orders, the relation
.. math:: H^{(2)}_{-v}(z) = H^{(2)}_v(z) \exp(-\imath\pi v)
is used.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
""")
add_newdoc("hankel2e",
r"""
hankel2e(v, z, out=None)
Exponentially scaled Hankel function of the second kind
Defined as::
hankel2e(v, z) = hankel2(v, z) * exp(1j * z)
Parameters
----------
v : array_like
Order (float).
z : array_like
Argument (float or complex).
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the exponentially scaled Hankel function of the second kind.
Notes
-----
A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the
computation using the relation,
.. math:: H^{(2)}_v(z) = -\frac{2}{\imath\pi}
\exp(\frac{\imath \pi v}{2}) K_v(z exp(\frac{\imath\pi}{2}))
where :math:`K_v` is the modified Bessel function of the second kind.
For negative orders, the relation
.. math:: H^{(2)}_{-v}(z) = H^{(2)}_v(z) \exp(-\imath\pi v)
is used.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
""")
add_newdoc("huber",
r"""
huber(delta, r, out=None)
Huber loss function.
.. math:: \text{huber}(\delta, r) = \begin{cases} \infty & \delta < 0 \\
\frac{1}{2}r^2 & 0 \le \delta, | r | \le \delta \\
\delta ( |r| - \frac{1}{2}\delta ) & \text{otherwise} \end{cases}
Parameters
----------
delta : ndarray
Input array, indicating the quadratic vs. linear loss changepoint.
r : ndarray
Input array, possibly representing residuals.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
The computed Huber loss function values.
See Also
--------
pseudo_huber : smooth approximation of this function
Notes
-----
`huber` is useful as a loss function in robust statistics or machine
learning to reduce the influence of outliers as compared to the common
squared error loss, residuals with a magnitude higher than `delta` are
not squared [1]_.
Typically, `r` represents residuals, the difference
between a model prediction and data. Then, for :math:`|r|\leq\delta`,
`huber` resembles the squared error and for :math:`|r|>\delta` the
absolute error. This way, the Huber loss often achieves
a fast convergence in model fitting for small residuals like the squared
error loss function and still reduces the influence of outliers
(:math:`|r|>\delta`) like the absolute error loss. As :math:`\delta` is
the cutoff between squared and absolute error regimes, it has
to be tuned carefully for each problem. `huber` is also
convex, making it suitable for gradient based optimization.
.. versionadded:: 0.15.0
References
----------
.. [1] Peter Huber. "Robust Estimation of a Location Parameter",
1964. Annals of Statistics. 53 (1): 73 - 101.
Examples
--------
Import all necessary modules.
>>> import numpy as np
>>> from scipy.special import huber
>>> import matplotlib.pyplot as plt
Compute the function for ``delta=1`` at ``r=2``
>>> huber(1., 2.)
1.5
Compute the function for different `delta` by providing a NumPy array or
list for `delta`.
>>> huber([1., 3., 5.], 4.)
array([3.5, 7.5, 8. ])
Compute the function at different points by providing a NumPy array or
list for `r`.
>>> huber(2., np.array([1., 1.5, 3.]))
array([0.5 , 1.125, 4. ])
The function can be calculated for different `delta` and `r` by
providing arrays for both with compatible shapes for broadcasting.
>>> r = np.array([1., 2.5, 8., 10.])
>>> deltas = np.array([[1.], [5.], [9.]])
>>> print(r.shape, deltas.shape)
(4,) (3, 1)
>>> huber(deltas, r)
array([[ 0.5 , 2. , 7.5 , 9.5 ],
[ 0.5 , 3.125, 27.5 , 37.5 ],
[ 0.5 , 3.125, 32. , 49.5 ]])
Plot the function for different `delta`.
>>> x = np.linspace(-4, 4, 500)
>>> deltas = [1, 2, 3]
>>> linestyles = ["dashed", "dotted", "dashdot"]
>>> fig, ax = plt.subplots()
>>> combined_plot_parameters = list(zip(deltas, linestyles))
>>> for delta, style in combined_plot_parameters:
... ax.plot(x, huber(delta, x), label=fr"$\delta={delta}$", ls=style)
>>> ax.legend(loc="upper center")
>>> ax.set_xlabel("$x$")
>>> ax.set_title(r"Huber loss function $h_{\delta}(x)$")
>>> ax.set_xlim(-4, 4)
>>> ax.set_ylim(0, 8)
>>> plt.show()
""")
add_newdoc("hyp0f1",
r"""
hyp0f1(v, z, out=None)
Confluent hypergeometric limit function 0F1.
Parameters
----------
v : array_like
Real-valued parameter
z : array_like
Real- or complex-valued argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The confluent hypergeometric limit function
Notes
-----
This function is defined as:
.. math:: _0F_1(v, z) = \sum_{k=0}^{\infty}\frac{z^k}{(v)_k k!}.
It's also the limit as :math:`q \to \infty` of :math:`_1F_1(q; v; z/q)`,
and satisfies the differential equation :math:`f''(z) + vf'(z) =
f(z)`. See [1]_ for more information.
References
----------
.. [1] Wolfram MathWorld, "Confluent Hypergeometric Limit Function",
http://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is one when `z` is zero.
>>> sc.hyp0f1(1, 0)
1.0
It is the limit of the confluent hypergeometric function as `q`
goes to infinity.
>>> q = np.array([1, 10, 100, 1000])
>>> v = 1
>>> z = 1
>>> sc.hyp1f1(q, v, z / q)
array([2.71828183, 2.31481985, 2.28303778, 2.27992985])
>>> sc.hyp0f1(v, z)
2.2795853023360673
It is related to Bessel functions.
>>> n = 1
>>> x = np.linspace(0, 1, 5)
>>> sc.jv(n, x)
array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])
>>> (0.5 * x)**n / sc.factorial(n) * sc.hyp0f1(n + 1, -0.25 * x**2)
array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])
""")
add_newdoc("hyp1f1",
r"""
hyp1f1(a, b, x, out=None)
Confluent hypergeometric function 1F1.
The confluent hypergeometric function is defined by the series
.. math::
{}_1F_1(a; b; x) = \sum_{k = 0}^\infty \frac{(a)_k}{(b)_k k!} x^k.
See [dlmf]_ for more details. Here :math:`(\cdot)_k` is the
Pochhammer symbol; see `poch`.
Parameters
----------
a, b : array_like
Real parameters
x : array_like
Real or complex argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the confluent hypergeometric function
See Also
--------
hyperu : another confluent hypergeometric function
hyp0f1 : confluent hypergeometric limit function
hyp2f1 : Gaussian hypergeometric function
References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/13.2#E2
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is one when `x` is zero:
>>> sc.hyp1f1(0.5, 0.5, 0)
1.0
It is singular when `b` is a nonpositive integer.
>>> sc.hyp1f1(0.5, -1, 0)
inf
It is a polynomial when `a` is a nonpositive integer.
>>> a, b, x = -1, 0.5, np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.hyp1f1(a, b, x)
array([-1., -3., -5., -7.])
>>> 1 + (a / b) * x
array([-1., -3., -5., -7.])
It reduces to the exponential function when `a = b`.
>>> sc.hyp1f1(2, 2, [1, 2, 3, 4])
array([ 2.71828183, 7.3890561 , 20.08553692, 54.59815003])
>>> np.exp([1, 2, 3, 4])
array([ 2.71828183, 7.3890561 , 20.08553692, 54.59815003])
""")
add_newdoc("hyp2f1",
r"""
hyp2f1(a, b, c, z, out=None)
Gauss hypergeometric function 2F1(a, b; c; z)
Parameters
----------
a, b, c : array_like
Arguments, should be real-valued.
z : array_like
Argument, real or complex.
out : ndarray, optional
Optional output array for the function values
Returns
-------
hyp2f1 : scalar or ndarray
The values of the gaussian hypergeometric function.
See Also
--------
hyp0f1 : confluent hypergeometric limit function.
hyp1f1 : Kummer's (confluent hypergeometric) function.
Notes
-----
This function is defined for :math:`|z| < 1` as
.. math::
\mathrm{hyp2f1}(a, b, c, z) = \sum_{n=0}^\infty
\frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!},
and defined on the rest of the complex z-plane by analytic
continuation [1]_.
Here :math:`(\cdot)_n` is the Pochhammer symbol; see `poch`. When
:math:`n` is an integer the result is a polynomial of degree :math:`n`.
The implementation for complex values of ``z`` is described in [2]_,
except for ``z`` in the region defined by
.. math::
0.9 <= \left|z\right| < 1.1,
\left|1 - z\right| >= 0.9,
\mathrm{real}(z) >= 0
in which the implementation follows [4]_.
References
----------
.. [1] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/15.2
.. [2] S. Zhang and J.M. Jin, "Computation of Special Functions", Wiley 1996
.. [3] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
.. [4] J.L. Lopez and N.M. Temme, "New series expansions of the Gauss
hypergeometric function", Adv Comput Math 39, 349-365 (2013).
https://doi.org/10.1007/s10444-012-9283-y
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It has poles when `c` is a negative integer.
>>> sc.hyp2f1(1, 1, -2, 1)
inf
It is a polynomial when `a` or `b` is a negative integer.
>>> a, b, c = -1, 1, 1.5
>>> z = np.linspace(0, 1, 5)
>>> sc.hyp2f1(a, b, c, z)
array([1. , 0.83333333, 0.66666667, 0.5 , 0.33333333])
>>> 1 + a * b * z / c
array([1. , 0.83333333, 0.66666667, 0.5 , 0.33333333])
It is symmetric in `a` and `b`.
>>> a = np.linspace(0, 1, 5)
>>> b = np.linspace(0, 1, 5)
>>> sc.hyp2f1(a, b, 1, 0.5)
array([1. , 1.03997334, 1.1803406 , 1.47074441, 2. ])
>>> sc.hyp2f1(b, a, 1, 0.5)
array([1. , 1.03997334, 1.1803406 , 1.47074441, 2. ])
It contains many other functions as special cases.
>>> z = 0.5
>>> sc.hyp2f1(1, 1, 2, z)
1.3862943611198901
>>> -np.log(1 - z) / z
1.3862943611198906
>>> sc.hyp2f1(0.5, 1, 1.5, z**2)
1.098612288668109
>>> np.log((1 + z) / (1 - z)) / (2 * z)
1.0986122886681098
>>> sc.hyp2f1(0.5, 1, 1.5, -z**2)
0.9272952180016117
>>> np.arctan(z) / z
0.9272952180016122
""")
add_newdoc("hyperu",
r"""
hyperu(a, b, x, out=None)
Confluent hypergeometric function U
It is defined as the solution to the equation
.. math::
x \frac{d^2w}{dx^2} + (b - x) \frac{dw}{dx} - aw = 0
which satisfies the property
.. math::
U(a, b, x) \sim x^{-a}
as :math:`x \to \infty`. See [dlmf]_ for more details.
Parameters
----------
a, b : array_like
Real-valued parameters
x : array_like
Real-valued argument
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of `U`
References
----------
.. [dlmf] NIST Digital Library of Mathematics Functions
https://dlmf.nist.gov/13.2#E6
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It has a branch cut along the negative `x` axis.
>>> x = np.linspace(-0.1, -10, 5)
>>> sc.hyperu(1, 1, x)
array([nan, nan, nan, nan, nan])
It approaches zero as `x` goes to infinity.
>>> x = np.array([1, 10, 100])
>>> sc.hyperu(1, 1, x)
array([0.59634736, 0.09156333, 0.00990194])
It satisfies Kummer's transformation.
>>> a, b, x = 2, 1, 1
>>> sc.hyperu(a, b, x)
0.1926947246463881
>>> x**(1 - b) * sc.hyperu(a - b + 1, 2 - b, x)
0.1926947246463881
""")
add_newdoc("i0",
r"""
i0(x, out=None)
Modified Bessel function of order 0.
Defined as,
.. math::
I_0(x) = \sum_{k=0}^\infty \frac{(x^2/4)^k}{(k!)^2} = J_0(\imath x),
where :math:`J_0` is the Bessel function of the first kind of order 0.
Parameters
----------
x : array_like
Argument (float)
out : ndarray, optional
Optional output array for the function values
Returns
-------
I : scalar or ndarray
Value of the modified Bessel function of order 0 at `x`.
See Also
--------
iv: Modified Bessel function of any order
i0e: Exponentially scaled modified Bessel function of order 0
Notes
-----
The range is partitioned into the two intervals [0, 8] and (8, infinity).
Chebyshev polynomial expansions are employed in each interval.
This function is a wrapper for the Cephes [1]_ routine `i0`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function at one point:
>>> from scipy.special import i0
>>> i0(1.)
1.2660658777520082
Calculate at several points:
>>> import numpy as np
>>> i0(np.array([-2., 0., 3.5]))
array([2.2795853 , 1. , 7.37820343])
Plot the function from -10 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i0(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("i0e",
"""
i0e(x, out=None)
Exponentially scaled modified Bessel function of order 0.
Defined as::
i0e(x) = exp(-abs(x)) * i0(x).
Parameters
----------
x : array_like
Argument (float)
out : ndarray, optional
Optional output array for the function values
Returns
-------
I : scalar or ndarray
Value of the exponentially scaled modified Bessel function of order 0
at `x`.
See Also
--------
iv: Modified Bessel function of the first kind
i0: Modified Bessel function of order 0
Notes
-----
The range is partitioned into the two intervals [0, 8] and (8, infinity).
Chebyshev polynomial expansions are employed in each interval. The
polynomial expansions used are the same as those in `i0`, but
they are not multiplied by the dominant exponential factor.
This function is a wrapper for the Cephes [1]_ routine `i0e`. `i0e`
is useful for large arguments `x`: for these, `i0` quickly overflows.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
In the following example `i0` returns infinity whereas `i0e` still returns
a finite number.
>>> from scipy.special import i0, i0e
>>> i0(1000.), i0e(1000.)
(inf, 0.012617240455891257)
Calculate the function at several points by providing a NumPy array or
list for `x`:
>>> import numpy as np
>>> i0e(np.array([-2., 0., 3.]))
array([0.30850832, 1. , 0.24300035])
Plot the function from -10 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i0e(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("i1",
r"""
i1(x, out=None)
Modified Bessel function of order 1.
Defined as,
.. math::
I_1(x) = \frac{1}{2}x \sum_{k=0}^\infty \frac{(x^2/4)^k}{k! (k + 1)!}
= -\imath J_1(\imath x),
where :math:`J_1` is the Bessel function of the first kind of order 1.
Parameters
----------
x : array_like
Argument (float)
out : ndarray, optional
Optional output array for the function values
Returns
-------
I : scalar or ndarray
Value of the modified Bessel function of order 1 at `x`.
See Also
--------
iv: Modified Bessel function of the first kind
i1e: Exponentially scaled modified Bessel function of order 1
Notes
-----
The range is partitioned into the two intervals [0, 8] and (8, infinity).
Chebyshev polynomial expansions are employed in each interval.
This function is a wrapper for the Cephes [1]_ routine `i1`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function at one point:
>>> from scipy.special import i1
>>> i1(1.)
0.5651591039924851
Calculate the function at several points:
>>> import numpy as np
>>> i1(np.array([-2., 0., 6.]))
array([-1.59063685, 0. , 61.34193678])
Plot the function between -10 and 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i1(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("i1e",
"""
i1e(x, out=None)
Exponentially scaled modified Bessel function of order 1.
Defined as::
i1e(x) = exp(-abs(x)) * i1(x)
Parameters
----------
x : array_like
Argument (float)
out : ndarray, optional
Optional output array for the function values
Returns
-------
I : scalar or ndarray
Value of the exponentially scaled modified Bessel function of order 1
at `x`.
See Also
--------
iv: Modified Bessel function of the first kind
i1: Modified Bessel function of order 1
Notes
-----
The range is partitioned into the two intervals [0, 8] and (8, infinity).
Chebyshev polynomial expansions are employed in each interval. The
polynomial expansions used are the same as those in `i1`, but
they are not multiplied by the dominant exponential factor.
This function is a wrapper for the Cephes [1]_ routine `i1e`. `i1e`
is useful for large arguments `x`: for these, `i1` quickly overflows.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
In the following example `i1` returns infinity whereas `i1e` still returns
a finite number.
>>> from scipy.special import i1, i1e
>>> i1(1000.), i1e(1000.)
(inf, 0.01261093025692863)
Calculate the function at several points by providing a NumPy array or
list for `x`:
>>> import numpy as np
>>> i1e(np.array([-2., 0., 6.]))
array([-0.21526929, 0. , 0.15205146])
Plot the function between -10 and 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i1e(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("_igam_fac",
"""
Internal function, do not use.
""")
add_newdoc("it2i0k0",
r"""
it2i0k0(x, out=None)
Integrals related to modified Bessel functions of order 0.
Computes the integrals
.. math::
\int_0^x \frac{I_0(t) - 1}{t} dt \\
\int_x^\infty \frac{K_0(t)}{t} dt.
Parameters
----------
x : array_like
Values at which to evaluate the integrals.
out : tuple of ndarrays, optional
Optional output arrays for the function results.
Returns
-------
ii0 : scalar or ndarray
The integral for `i0`
ik0 : scalar or ndarray
The integral for `k0`
References
----------
.. [1] S. Zhang and J.M. Jin, "Computation of Special Functions",
Wiley 1996
Examples
--------
Evaluate the functions at one point.
>>> from scipy.special import it2i0k0
>>> int_i, int_k = it2i0k0(1.)
>>> int_i, int_k
(0.12897944249456852, 0.2085182909001295)
Evaluate the functions at several points.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> int_i, int_k = it2i0k0(points)
>>> int_i, int_k
(array([0.03149527, 0.30187149, 1.50012461]),
array([0.66575102, 0.0823715 , 0.00823631]))
Plot the functions from 0 to 5.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 5., 1000)
>>> int_i, int_k = it2i0k0(x)
>>> ax.plot(x, int_i, label=r"$\int_0^x \frac{I_0(t)-1}{t}\,dt$")
>>> ax.plot(x, int_k, label=r"$\int_x^{\infty} \frac{K_0(t)}{t}\,dt$")
>>> ax.legend()
>>> ax.set_ylim(0, 10)
>>> plt.show()
""")
add_newdoc("it2j0y0",
r"""
it2j0y0(x, out=None)
Integrals related to Bessel functions of the first kind of order 0.
Computes the integrals
.. math::
\int_0^x \frac{1 - J_0(t)}{t} dt \\
\int_x^\infty \frac{Y_0(t)}{t} dt.
For more on :math:`J_0` and :math:`Y_0` see `j0` and `y0`.
Parameters
----------
x : array_like
Values at which to evaluate the integrals.
out : tuple of ndarrays, optional
Optional output arrays for the function results.
Returns
-------
ij0 : scalar or ndarray
The integral for `j0`
iy0 : scalar or ndarray
The integral for `y0`
References
----------
.. [1] S. Zhang and J.M. Jin, "Computation of Special Functions",
Wiley 1996
Examples
--------
Evaluate the functions at one point.
>>> from scipy.special import it2j0y0
>>> int_j, int_y = it2j0y0(1.)
>>> int_j, int_y
(0.12116524699506871, 0.39527290169929336)
Evaluate the functions at several points.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> int_j, int_y = it2j0y0(points)
>>> int_j, int_y
(array([0.03100699, 0.26227724, 0.85614669]),
array([ 0.26968854, 0.29769696, -0.02987272]))
Plot the functions from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> int_j, int_y = it2j0y0(x)
>>> ax.plot(x, int_j, label=r"$\int_0^x \frac{1-J_0(t)}{t}\,dt$")
>>> ax.plot(x, int_y, label=r"$\int_x^{\infty} \frac{Y_0(t)}{t}\,dt$")
>>> ax.legend()
>>> ax.set_ylim(-2.5, 2.5)
>>> plt.show()
""")
add_newdoc("it2struve0",
r"""
it2struve0(x, out=None)
Integral related to the Struve function of order 0.
Returns the integral,
.. math::
\int_x^\infty \frac{H_0(t)}{t}\,dt
where :math:`H_0` is the Struve function of order 0.
Parameters
----------
x : array_like
Lower limit of integration.
out : ndarray, optional
Optional output array for the function values
Returns
-------
I : scalar or ndarray
The value of the integral.
See Also
--------
struve
Notes
-----
Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html
Examples
--------
Evaluate the function at one point.
>>> import numpy as np
>>> from scipy.special import it2struve0
>>> it2struve0(1.)
0.9571973506383524
Evaluate the function at several points by supplying
an array for `x`.
>>> points = np.array([1., 2., 3.5])
>>> it2struve0(points)
array([0.95719735, 0.46909296, 0.10366042])
Plot the function from -10 to 10.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-10., 10., 1000)
>>> it2struve0_values = it2struve0(x)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, it2struve0_values)
>>> ax.set_xlabel(r'$x$')
>>> ax.set_ylabel(r'$\int_x^{\infty}\frac{H_0(t)}{t}\,dt$')
>>> plt.show()
""")
add_newdoc(
"itairy",
r"""
itairy(x, out=None)
Integrals of Airy functions
Calculates the integrals of Airy functions from 0 to `x`.
Parameters
----------
x : array_like
Upper limit of integration (float).
out : tuple of ndarray, optional
Optional output arrays for the function values
Returns
-------
Apt : scalar or ndarray
Integral of Ai(t) from 0 to x.
Bpt : scalar or ndarray
Integral of Bi(t) from 0 to x.
Ant : scalar or ndarray
Integral of Ai(-t) from 0 to x.
Bnt : scalar or ndarray
Integral of Bi(-t) from 0 to x.
Notes
-----
Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html
Examples
--------
Compute the functions at ``x=1.``.
>>> import numpy as np
>>> from scipy.special import itairy
>>> import matplotlib.pyplot as plt
>>> apt, bpt, ant, bnt = itairy(1.)
>>> apt, bpt, ant, bnt
(0.23631734191710949,
0.8727691167380077,
0.46567398346706845,
0.3730050096342943)
Compute the functions at several points by providing a NumPy array for `x`.
>>> x = np.array([1., 1.5, 2.5, 5])
>>> apt, bpt, ant, bnt = itairy(x)
>>> apt, bpt, ant, bnt
(array([0.23631734, 0.28678675, 0.324638 , 0.33328759]),
array([ 0.87276912, 1.62470809, 5.20906691, 321.47831857]),
array([0.46567398, 0.72232876, 0.93187776, 0.7178822 ]),
array([ 0.37300501, 0.35038814, -0.02812939, 0.15873094]))
Plot the functions from -10 to 10.
>>> x = np.linspace(-10, 10, 500)
>>> apt, bpt, ant, bnt = itairy(x)
>>> fig, ax = plt.subplots(figsize=(6, 5))
>>> ax.plot(x, apt, label=r"$\int_0^x\, Ai(t)\, dt$")
>>> ax.plot(x, bpt, ls="dashed", label=r"$\int_0^x\, Bi(t)\, dt$")
>>> ax.plot(x, ant, ls="dashdot", label=r"$\int_0^x\, Ai(-t)\, dt$")
>>> ax.plot(x, bnt, ls="dotted", label=r"$\int_0^x\, Bi(-t)\, dt$")
>>> ax.set_ylim(-2, 1.5)
>>> ax.legend(loc="lower right")
>>> plt.show()
""")
add_newdoc("iti0k0",
r"""
iti0k0(x, out=None)
Integrals of modified Bessel functions of order 0.
Computes the integrals
.. math::
\int_0^x I_0(t) dt \\
\int_0^x K_0(t) dt.
For more on :math:`I_0` and :math:`K_0` see `i0` and `k0`.
Parameters
----------
x : array_like
Values at which to evaluate the integrals.
out : tuple of ndarrays, optional
Optional output arrays for the function results.
Returns
-------
ii0 : scalar or ndarray
The integral for `i0`
ik0 : scalar or ndarray
The integral for `k0`
References
----------
.. [1] S. Zhang and J.M. Jin, "Computation of Special Functions",
Wiley 1996
Examples
--------
Evaluate the functions at one point.
>>> from scipy.special import iti0k0
>>> int_i, int_k = iti0k0(1.)
>>> int_i, int_k
(1.0865210970235892, 1.2425098486237771)
Evaluate the functions at several points.
>>> import numpy as np
>>> points = np.array([0., 1.5, 3.])
>>> int_i, int_k = iti0k0(points)
>>> int_i, int_k
(array([0. , 1.80606937, 6.16096149]),
array([0. , 1.39458246, 1.53994809]))
Plot the functions from 0 to 5.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 5., 1000)
>>> int_i, int_k = iti0k0(x)
>>> ax.plot(x, int_i, label=r"$\int_0^x I_0(t)\,dt$")
>>> ax.plot(x, int_k, label=r"$\int_0^x K_0(t)\,dt$")
>>> ax.legend()
>>> plt.show()
""")
add_newdoc("itj0y0",
r"""
itj0y0(x, out=None)
Integrals of Bessel functions of the first kind of order 0.
Computes the integrals
.. math::
\int_0^x J_0(t) dt \\
\int_0^x Y_0(t) dt.
For more on :math:`J_0` and :math:`Y_0` see `j0` and `y0`.
Parameters
----------
x : array_like
Values at which to evaluate the integrals.
out : tuple of ndarrays, optional
Optional output arrays for the function results.
Returns
-------
ij0 : scalar or ndarray
The integral of `j0`
iy0 : scalar or ndarray
The integral of `y0`
References
----------
.. [1] S. Zhang and J.M. Jin, "Computation of Special Functions",
Wiley 1996
Examples
--------
Evaluate the functions at one point.
>>> from scipy.special import itj0y0
>>> int_j, int_y = itj0y0(1.)
>>> int_j, int_y
(0.9197304100897596, -0.637069376607422)
Evaluate the functions at several points.
>>> import numpy as np
>>> points = np.array([0., 1.5, 3.])
>>> int_j, int_y = itj0y0(points)
>>> int_j, int_y
(array([0. , 1.24144951, 1.38756725]),
array([ 0. , -0.51175903, 0.19765826]))
Plot the functions from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> int_j, int_y = itj0y0(x)
>>> ax.plot(x, int_j, label=r"$\int_0^x J_0(t)\,dt$")
>>> ax.plot(x, int_y, label=r"$\int_0^x Y_0(t)\,dt$")
>>> ax.legend()
>>> plt.show()
""")
add_newdoc("itmodstruve0",
r"""
itmodstruve0(x, out=None)
Integral of the modified Struve function of order 0.
.. math::
I = \int_0^x L_0(t)\,dt
Parameters
----------
x : array_like
Upper limit of integration (float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
I : scalar or ndarray
The integral of :math:`L_0` from 0 to `x`.
See Also
--------
modstruve: Modified Struve function which is integrated by this function
Notes
-----
Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html
Examples
--------
Evaluate the function at one point.
>>> import numpy as np
>>> from scipy.special import itmodstruve0
>>> itmodstruve0(1.)
0.3364726286440384
Evaluate the function at several points by supplying
an array for `x`.
>>> points = np.array([1., 2., 3.5])
>>> itmodstruve0(points)
array([0.33647263, 1.588285 , 7.60382578])
Plot the function from -10 to 10.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-10., 10., 1000)
>>> itmodstruve0_values = itmodstruve0(x)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, itmodstruve0_values)
>>> ax.set_xlabel(r'$x$')
>>> ax.set_ylabel(r'$\int_0^xL_0(t)\,dt$')
>>> plt.show()
""")
add_newdoc("itstruve0",
r"""
itstruve0(x, out=None)
Integral of the Struve function of order 0.
.. math::
I = \int_0^x H_0(t)\,dt
Parameters
----------
x : array_like
Upper limit of integration (float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
I : scalar or ndarray
The integral of :math:`H_0` from 0 to `x`.
See Also
--------
struve: Function which is integrated by this function
Notes
-----
Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html
Examples
--------
Evaluate the function at one point.
>>> import numpy as np
>>> from scipy.special import itstruve0
>>> itstruve0(1.)
0.30109042670805547
Evaluate the function at several points by supplying
an array for `x`.
>>> points = np.array([1., 2., 3.5])
>>> itstruve0(points)
array([0.30109043, 1.01870116, 1.96804581])
Plot the function from -20 to 20.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-20., 20., 1000)
>>> istruve0_values = itstruve0(x)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, istruve0_values)
>>> ax.set_xlabel(r'$x$')
>>> ax.set_ylabel(r'$\int_0^{x}H_0(t)\,dt$')
>>> plt.show()
""")
add_newdoc("iv",
r"""
iv(v, z, out=None)
Modified Bessel function of the first kind of real order.
Parameters
----------
v : array_like
Order. If `z` is of real type and negative, `v` must be integer
valued.
z : array_like of float or complex
Argument.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the modified Bessel function.
See Also
--------
ive : This function with leading exponential behavior stripped off.
i0 : Faster version of this function for order 0.
i1 : Faster version of this function for order 1.
Notes
-----
For real `z` and :math:`v \in [-50, 50]`, the evaluation is carried out
using Temme's method [1]_. For larger orders, uniform asymptotic
expansions are applied.
For complex `z` and positive `v`, the AMOS [2]_ `zbesi` routine is
called. It uses a power series for small `z`, the asymptotic expansion
for large `abs(z)`, the Miller algorithm normalized by the Wronskian
and a Neumann series for intermediate magnitudes, and the uniform
asymptotic expansions for :math:`I_v(z)` and :math:`J_v(z)` for large
orders. Backward recurrence is used to generate sequences or reduce
orders when necessary.
The calculations above are done in the right half plane and continued
into the left half plane by the formula,
.. math:: I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z)
(valid when the real part of `z` is positive). For negative `v`, the
formula
.. math:: I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z)
is used, where :math:`K_v(z)` is the modified Bessel function of the
second kind, evaluated using the AMOS routine `zbesk`.
References
----------
.. [1] Temme, Journal of Computational Physics, vol 21, 343 (1976)
.. [2] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
Examples
--------
Evaluate the function of order 0 at one point.
>>> from scipy.special import iv
>>> iv(0, 1.)
1.2660658777520084
Evaluate the function at one point for different orders.
>>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)
(1.2660658777520084, 0.565159103992485, 0.2935253263474798)
The evaluation for different orders can be carried out in one call by
providing a list or NumPy array as argument for the `v` parameter:
>>> iv([0, 1, 1.5], 1.)
array([1.26606588, 0.5651591 , 0.29352533])
Evaluate the function at several points for order 0 by providing an
array for `z`.
>>> import numpy as np
>>> points = np.array([-2., 0., 3.])
>>> iv(0, points)
array([2.2795853 , 1. , 4.88079259])
If `z` is an array, the order parameter `v` must be broadcastable to
the correct shape if different orders shall be computed in one call.
To calculate the orders 0 and 1 for an 1D array:
>>> orders = np.array([[0], [1]])
>>> orders.shape
(2, 1)
>>> iv(orders, points)
array([[ 2.2795853 , 1. , 4.88079259],
[-1.59063685, 0. , 3.95337022]])
Plot the functions of order 0 to 3 from -5 to 5.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-5., 5., 1000)
>>> for i in range(4):
... ax.plot(x, iv(i, x), label=f'$I_{i!r}$')
>>> ax.legend()
>>> plt.show()
""")
add_newdoc("ive",
r"""
ive(v, z, out=None)
Exponentially scaled modified Bessel function of the first kind.
Defined as::
ive(v, z) = iv(v, z) * exp(-abs(z.real))
For imaginary numbers without a real part, returns the unscaled
Bessel function of the first kind `iv`.
Parameters
----------
v : array_like of float
Order.
z : array_like of float or complex
Argument.
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
Values of the exponentially scaled modified Bessel function.
See Also
--------
iv: Modified Bessel function of the first kind
i0e: Faster implementation of this function for order 0
i1e: Faster implementation of this function for order 1
Notes
-----
For positive `v`, the AMOS [1]_ `zbesi` routine is called. It uses a
power series for small `z`, the asymptotic expansion for large
`abs(z)`, the Miller algorithm normalized by the Wronskian and a
Neumann series for intermediate magnitudes, and the uniform asymptotic
expansions for :math:`I_v(z)` and :math:`J_v(z)` for large orders.
Backward recurrence is used to generate sequences or reduce orders when
necessary.
The calculations above are done in the right half plane and continued
into the left half plane by the formula,
.. math:: I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z)
(valid when the real part of `z` is positive). For negative `v`, the
formula
.. math:: I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z)
is used, where :math:`K_v(z)` is the modified Bessel function of the
second kind, evaluated using the AMOS routine `zbesk`.
`ive` is useful for large arguments `z`: for these, `iv` easily overflows,
while `ive` does not due to the exponential scaling.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
Examples
--------
In the following example `iv` returns infinity whereas `ive` still returns
a finite number.
>>> from scipy.special import iv, ive
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> iv(3, 1000.), ive(3, 1000.)
(inf, 0.01256056218254712)
Evaluate the function at one point for different orders by
providing a list or NumPy array as argument for the `v` parameter:
>>> ive([0, 1, 1.5], 1.)
array([0.46575961, 0.20791042, 0.10798193])
Evaluate the function at several points for order 0 by providing an
array for `z`.
>>> points = np.array([-2., 0., 3.])
>>> ive(0, points)
array([0.30850832, 1. , 0.24300035])
Evaluate the function at several points for different orders by
providing arrays for both `v` for `z`. Both arrays have to be
broadcastable to the correct shape. To calculate the orders 0, 1
and 2 for a 1D array of points:
>>> ive([[0], [1], [2]], points)
array([[ 0.30850832, 1. , 0.24300035],
[-0.21526929, 0. , 0.19682671],
[ 0.09323903, 0. , 0.11178255]])
Plot the functions of order 0 to 3 from -5 to 5.
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-5., 5., 1000)
>>> for i in range(4):
... ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\cdot e^{{-|z|}}$')
>>> ax.legend()
>>> ax.set_xlabel(r"$z$")
>>> plt.show()
""")
add_newdoc("j0",
r"""
j0(x, out=None)
Bessel function of the first kind of order 0.
Parameters
----------
x : array_like
Argument (float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
J : scalar or ndarray
Value of the Bessel function of the first kind of order 0 at `x`.
See Also
--------
jv : Bessel function of real order and complex argument.
spherical_jn : spherical Bessel functions.
Notes
-----
The domain is divided into the intervals [0, 5] and (5, infinity). In the
first interval the following rational approximation is used:
.. math::
J_0(x) \approx (w - r_1^2)(w - r_2^2) \frac{P_3(w)}{Q_8(w)},
where :math:`w = x^2` and :math:`r_1`, :math:`r_2` are the zeros of
:math:`J_0`, and :math:`P_3` and :math:`Q_8` are polynomials of degrees 3
and 8, respectively.
In the second interval, the Hankel asymptotic expansion is employed with
two rational functions of degree 6/6 and 7/7.
This function is a wrapper for the Cephes [1]_ routine `j0`.
It should not be confused with the spherical Bessel functions (see
`spherical_jn`).
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function at one point:
>>> from scipy.special import j0
>>> j0(1.)
0.7651976865579665
Calculate the function at several points:
>>> import numpy as np
>>> j0(np.array([-2., 0., 4.]))
array([ 0.22389078, 1. , -0.39714981])
Plot the function from -20 to 20.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-20., 20., 1000)
>>> y = j0(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("j1",
"""
j1(x, out=None)
Bessel function of the first kind of order 1.
Parameters
----------
x : array_like
Argument (float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
J : scalar or ndarray
Value of the Bessel function of the first kind of order 1 at `x`.
See Also
--------
jv: Bessel function of the first kind
spherical_jn: spherical Bessel functions.
Notes
-----
The domain is divided into the intervals [0, 8] and (8, infinity). In the
first interval a 24 term Chebyshev expansion is used. In the second, the
asymptotic trigonometric representation is employed using two rational
functions of degree 5/5.
This function is a wrapper for the Cephes [1]_ routine `j1`.
It should not be confused with the spherical Bessel functions (see
`spherical_jn`).
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function at one point:
>>> from scipy.special import j1
>>> j1(1.)
0.44005058574493355
Calculate the function at several points:
>>> import numpy as np
>>> j1(np.array([-2., 0., 4.]))
array([-0.57672481, 0. , -0.06604333])
Plot the function from -20 to 20.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-20., 20., 1000)
>>> y = j1(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("jn",
"""
jn(n, x, out=None)
Bessel function of the first kind of integer order and real argument.
Parameters
----------
n : array_like
order of the Bessel function
x : array_like
argument of the Bessel function
out : ndarray, optional
Optional output array for the function values
Returns
-------
scalar or ndarray
The value of the bessel function
See Also
--------
jv
spherical_jn : spherical Bessel functions.
Notes
-----
`jn` is an alias of `jv`.
Not to be confused with the spherical Bessel functions (see
`spherical_jn`).
""")
add_newdoc("jv",
r"""
jv(v, z, out=None)
Bessel function of the first kind of real order and complex argument.
Parameters
----------
v : array_like
Order (float).
z : array_like
Argument (float or complex).
out : ndarray, optional
Optional output array for the function values
Returns
-------
J : scalar or ndarray
Value of the Bessel function, :math:`J_v(z)`.
See Also
--------
jve : :math:`J_v` with leading exponential behavior stripped off.
spherical_jn : spherical Bessel functions.
j0 : faster version of this function for order 0.
j1 : faster version of this function for order 1.
Notes
-----
For positive `v` values, the computation is carried out using the AMOS
[1]_ `zbesj` routine, which exploits the connection to the modified
Bessel function :math:`I_v`,
.. math::
J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)
J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)
For negative `v` values the formula,
.. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v)
is used, where :math:`Y_v(z)` is the Bessel function of the second
kind, computed using the AMOS routine `zbesy`. Note that the second
term is exactly zero for integer `v`; to improve accuracy the second
term is explicitly omitted for `v` values such that `v = floor(v)`.
Not to be confused with the spherical Bessel functions (see `spherical_jn`).
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
Examples
--------
Evaluate the function of order 0 at one point.
>>> from scipy.special import jv
>>> jv(0, 1.)
0.7651976865579666
Evaluate the function at one point for different orders.
>>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)
(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)
The evaluation for different orders can be carried out in one call by
providing a list or NumPy array as argument for the `v` parameter:
>>> jv([0, 1, 1.5], 1.)
array([0.76519769, 0.44005059, 0.24029784])
Evaluate the function at several points for order 0 by providing an
array for `z`.
>>> import numpy as np
>>> points = np.array([-2., 0., 3.])
>>> jv(0, points)
array([ 0.22389078, 1. , -0.26005195])
If `z` is an array, the order parameter `v` must be broadcastable to
the correct shape if different orders shall be computed in one call.
To calculate the orders 0 and 1 for an 1D array:
>>> orders = np.array([[0], [1]])
>>> orders.shape
(2, 1)
>>> jv(orders, points)
array([[ 0.22389078, 1. , -0.26005195],
[-0.57672481, 0. , 0.33905896]])
Plot the functions of order 0 to 3 from -10 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> for i in range(4):
... ax.plot(x, jv(i, x), label=f'$J_{i!r}$')
>>> ax.legend()
>>> plt.show()
""")
add_newdoc("jve",
r"""
jve(v, z, out=None)
Exponentially scaled Bessel function of the first kind of order `v`.
Defined as::
jve(v, z) = jv(v, z) * exp(-abs(z.imag))
Parameters
----------
v : array_like
Order (float).
z : array_like
Argument (float or complex).
out : ndarray, optional
Optional output array for the function values
Returns
-------
J : scalar or ndarray
Value of the exponentially scaled Bessel function.
See Also
--------
jv: Unscaled Bessel function of the first kind
Notes
-----
For positive `v` values, the computation is carried out using the AMOS
[1]_ `zbesj` routine, which exploits the connection to the modified
Bessel function :math:`I_v`,
.. math::
J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)
J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)
For negative `v` values the formula,
.. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v)
is used, where :math:`Y_v(z)` is the Bessel function of the second
kind, computed using the AMOS routine `zbesy`. Note that the second
term is exactly zero for integer `v`; to improve accuracy the second
term is explicitly omitted for `v` values such that `v = floor(v)`.
Exponentially scaled Bessel functions are useful for large arguments `z`:
for these, the unscaled Bessel functions can easily under-or overflow.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
Examples
--------
Compare the output of `jv` and `jve` for large complex arguments for `z`
by computing their values for order ``v=1`` at ``z=1000j``. We see that
`jv` overflows but `jve` returns a finite number:
>>> import numpy as np
>>> from scipy.special import jv, jve
>>> v = 1
>>> z = 1000j
>>> jv(v, z), jve(v, z)
((inf+infj), (7.721967686709077e-19+0.012610930256928629j))
For real arguments for `z`, `jve` returns the same as `jv`.
>>> v, z = 1, 1000
>>> jv(v, z), jve(v, z)
(0.004728311907089523, 0.004728311907089523)
The function can be evaluated for several orders at the same time by
providing a list or NumPy array for `v`:
>>> jve([1, 3, 5], 1j)
array([1.27304208e-17+2.07910415e-01j, -4.99352086e-19-8.15530777e-03j,
6.11480940e-21+9.98657141e-05j])
In the same way, the function can be evaluated at several points in one
call by providing a list or NumPy array for `z`:
>>> jve(1, np.array([1j, 2j, 3j]))
array([1.27308412e-17+0.20791042j, 1.31814423e-17+0.21526929j,
1.20521602e-17+0.19682671j])
It is also possible to evaluate several orders at several points
at the same time by providing arrays for `v` and `z` with
compatible shapes for broadcasting. Compute `jve` for two different orders
`v` and three points `z` resulting in a 2x3 array.
>>> v = np.array([[1], [3]])
>>> z = np.array([1j, 2j, 3j])
>>> v.shape, z.shape
((2, 1), (3,))
>>> jve(v, z)
array([[1.27304208e-17+0.20791042j, 1.31810070e-17+0.21526929j,
1.20517622e-17+0.19682671j],
[-4.99352086e-19-0.00815531j, -1.76289571e-18-0.02879122j,
-2.92578784e-18-0.04778332j]])
""")
add_newdoc("k0",
r"""
k0(x, out=None)
Modified Bessel function of the second kind of order 0, :math:`K_0`.
This function is also sometimes referred to as the modified Bessel
function of the third kind of order 0.
Parameters
----------
x : array_like
Argument (float).
out : ndarray, optional
Optional output array for the function values
Returns
-------
K : scalar or ndarray
Value of the modified Bessel function :math:`K_0` at `x`.
See Also
--------
kv: Modified Bessel function of the second kind of any order
k0e: Exponentially scaled modified Bessel function of the second kind
Notes
-----
The range is partitioned into the two intervals [0, 2] and (2, infinity).
Chebyshev polynomial expansions are employed in each interval.
This function is a wrapper for the Cephes [1]_ routine `k0`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function at one point:
>>> from scipy.special import k0
>>> k0(1.)
0.42102443824070823
Calculate the function at several points:
>>> import numpy as np
>>> k0(np.array([0.5, 2., 3.]))
array([0.92441907, 0.11389387, 0.0347395 ])
Plot the function from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k0(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("k0e",
"""
k0e(x, out=None)
Exponentially scaled modified Bessel function K of order 0
Defined as::
k0e(x) = exp(x) * k0(x).
Parameters
----------
x : array_like
Argument (float)
out : ndarray, optional
Optional output array for the function values
Returns
-------
K : scalar or ndarray
Value of the exponentially scaled modified Bessel function K of order
0 at `x`.
See Also
--------
kv: Modified Bessel function of the second kind of any order
k0: Modified Bessel function of the second kind
Notes
-----
The range is partitioned into the two intervals [0, 2] and (2, infinity).
Chebyshev polynomial expansions are employed in each interval.
This function is a wrapper for the Cephes [1]_ routine `k0e`. `k0e` is
useful for large arguments: for these, `k0` easily underflows.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
In the following example `k0` returns 0 whereas `k0e` still returns a
useful finite number:
>>> from scipy.special import k0, k0e
>>> k0(1000.), k0e(1000)
(0., 0.03962832160075422)
Calculate the function at several points by providing a NumPy array or
list for `x`:
>>> import numpy as np
>>> k0e(np.array([0.5, 2., 3.]))
array([1.52410939, 0.84156822, 0.6977616 ])
Plot the function from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k0e(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("k1",
"""
k1(x, out=None)
Modified Bessel function of the second kind of order 1, :math:`K_1(x)`.
Parameters
----------
x : array_like
Argument (float)
out : ndarray, optional
Optional output array for the function values
Returns
-------
K : scalar or ndarray
Value of the modified Bessel function K of order 1 at `x`.
See Also
--------
kv: Modified Bessel function of the second kind of any order
k1e: Exponentially scaled modified Bessel function K of order 1
Notes
-----
The range is partitioned into the two intervals [0, 2] and (2, infinity).
Chebyshev polynomial expansions are employed in each interval.
This function is a wrapper for the Cephes [1]_ routine `k1`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function at one point:
>>> from scipy.special import k1
>>> k1(1.)
0.6019072301972346
Calculate the function at several points:
>>> import numpy as np
>>> k1(np.array([0.5, 2., 3.]))
array([1.65644112, 0.13986588, 0.04015643])
Plot the function from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k1(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("k1e",
"""
k1e(x, out=None)
Exponentially scaled modified Bessel function K of order 1
Defined as::
k1e(x) = exp(x) * k1(x)
Parameters
----------
x : array_like
Argument (float)
out : ndarray, optional
Optional output array for the function values
Returns
-------
K : scalar or ndarray
Value of the exponentially scaled modified Bessel function K of order
1 at `x`.
See Also
--------
kv: Modified Bessel function of the second kind of any order
k1: Modified Bessel function of the second kind of order 1
Notes
-----
The range is partitioned into the two intervals [0, 2] and (2, infinity).
Chebyshev polynomial expansions are employed in each interval.
This function is a wrapper for the Cephes [1]_ routine `k1e`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
In the following example `k1` returns 0 whereas `k1e` still returns a
useful floating point number.
>>> from scipy.special import k1, k1e
>>> k1(1000.), k1e(1000.)
(0., 0.03964813081296021)
Calculate the function at several points by providing a NumPy array or
list for `x`:
>>> import numpy as np
>>> k1e(np.array([0.5, 2., 3.]))
array([2.73100971, 1.03347685, 0.80656348])
Plot the function from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k1e(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("kei",
r"""
kei(x, out=None)
Kelvin function kei.
Defined as
.. math::
\mathrm{kei}(x) = \Im[K_0(x e^{\pi i / 4})]
where :math:`K_0` is the modified Bessel function of the second
kind (see `kv`). See [dlmf]_ for more details.
Parameters
----------
x : array_like
Real argument.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of the Kelvin function.
See Also
--------
ker : the corresponding real part
keip : the derivative of kei
kv : modified Bessel function of the second kind
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/10.61
Examples
--------
It can be expressed using the modified Bessel function of the
second kind.
>>> import numpy as np
>>> import scipy.special as sc
>>> x = np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.kv(0, x * np.exp(np.pi * 1j / 4)).imag
array([-0.49499464, -0.20240007, -0.05112188, 0.0021984 ])
>>> sc.kei(x)
array([-0.49499464, -0.20240007, -0.05112188, 0.0021984 ])
""")
add_newdoc("keip",
r"""
keip(x, out=None)
Derivative of the Kelvin function kei.
Parameters
----------
x : array_like
Real argument.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
The values of the derivative of kei.
See Also
--------
kei
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/10#PT5
""")
add_newdoc("kelvin",
"""
kelvin(x, out=None)
Kelvin functions as complex numbers
Parameters
----------
x : array_like
Argument
out : tuple of ndarray, optional
Optional output arrays for the function values
Returns
-------
Be, Ke, Bep, Kep : 4-tuple of scalar or ndarray
The tuple (Be, Ke, Bep, Kep) contains complex numbers
representing the real and imaginary Kelvin functions and their
derivatives evaluated at `x`. For example, kelvin(x)[0].real =
ber x and kelvin(x)[0].imag = bei x with similar relationships
for ker and kei.
""")
add_newdoc("ker",
r"""
ker(x, out=None)
Kelvin function ker.
Defined as
.. math::
\mathrm{ker}(x) = \Re[K_0(x e^{\pi i / 4})]
Where :math:`K_0` is the modified Bessel function of the second
kind (see `kv`). See [dlmf]_ for more details.
Parameters
----------
x : array_like
Real argument.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of the Kelvin function.
See Also
--------
kei : the corresponding imaginary part
kerp : the derivative of ker
kv : modified Bessel function of the second kind
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/10.61
Examples
--------
It can be expressed using the modified Bessel function of the
second kind.
>>> import numpy as np
>>> import scipy.special as sc
>>> x = np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.kv(0, x * np.exp(np.pi * 1j / 4)).real
array([ 0.28670621, -0.04166451, -0.06702923, -0.03617885])
>>> sc.ker(x)
array([ 0.28670621, -0.04166451, -0.06702923, -0.03617885])
""")
add_newdoc("kerp",
r"""
kerp(x, out=None)
Derivative of the Kelvin function ker.
Parameters
----------
x : array_like
Real argument.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of the derivative of ker.
See Also
--------
ker
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/10#PT5
""")
add_newdoc("kl_div",
r"""
kl_div(x, y, out=None)
Elementwise function for computing Kullback-Leibler divergence.
.. math::
\mathrm{kl\_div}(x, y) =
\begin{cases}
x \log(x / y) - x + y & x > 0, y > 0 \\
y & x = 0, y \ge 0 \\
\infty & \text{otherwise}
\end{cases}
Parameters
----------
x, y : array_like
Real arguments
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the Kullback-Liebler divergence.
See Also
--------
entr, rel_entr, scipy.stats.entropy
Notes
-----
.. versionadded:: 0.15.0
This function is non-negative and is jointly convex in `x` and `y`.
The origin of this function is in convex programming; see [1]_ for
details. This is why the function contains the extra :math:`-x
+ y` terms over what might be expected from the Kullback-Leibler
divergence. For a version of the function without the extra terms,
see `rel_entr`.
References
----------
.. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*.
Cambridge University Press, 2004.
:doi:`https://doi.org/10.1017/CBO9780511804441`
""")
add_newdoc("kn",
r"""
kn(n, x, out=None)
Modified Bessel function of the second kind of integer order `n`
Returns the modified Bessel function of the second kind for integer order
`n` at real `z`.
These are also sometimes called functions of the third kind, Basset
functions, or Macdonald functions.
Parameters
----------
n : array_like of int
Order of Bessel functions (floats will truncate with a warning)
x : array_like of float
Argument at which to evaluate the Bessel functions
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Value of the Modified Bessel function of the second kind,
:math:`K_n(x)`.
See Also
--------
kv : Same function, but accepts real order and complex argument
kvp : Derivative of this function
Notes
-----
Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the
algorithm used, see [2]_ and the references therein.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
.. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel
functions of a complex argument and nonnegative order", ACM
TOMS Vol. 12 Issue 3, Sept. 1986, p. 265
Examples
--------
Plot the function of several orders for real input:
>>> import numpy as np
>>> from scipy.special import kn
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> for N in range(6):
... plt.plot(x, kn(N, x), label='$K_{}(x)$'.format(N))
>>> plt.ylim(0, 10)
>>> plt.legend()
>>> plt.title(r'Modified Bessel function of the second kind $K_n(x)$')
>>> plt.show()
Calculate for a single value at multiple orders:
>>> kn([4, 5, 6], 1)
array([ 44.23241585, 360.9605896 , 3653.83831186])
""")
add_newdoc("kolmogi",
"""
kolmogi(p, out=None)
Inverse Survival Function of Kolmogorov distribution
It is the inverse function to `kolmogorov`.
Returns y such that ``kolmogorov(y) == p``.
Parameters
----------
p : float array_like
Probability
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The value(s) of kolmogi(p)
See Also
--------
kolmogorov : The Survival Function for the distribution
scipy.stats.kstwobign : Provides the functionality as a continuous distribution
smirnov, smirnovi : Functions for the one-sided distribution
Notes
-----
`kolmogorov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this
function is exposed in `scpy.special`, but the recommended way to achieve
the most accurate CDF/SF/PDF/PPF/ISF computations is to use the
`stats.kstwobign` distribution.
Examples
--------
>>> from scipy.special import kolmogi
>>> kolmogi([0, 0.1, 0.25, 0.5, 0.75, 0.9, 1.0])
array([ inf, 1.22384787, 1.01918472, 0.82757356, 0.67644769,
0.57117327, 0. ])
""")
add_newdoc("kolmogorov",
r"""
kolmogorov(y, out=None)
Complementary cumulative distribution (Survival Function) function of
Kolmogorov distribution.
Returns the complementary cumulative distribution function of
Kolmogorov's limiting distribution (``D_n*\sqrt(n)`` as n goes to infinity)
of a two-sided test for equality between an empirical and a theoretical
distribution. It is equal to the (limit as n->infinity of the)
probability that ``sqrt(n) * max absolute deviation > y``.
Parameters
----------
y : float array_like
Absolute deviation between the Empirical CDF (ECDF) and the target CDF,
multiplied by sqrt(n).
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The value(s) of kolmogorov(y)
See Also
--------
kolmogi : The Inverse Survival Function for the distribution
scipy.stats.kstwobign : Provides the functionality as a continuous distribution
smirnov, smirnovi : Functions for the one-sided distribution
Notes
-----
`kolmogorov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this
function is exposed in `scpy.special`, but the recommended way to achieve
the most accurate CDF/SF/PDF/PPF/ISF computations is to use the
`stats.kstwobign` distribution.
Examples
--------
Show the probability of a gap at least as big as 0, 0.5 and 1.0.
>>> import numpy as np
>>> from scipy.special import kolmogorov
>>> from scipy.stats import kstwobign
>>> kolmogorov([0, 0.5, 1.0])
array([ 1. , 0.96394524, 0.26999967])
Compare a sample of size 1000 drawn from a Laplace(0, 1) distribution against
the target distribution, a Normal(0, 1) distribution.
>>> from scipy.stats import norm, laplace
>>> rng = np.random.default_rng()
>>> n = 1000
>>> lap01 = laplace(0, 1)
>>> x = np.sort(lap01.rvs(n, random_state=rng))
>>> np.mean(x), np.std(x)
(-0.05841730131499543, 1.3968109101997568)
Construct the Empirical CDF and the K-S statistic Dn.
>>> target = norm(0,1) # Normal mean 0, stddev 1
>>> cdfs = target.cdf(x)
>>> ecdfs = np.arange(n+1, dtype=float)/n
>>> gaps = np.column_stack([cdfs - ecdfs[:n], ecdfs[1:] - cdfs])
>>> Dn = np.max(gaps)
>>> Kn = np.sqrt(n) * Dn
>>> print('Dn=%f, sqrt(n)*Dn=%f' % (Dn, Kn))
Dn=0.043363, sqrt(n)*Dn=1.371265
>>> print(chr(10).join(['For a sample of size n drawn from a N(0, 1) distribution:',
... ' the approximate Kolmogorov probability that sqrt(n)*Dn>=%f is %f' %
... (Kn, kolmogorov(Kn)),
... ' the approximate Kolmogorov probability that sqrt(n)*Dn<=%f is %f' %
... (Kn, kstwobign.cdf(Kn))]))
For a sample of size n drawn from a N(0, 1) distribution:
the approximate Kolmogorov probability that sqrt(n)*Dn>=1.371265 is 0.046533
the approximate Kolmogorov probability that sqrt(n)*Dn<=1.371265 is 0.953467
Plot the Empirical CDF against the target N(0, 1) CDF.
>>> import matplotlib.pyplot as plt
>>> plt.step(np.concatenate([[-3], x]), ecdfs, where='post', label='Empirical CDF')
>>> x3 = np.linspace(-3, 3, 100)
>>> plt.plot(x3, target.cdf(x3), label='CDF for N(0, 1)')
>>> plt.ylim([0, 1]); plt.grid(True); plt.legend();
>>> # Add vertical lines marking Dn+ and Dn-
>>> iminus, iplus = np.argmax(gaps, axis=0)
>>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus],
... color='r', linestyle='dashed', lw=4)
>>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1],
... color='r', linestyle='dashed', lw=4)
>>> plt.show()
""")
add_newdoc("_kolmogc",
r"""
Internal function, do not use.
""")
add_newdoc("_kolmogci",
r"""
Internal function, do not use.
""")
add_newdoc("_kolmogp",
r"""
Internal function, do not use.
""")
add_newdoc("kv",
r"""
kv(v, z, out=None)
Modified Bessel function of the second kind of real order `v`
Returns the modified Bessel function of the second kind for real order
`v` at complex `z`.
These are also sometimes called functions of the third kind, Basset
functions, or Macdonald functions. They are defined as those solutions
of the modified Bessel equation for which,
.. math::
K_v(x) \sim \sqrt{\pi/(2x)} \exp(-x)
as :math:`x \to \infty` [3]_.
Parameters
----------
v : array_like of float
Order of Bessel functions
z : array_like of complex
Argument at which to evaluate the Bessel functions
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The results. Note that input must be of complex type to get complex
output, e.g. ``kv(3, -2+0j)`` instead of ``kv(3, -2)``.
See Also
--------
kve : This function with leading exponential behavior stripped off.
kvp : Derivative of this function
Notes
-----
Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the
algorithm used, see [2]_ and the references therein.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
.. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel
functions of a complex argument and nonnegative order", ACM
TOMS Vol. 12 Issue 3, Sept. 1986, p. 265
.. [3] NIST Digital Library of Mathematical Functions,
Eq. 10.25.E3. https://dlmf.nist.gov/10.25.E3
Examples
--------
Plot the function of several orders for real input:
>>> import numpy as np
>>> from scipy.special import kv
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> for N in np.linspace(0, 6, 5):
... plt.plot(x, kv(N, x), label='$K_{{{}}}(x)$'.format(N))
>>> plt.ylim(0, 10)
>>> plt.legend()
>>> plt.title(r'Modified Bessel function of the second kind $K_\nu(x)$')
>>> plt.show()
Calculate for a single value at multiple orders:
>>> kv([4, 4.5, 5], 1+2j)
array([ 0.1992+2.3892j, 2.3493+3.6j , 7.2827+3.8104j])
""")
add_newdoc("kve",
r"""
kve(v, z, out=None)
Exponentially scaled modified Bessel function of the second kind.
Returns the exponentially scaled, modified Bessel function of the
second kind (sometimes called the third kind) for real order `v` at
complex `z`::
kve(v, z) = kv(v, z) * exp(z)
Parameters
----------
v : array_like of float
Order of Bessel functions
z : array_like of complex
Argument at which to evaluate the Bessel functions
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The exponentially scaled modified Bessel function of the second kind.
See Also
--------
kv : This function without exponential scaling.
k0e : Faster version of this function for order 0.
k1e : Faster version of this function for order 1.
Notes
-----
Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the
algorithm used, see [2]_ and the references therein.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
.. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel
functions of a complex argument and nonnegative order", ACM
TOMS Vol. 12 Issue 3, Sept. 1986, p. 265
Examples
--------
In the following example `kv` returns 0 whereas `kve` still returns
a useful finite number.
>>> import numpy as np
>>> from scipy.special import kv, kve
>>> import matplotlib.pyplot as plt
>>> kv(3, 1000.), kve(3, 1000.)
(0.0, 0.03980696128440973)
Evaluate the function at one point for different orders by
providing a list or NumPy array as argument for the `v` parameter:
>>> kve([0, 1, 1.5], 1.)
array([1.14446308, 1.63615349, 2.50662827])
Evaluate the function at several points for order 0 by providing an
array for `z`.
>>> points = np.array([1., 3., 10.])
>>> kve(0, points)
array([1.14446308, 0.6977616 , 0.39163193])
Evaluate the function at several points for different orders by
providing arrays for both `v` for `z`. Both arrays have to be
broadcastable to the correct shape. To calculate the orders 0, 1
and 2 for a 1D array of points:
>>> kve([[0], [1], [2]], points)
array([[1.14446308, 0.6977616 , 0.39163193],
[1.63615349, 0.80656348, 0.41076657],
[4.41677005, 1.23547058, 0.47378525]])
Plot the functions of order 0 to 3 from 0 to 5.
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 5., 1000)
>>> for i in range(4):
... ax.plot(x, kve(i, x), label=fr'$K_{i!r}(z)\cdot e^z$')
>>> ax.legend()
>>> ax.set_xlabel(r"$z$")
>>> ax.set_ylim(0, 4)
>>> ax.set_xlim(0, 5)
>>> plt.show()
""")
add_newdoc("_lanczos_sum_expg_scaled",
"""
Internal function, do not use.
""")
add_newdoc("_lgam1p",
"""
Internal function, do not use.
""")
add_newdoc("log1p",
"""
log1p(x, out=None)
Calculates log(1 + x) for use when `x` is near zero.
Parameters
----------
x : array_like
Real or complex valued input.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of ``log(1 + x)``.
See Also
--------
expm1, cosm1
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is more accurate than using ``log(1 + x)`` directly for ``x``
near 0. Note that in the below example ``1 + 1e-17 == 1`` to
double precision.
>>> sc.log1p(1e-17)
1e-17
>>> np.log(1 + 1e-17)
0.0
""")
add_newdoc("_log1pmx",
"""
Internal function, do not use.
""")
add_newdoc('log_expit',
"""
log_expit(x, out=None)
Logarithm of the logistic sigmoid function.
The SciPy implementation of the logistic sigmoid function is
`scipy.special.expit`, so this function is called ``log_expit``.
The function is mathematically equivalent to ``log(expit(x))``, but
is formulated to avoid loss of precision for inputs with large
(positive or negative) magnitude.
Parameters
----------
x : array_like
The values to apply ``log_expit`` to element-wise.
out : ndarray, optional
Optional output array for the function results
Returns
-------
out : scalar or ndarray
The computed values, an ndarray of the same shape as ``x``.
See Also
--------
expit
Notes
-----
As a ufunc, ``log_expit`` takes a number of optional keyword arguments.
For more information see
`ufuncs <https://docs.scipy.org/doc/numpy/reference/ufuncs.html>`_
.. versionadded:: 1.8.0
Examples
--------
>>> import numpy as np
>>> from scipy.special import log_expit, expit
>>> log_expit([-3.0, 0.25, 2.5, 5.0])
array([-3.04858735, -0.57593942, -0.07888973, -0.00671535])
Large negative values:
>>> log_expit([-100, -500, -1000])
array([ -100., -500., -1000.])
Note that ``expit(-1000)`` returns 0, so the naive implementation
``log(expit(-1000))`` return ``-inf``.
Large positive values:
>>> log_expit([29, 120, 400])
array([-2.54366565e-013, -7.66764807e-053, -1.91516960e-174])
Compare that to the naive implementation:
>>> np.log(expit([29, 120, 400]))
array([-2.54463117e-13, 0.00000000e+00, 0.00000000e+00])
The first value is accurate to only 3 digits, and the larger inputs
lose all precision and return 0.
""")
add_newdoc('logit',
"""
logit(x, out=None)
Logit ufunc for ndarrays.
The logit function is defined as logit(p) = log(p/(1-p)).
Note that logit(0) = -inf, logit(1) = inf, and logit(p)
for p<0 or p>1 yields nan.
Parameters
----------
x : ndarray
The ndarray to apply logit to element-wise.
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
An ndarray of the same shape as x. Its entries
are logit of the corresponding entry of x.
See Also
--------
expit
Notes
-----
As a ufunc logit takes a number of optional
keyword arguments. For more information
see `ufuncs <https://docs.scipy.org/doc/numpy/reference/ufuncs.html>`_
.. versionadded:: 0.10.0
Examples
--------
>>> import numpy as np
>>> from scipy.special import logit, expit
>>> logit([0, 0.25, 0.5, 0.75, 1])
array([ -inf, -1.09861229, 0. , 1.09861229, inf])
`expit` is the inverse of `logit`:
>>> expit(logit([0.1, 0.75, 0.999]))
array([ 0.1 , 0.75 , 0.999])
Plot logit(x) for x in [0, 1]:
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 1, 501)
>>> y = logit(x)
>>> plt.plot(x, y)
>>> plt.grid()
>>> plt.ylim(-6, 6)
>>> plt.xlabel('x')
>>> plt.title('logit(x)')
>>> plt.show()
""")
add_newdoc("lpmv",
r"""
lpmv(m, v, x, out=None)
Associated Legendre function of integer order and real degree.
Defined as
.. math::
P_v^m = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_v(x)
where
.. math::
P_v = \sum_{k = 0}^\infty \frac{(-v)_k (v + 1)_k}{(k!)^2}
\left(\frac{1 - x}{2}\right)^k
is the Legendre function of the first kind. Here :math:`(\cdot)_k`
is the Pochhammer symbol; see `poch`.
Parameters
----------
m : array_like
Order (int or float). If passed a float not equal to an
integer the function returns NaN.
v : array_like
Degree (float).
x : array_like
Argument (float). Must have ``|x| <= 1``.
out : ndarray, optional
Optional output array for the function results
Returns
-------
pmv : scalar or ndarray
Value of the associated Legendre function.
See Also
--------
lpmn : Compute the associated Legendre function for all orders
``0, ..., m`` and degrees ``0, ..., n``.
clpmn : Compute the associated Legendre function at complex
arguments.
Notes
-----
Note that this implementation includes the Condon-Shortley phase.
References
----------
.. [1] Zhang, Jin, "Computation of Special Functions", John Wiley
and Sons, Inc, 1996.
""")
add_newdoc("mathieu_a",
"""
mathieu_a(m, q, out=None)
Characteristic value of even Mathieu functions
Parameters
----------
m : array_like
Order of the function
q : array_like
Parameter of the function
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Characteristic value for the even solution, ``ce_m(z, q)``, of
Mathieu's equation.
See Also
--------
mathieu_b, mathieu_cem, mathieu_sem
""")
add_newdoc("mathieu_b",
"""
mathieu_b(m, q, out=None)
Characteristic value of odd Mathieu functions
Parameters
----------
m : array_like
Order of the function
q : array_like
Parameter of the function
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Characteristic value for the odd solution, ``se_m(z, q)``, of Mathieu's
equation.
See Also
--------
mathieu_a, mathieu_cem, mathieu_sem
""")
add_newdoc("mathieu_cem",
"""
mathieu_cem(m, q, x, out=None)
Even Mathieu function and its derivative
Returns the even Mathieu function, ``ce_m(x, q)``, of order `m` and
parameter `q` evaluated at `x` (given in degrees). Also returns the
derivative with respect to `x` of ce_m(x, q)
Parameters
----------
m : array_like
Order of the function
q : array_like
Parameter of the function
x : array_like
Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
y : scalar or ndarray
Value of the function
yp : scalar or ndarray
Value of the derivative vs x
See Also
--------
mathieu_a, mathieu_b, mathieu_sem
""")
add_newdoc("mathieu_modcem1",
"""
mathieu_modcem1(m, q, x, out=None)
Even modified Mathieu function of the first kind and its derivative
Evaluates the even modified Mathieu function of the first kind,
``Mc1m(x, q)``, and its derivative at `x` for order `m` and parameter
`q`.
Parameters
----------
m : array_like
Order of the function
q : array_like
Parameter of the function
x : array_like
Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
y : scalar or ndarray
Value of the function
yp : scalar or ndarray
Value of the derivative vs x
See Also
--------
mathieu_modsem1
""")
add_newdoc("mathieu_modcem2",
"""
mathieu_modcem2(m, q, x, out=None)
Even modified Mathieu function of the second kind and its derivative
Evaluates the even modified Mathieu function of the second kind,
Mc2m(x, q), and its derivative at `x` (given in degrees) for order `m`
and parameter `q`.
Parameters
----------
m : array_like
Order of the function
q : array_like
Parameter of the function
x : array_like
Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
y : scalar or ndarray
Value of the function
yp : scalar or ndarray
Value of the derivative vs x
See Also
--------
mathieu_modsem2
""")
add_newdoc("mathieu_modsem1",
"""
mathieu_modsem1(m, q, x, out=None)
Odd modified Mathieu function of the first kind and its derivative
Evaluates the odd modified Mathieu function of the first kind,
Ms1m(x, q), and its derivative at `x` (given in degrees) for order `m`
and parameter `q`.
Parameters
----------
m : array_like
Order of the function
q : array_like
Parameter of the function
x : array_like
Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
y : scalar or ndarray
Value of the function
yp : scalar or ndarray
Value of the derivative vs x
See Also
--------
mathieu_modcem1
""")
add_newdoc("mathieu_modsem2",
"""
mathieu_modsem2(m, q, x, out=None)
Odd modified Mathieu function of the second kind and its derivative
Evaluates the odd modified Mathieu function of the second kind,
Ms2m(x, q), and its derivative at `x` (given in degrees) for order `m`
and parameter q.
Parameters
----------
m : array_like
Order of the function
q : array_like
Parameter of the function
x : array_like
Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
y : scalar or ndarray
Value of the function
yp : scalar or ndarray
Value of the derivative vs x
See Also
--------
mathieu_modcem2
""")
add_newdoc(
"mathieu_sem",
"""
mathieu_sem(m, q, x, out=None)
Odd Mathieu function and its derivative
Returns the odd Mathieu function, se_m(x, q), of order `m` and
parameter `q` evaluated at `x` (given in degrees). Also returns the
derivative with respect to `x` of se_m(x, q).
Parameters
----------
m : array_like
Order of the function
q : array_like
Parameter of the function
x : array_like
Argument of the function, *given in degrees, not radians*.
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
y : scalar or ndarray
Value of the function
yp : scalar or ndarray
Value of the derivative vs x
See Also
--------
mathieu_a, mathieu_b, mathieu_cem
""")
add_newdoc("modfresnelm",
"""
modfresnelm(x, out=None)
Modified Fresnel negative integrals
Parameters
----------
x : array_like
Function argument
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
fm : scalar or ndarray
Integral ``F_-(x)``: ``integral(exp(-1j*t*t), t=x..inf)``
km : scalar or ndarray
Integral ``K_-(x)``: ``1/sqrt(pi)*exp(1j*(x*x+pi/4))*fp``
See Also
--------
modfresnelp
""")
add_newdoc("modfresnelp",
"""
modfresnelp(x, out=None)
Modified Fresnel positive integrals
Parameters
----------
x : array_like
Function argument
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
fp : scalar or ndarray
Integral ``F_+(x)``: ``integral(exp(1j*t*t), t=x..inf)``
kp : scalar or ndarray
Integral ``K_+(x)``: ``1/sqrt(pi)*exp(-1j*(x*x+pi/4))*fp``
See Also
--------
modfresnelm
""")
add_newdoc("modstruve",
r"""
modstruve(v, x, out=None)
Modified Struve function.
Return the value of the modified Struve function of order `v` at `x`. The
modified Struve function is defined as,
.. math::
L_v(x) = -\imath \exp(-\pi\imath v/2) H_v(\imath x),
where :math:`H_v` is the Struve function.
Parameters
----------
v : array_like
Order of the modified Struve function (float).
x : array_like
Argument of the Struve function (float; must be positive unless `v` is
an integer).
out : ndarray, optional
Optional output array for the function results
Returns
-------
L : scalar or ndarray
Value of the modified Struve function of order `v` at `x`.
See Also
--------
struve
Notes
-----
Three methods discussed in [1]_ are used to evaluate the function:
- power series
- expansion in Bessel functions (if :math:`|x| < |v| + 20`)
- asymptotic large-x expansion (if :math:`x \geq 0.7v + 12`)
Rounding errors are estimated based on the largest terms in the sums, and
the result associated with the smallest error is returned.
References
----------
.. [1] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/11
Examples
--------
Calculate the modified Struve function of order 1 at 2.
>>> import numpy as np
>>> from scipy.special import modstruve
>>> import matplotlib.pyplot as plt
>>> modstruve(1, 2.)
1.102759787367716
Calculate the modified Struve function at 2 for orders 1, 2 and 3 by
providing a list for the order parameter `v`.
>>> modstruve([1, 2, 3], 2.)
array([1.10275979, 0.41026079, 0.11247294])
Calculate the modified Struve function of order 1 for several points
by providing an array for `x`.
>>> points = np.array([2., 5., 8.])
>>> modstruve(1, points)
array([ 1.10275979, 23.72821578, 399.24709139])
Compute the modified Struve function for several orders at several
points by providing arrays for `v` and `z`. The arrays have to be
broadcastable to the correct shapes.
>>> orders = np.array([[1], [2], [3]])
>>> points.shape, orders.shape
((3,), (3, 1))
>>> modstruve(orders, points)
array([[1.10275979e+00, 2.37282158e+01, 3.99247091e+02],
[4.10260789e-01, 1.65535979e+01, 3.25973609e+02],
[1.12472937e-01, 9.42430454e+00, 2.33544042e+02]])
Plot the modified Struve functions of order 0 to 3 from -5 to 5.
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-5., 5., 1000)
>>> for i in range(4):
... ax.plot(x, modstruve(i, x), label=f'$L_{i!r}$')
>>> ax.legend(ncol=2)
>>> ax.set_xlim(-5, 5)
>>> ax.set_title(r"Modified Struve functions $L_{\nu}$")
>>> plt.show()
""")
add_newdoc("nbdtr",
r"""
nbdtr(k, n, p, out=None)
Negative binomial cumulative distribution function.
Returns the sum of the terms 0 through `k` of the negative binomial
distribution probability mass function,
.. math::
F = \sum_{j=0}^k {{n + j - 1}\choose{j}} p^n (1 - p)^j.
In a sequence of Bernoulli trials with individual success probabilities
`p`, this is the probability that `k` or fewer failures precede the nth
success.
Parameters
----------
k : array_like
The maximum number of allowed failures (nonnegative int).
n : array_like
The target number of successes (positive int).
p : array_like
Probability of success in a single event (float).
out : ndarray, optional
Optional output array for the function results
Returns
-------
F : scalar or ndarray
The probability of `k` or fewer failures before `n` successes in a
sequence of events with individual success probability `p`.
See Also
--------
nbdtrc : Negative binomial survival function
nbdtrik : Negative binomial quantile function
scipy.stats.nbinom : Negative binomial distribution
Notes
-----
If floating point values are passed for `k` or `n`, they will be truncated
to integers.
The terms are not summed directly; instead the regularized incomplete beta
function is employed, according to the formula,
.. math::
\mathrm{nbdtr}(k, n, p) = I_{p}(n, k + 1).
Wrapper for the Cephes [1]_ routine `nbdtr`.
The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtr` directly can improve performance
compared to the ``cdf`` method of `scipy.stats.nbinom` (see last example).
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``.
>>> import numpy as np
>>> from scipy.special import nbdtr
>>> nbdtr(10, 5, 0.5)
0.940765380859375
Compute the function for ``n=10`` and ``p=0.5`` at several points by
providing a NumPy array or list for `k`.
>>> nbdtr([5, 10, 15], 10, 0.5)
array([0.15087891, 0.58809853, 0.88523853])
Plot the function for four different parameter sets.
>>> import matplotlib.pyplot as plt
>>> k = np.arange(130)
>>> n_parameters = [20, 20, 20, 80]
>>> p_parameters = [0.2, 0.5, 0.8, 0.5]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(p_parameters, n_parameters,
... linestyles))
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
... p, n, style = parameter_set
... nbdtr_vals = nbdtr(k, n, p)
... ax.plot(k, nbdtr_vals, label=rf"$n={n},\, p={p}$",
... ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$k$")
>>> ax.set_title("Negative binomial cumulative distribution function")
>>> plt.show()
The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtr` directly can be much faster than
calling the ``cdf`` method of `scipy.stats.nbinom`, especially for small
arrays or individual values. To get the same results one must use the
following parametrization: ``nbinom(n, p).cdf(k)=nbdtr(k, n, p)``.
>>> from scipy.stats import nbinom
>>> k, n, p = 5, 3, 0.5
>>> nbdtr_res = nbdtr(k, n, p) # this will often be faster than below
>>> stats_res = nbinom(n, p).cdf(k)
>>> stats_res, nbdtr_res # test that results are equal
(0.85546875, 0.85546875)
`nbdtr` can evaluate different parameter sets by providing arrays with
shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute
the function for three different `k` at four locations `p`, resulting in
a 3x4 array.
>>> k = np.array([[5], [10], [15]])
>>> p = np.array([0.3, 0.5, 0.7, 0.9])
>>> k.shape, p.shape
((3, 1), (4,))
>>> nbdtr(k, 5, p)
array([[0.15026833, 0.62304687, 0.95265101, 0.9998531 ],
[0.48450894, 0.94076538, 0.99932777, 0.99999999],
[0.76249222, 0.99409103, 0.99999445, 1. ]])
""")
add_newdoc("nbdtrc",
r"""
nbdtrc(k, n, p, out=None)
Negative binomial survival function.
Returns the sum of the terms `k + 1` to infinity of the negative binomial
distribution probability mass function,
.. math::
F = \sum_{j=k + 1}^\infty {{n + j - 1}\choose{j}} p^n (1 - p)^j.
In a sequence of Bernoulli trials with individual success probabilities
`p`, this is the probability that more than `k` failures precede the nth
success.
Parameters
----------
k : array_like
The maximum number of allowed failures (nonnegative int).
n : array_like
The target number of successes (positive int).
p : array_like
Probability of success in a single event (float).
out : ndarray, optional
Optional output array for the function results
Returns
-------
F : scalar or ndarray
The probability of `k + 1` or more failures before `n` successes in a
sequence of events with individual success probability `p`.
See Also
--------
nbdtr : Negative binomial cumulative distribution function
nbdtrik : Negative binomial percentile function
scipy.stats.nbinom : Negative binomial distribution
Notes
-----
If floating point values are passed for `k` or `n`, they will be truncated
to integers.
The terms are not summed directly; instead the regularized incomplete beta
function is employed, according to the formula,
.. math::
\mathrm{nbdtrc}(k, n, p) = I_{1 - p}(k + 1, n).
Wrapper for the Cephes [1]_ routine `nbdtrc`.
The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtrc` directly can improve performance
compared to the ``sf`` method of `scipy.stats.nbinom` (see last example).
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``.
>>> import numpy as np
>>> from scipy.special import nbdtrc
>>> nbdtrc(10, 5, 0.5)
0.059234619140624986
Compute the function for ``n=10`` and ``p=0.5`` at several points by
providing a NumPy array or list for `k`.
>>> nbdtrc([5, 10, 15], 10, 0.5)
array([0.84912109, 0.41190147, 0.11476147])
Plot the function for four different parameter sets.
>>> import matplotlib.pyplot as plt
>>> k = np.arange(130)
>>> n_parameters = [20, 20, 20, 80]
>>> p_parameters = [0.2, 0.5, 0.8, 0.5]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(p_parameters, n_parameters,
... linestyles))
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
... p, n, style = parameter_set
... nbdtrc_vals = nbdtrc(k, n, p)
... ax.plot(k, nbdtrc_vals, label=rf"$n={n},\, p={p}$",
... ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$k$")
>>> ax.set_title("Negative binomial distribution survival function")
>>> plt.show()
The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtrc` directly can be much faster than
calling the ``sf`` method of `scipy.stats.nbinom`, especially for small
arrays or individual values. To get the same results one must use the
following parametrization: ``nbinom(n, p).sf(k)=nbdtrc(k, n, p)``.
>>> from scipy.stats import nbinom
>>> k, n, p = 3, 5, 0.5
>>> nbdtr_res = nbdtrc(k, n, p) # this will often be faster than below
>>> stats_res = nbinom(n, p).sf(k)
>>> stats_res, nbdtr_res # test that results are equal
(0.6367187499999999, 0.6367187499999999)
`nbdtrc` can evaluate different parameter sets by providing arrays with
shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute
the function for three different `k` at four locations `p`, resulting in
a 3x4 array.
>>> k = np.array([[5], [10], [15]])
>>> p = np.array([0.3, 0.5, 0.7, 0.9])
>>> k.shape, p.shape
((3, 1), (4,))
>>> nbdtrc(k, 5, p)
array([[8.49731667e-01, 3.76953125e-01, 4.73489874e-02, 1.46902600e-04],
[5.15491059e-01, 5.92346191e-02, 6.72234070e-04, 9.29610100e-09],
[2.37507779e-01, 5.90896606e-03, 5.55025308e-06, 3.26346760e-13]])
""")
add_newdoc(
"nbdtri",
r"""
nbdtri(k, n, y, out=None)
Returns the inverse with respect to the parameter `p` of
`y = nbdtr(k, n, p)`, the negative binomial cumulative distribution
function.
Parameters
----------
k : array_like
The maximum number of allowed failures (nonnegative int).
n : array_like
The target number of successes (positive int).
y : array_like
The probability of `k` or fewer failures before `n` successes (float).
out : ndarray, optional
Optional output array for the function results
Returns
-------
p : scalar or ndarray
Probability of success in a single event (float) such that
`nbdtr(k, n, p) = y`.
See Also
--------
nbdtr : Cumulative distribution function of the negative binomial.
nbdtrc : Negative binomial survival function.
scipy.stats.nbinom : negative binomial distribution.
nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`.
nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`.
scipy.stats.nbinom : Negative binomial distribution
Notes
-----
Wrapper for the Cephes [1]_ routine `nbdtri`.
The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtri` directly can improve performance
compared to the ``ppf`` method of `scipy.stats.nbinom`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
`nbdtri` is the inverse of `nbdtr` with respect to `p`.
Up to floating point errors the following holds:
``nbdtri(k, n, nbdtr(k, n, p))=p``.
>>> import numpy as np
>>> from scipy.special import nbdtri, nbdtr
>>> k, n, y = 5, 10, 0.2
>>> cdf_val = nbdtr(k, n, y)
>>> nbdtri(k, n, cdf_val)
0.20000000000000004
Compute the function for ``k=10`` and ``n=5`` at several points by
providing a NumPy array or list for `y`.
>>> y = np.array([0.1, 0.4, 0.8])
>>> nbdtri(3, 5, y)
array([0.34462319, 0.51653095, 0.69677416])
Plot the function for three different parameter sets.
>>> import matplotlib.pyplot as plt
>>> n_parameters = [5, 20, 30, 30]
>>> k_parameters = [20, 20, 60, 80]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(n_parameters, k_parameters, linestyles))
>>> cdf_vals = np.linspace(0, 1, 1000)
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
... n, k, style = parameter_set
... nbdtri_vals = nbdtri(k, n, cdf_vals)
... ax.plot(cdf_vals, nbdtri_vals, label=rf"$k={k},\ n={n}$",
... ls=style)
>>> ax.legend()
>>> ax.set_ylabel("$p$")
>>> ax.set_xlabel("$CDF$")
>>> title = "nbdtri: inverse of negative binomial CDF with respect to $p$"
>>> ax.set_title(title)
>>> plt.show()
`nbdtri` can evaluate different parameter sets by providing arrays with
shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute
the function for three different `k` at four locations `p`, resulting in
a 3x4 array.
>>> k = np.array([[5], [10], [15]])
>>> y = np.array([0.3, 0.5, 0.7, 0.9])
>>> k.shape, y.shape
((3, 1), (4,))
>>> nbdtri(k, 5, y)
array([[0.37258157, 0.45169416, 0.53249956, 0.64578407],
[0.24588501, 0.30451981, 0.36778453, 0.46397088],
[0.18362101, 0.22966758, 0.28054743, 0.36066188]])
""")
add_newdoc("nbdtrik",
r"""
nbdtrik(y, n, p, out=None)
Negative binomial percentile function.
Returns the inverse with respect to the parameter `k` of
`y = nbdtr(k, n, p)`, the negative binomial cumulative distribution
function.
Parameters
----------
y : array_like
The probability of `k` or fewer failures before `n` successes (float).
n : array_like
The target number of successes (positive int).
p : array_like
Probability of success in a single event (float).
out : ndarray, optional
Optional output array for the function results
Returns
-------
k : scalar or ndarray
The maximum number of allowed failures such that `nbdtr(k, n, p) = y`.
See Also
--------
nbdtr : Cumulative distribution function of the negative binomial.
nbdtrc : Survival function of the negative binomial.
nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`.
nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`.
scipy.stats.nbinom : Negative binomial distribution
Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`.
Formula 26.5.26 of [2]_,
.. math::
\sum_{j=k + 1}^\infty {{n + j - 1}
\choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n),
is used to reduce calculation of the cumulative distribution function to
that of a regularized incomplete beta :math:`I`.
Computation of `k` involves a search for a value that produces the desired
value of `y`. The search relies on the monotonicity of `y` with `k`.
References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
Examples
--------
Compute the negative binomial cumulative distribution function for an
exemplary parameter set.
>>> import numpy as np
>>> from scipy.special import nbdtr, nbdtrik
>>> k, n, p = 5, 2, 0.5
>>> cdf_value = nbdtr(k, n, p)
>>> cdf_value
0.9375
Verify that `nbdtrik` recovers the original value for `k`.
>>> nbdtrik(cdf_value, n, p)
5.0
Plot the function for different parameter sets.
>>> import matplotlib.pyplot as plt
>>> p_parameters = [0.2, 0.5, 0.7, 0.5]
>>> n_parameters = [30, 30, 30, 80]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(p_parameters, n_parameters, linestyles))
>>> cdf_vals = np.linspace(0, 1, 1000)
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
... p, n, style = parameter_set
... nbdtrik_vals = nbdtrik(cdf_vals, n, p)
... ax.plot(cdf_vals, nbdtrik_vals, label=rf"$n={n},\ p={p}$",
... ls=style)
>>> ax.legend()
>>> ax.set_ylabel("$k$")
>>> ax.set_xlabel("$CDF$")
>>> ax.set_title("Negative binomial percentile function")
>>> plt.show()
The negative binomial distribution is also available as
`scipy.stats.nbinom`. The percentile function method ``ppf``
returns the result of `nbdtrik` rounded up to integers:
>>> from scipy.stats import nbinom
>>> q, n, p = 0.6, 5, 0.5
>>> nbinom.ppf(q, n, p), nbdtrik(q, n, p)
(5.0, 4.800428460273882)
""")
add_newdoc("nbdtrin",
r"""
nbdtrin(k, y, p, out=None)
Inverse of `nbdtr` vs `n`.
Returns the inverse with respect to the parameter `n` of
`y = nbdtr(k, n, p)`, the negative binomial cumulative distribution
function.
Parameters
----------
k : array_like
The maximum number of allowed failures (nonnegative int).
y : array_like
The probability of `k` or fewer failures before `n` successes (float).
p : array_like
Probability of success in a single event (float).
out : ndarray, optional
Optional output array for the function results
Returns
-------
n : scalar or ndarray
The number of successes `n` such that `nbdtr(k, n, p) = y`.
See Also
--------
nbdtr : Cumulative distribution function of the negative binomial.
nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`.
nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`.
Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`.
Formula 26.5.26 of [2]_,
.. math::
\sum_{j=k + 1}^\infty {{n + j - 1}
\choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n),
is used to reduce calculation of the cumulative distribution function to
that of a regularized incomplete beta :math:`I`.
Computation of `n` involves a search for a value that produces the desired
value of `y`. The search relies on the monotonicity of `y` with `n`.
References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
Examples
--------
Compute the negative binomial cumulative distribution function for an
exemplary parameter set.
>>> from scipy.special import nbdtr, nbdtrin
>>> k, n, p = 5, 2, 0.5
>>> cdf_value = nbdtr(k, n, p)
>>> cdf_value
0.9375
Verify that `nbdtrin` recovers the original value for `n` up to floating
point accuracy.
>>> nbdtrin(k, cdf_value, p)
1.999999999998137
""")
add_newdoc("ncfdtr",
r"""
ncfdtr(dfn, dfd, nc, f, out=None)
Cumulative distribution function of the non-central F distribution.
The non-central F describes the distribution of,
.. math::
Z = \frac{X/d_n}{Y/d_d}
where :math:`X` and :math:`Y` are independently distributed, with
:math:`X` distributed non-central :math:`\chi^2` with noncentrality
parameter `nc` and :math:`d_n` degrees of freedom, and :math:`Y`
distributed :math:`\chi^2` with :math:`d_d` degrees of freedom.
Parameters
----------
dfn : array_like
Degrees of freedom of the numerator sum of squares. Range (0, inf).
dfd : array_like
Degrees of freedom of the denominator sum of squares. Range (0, inf).
nc : array_like
Noncentrality parameter. Should be in range (0, 1e4).
f : array_like
Quantiles, i.e. the upper limit of integration.
out : ndarray, optional
Optional output array for the function results
Returns
-------
cdf : scalar or ndarray
The calculated CDF. If all inputs are scalar, the return will be a
float. Otherwise it will be an array.
See Also
--------
ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.
ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.
ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.
ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.
Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdffnc`.
The cumulative distribution function is computed using Formula 26.6.20 of
[2]_:
.. math::
F(d_n, d_d, n_c, f) = \sum_{j=0}^\infty e^{-n_c/2}
\frac{(n_c/2)^j}{j!} I_{x}(\frac{d_n}{2} + j, \frac{d_d}{2}),
where :math:`I` is the regularized incomplete beta function, and
:math:`x = f d_n/(f d_n + d_d)`.
The computation time required for this routine is proportional to the
noncentrality parameter `nc`. Very large values of this parameter can
consume immense computer resources. This is why the search range is
bounded by 10,000.
References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
CDFLIB: Library of Fortran Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters.
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
Plot the CDF of the non-central F distribution, for nc=0. Compare with the
F-distribution from scipy.stats:
>>> x = np.linspace(-1, 8, num=500)
>>> dfn = 3
>>> dfd = 2
>>> ncf_stats = stats.f.cdf(x, dfn, dfd)
>>> ncf_special = special.ncfdtr(dfn, dfd, 0, x)
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, ncf_stats, 'b-', lw=3)
>>> ax.plot(x, ncf_special, 'r-')
>>> plt.show()
""")
add_newdoc("ncfdtri",
"""
ncfdtri(dfn, dfd, nc, p, out=None)
Inverse with respect to `f` of the CDF of the non-central F distribution.
See `ncfdtr` for more details.
Parameters
----------
dfn : array_like
Degrees of freedom of the numerator sum of squares. Range (0, inf).
dfd : array_like
Degrees of freedom of the denominator sum of squares. Range (0, inf).
nc : array_like
Noncentrality parameter. Should be in range (0, 1e4).
p : array_like
Value of the cumulative distribution function. Must be in the
range [0, 1].
out : ndarray, optional
Optional output array for the function results
Returns
-------
f : scalar or ndarray
Quantiles, i.e., the upper limit of integration.
See Also
--------
ncfdtr : CDF of the non-central F distribution.
ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.
ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.
ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.
Examples
--------
>>> from scipy.special import ncfdtr, ncfdtri
Compute the CDF for several values of `f`:
>>> f = [0.5, 1, 1.5]
>>> p = ncfdtr(2, 3, 1.5, f)
>>> p
array([ 0.20782291, 0.36107392, 0.47345752])
Compute the inverse. We recover the values of `f`, as expected:
>>> ncfdtri(2, 3, 1.5, p)
array([ 0.5, 1. , 1.5])
""")
add_newdoc("ncfdtridfd",
"""
ncfdtridfd(dfn, p, nc, f, out=None)
Calculate degrees of freedom (denominator) for the noncentral F-distribution.
This is the inverse with respect to `dfd` of `ncfdtr`.
See `ncfdtr` for more details.
Parameters
----------
dfn : array_like
Degrees of freedom of the numerator sum of squares. Range (0, inf).
p : array_like
Value of the cumulative distribution function. Must be in the
range [0, 1].
nc : array_like
Noncentrality parameter. Should be in range (0, 1e4).
f : array_like
Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
Optional output array for the function results
Returns
-------
dfd : scalar or ndarray
Degrees of freedom of the denominator sum of squares.
See Also
--------
ncfdtr : CDF of the non-central F distribution.
ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.
ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.
ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.
Notes
-----
The value of the cumulative noncentral F distribution is not necessarily
monotone in either degrees of freedom. There thus may be two values that
provide a given CDF value. This routine assumes monotonicity and will
find an arbitrary one of the two values.
Examples
--------
>>> from scipy.special import ncfdtr, ncfdtridfd
Compute the CDF for several values of `dfd`:
>>> dfd = [1, 2, 3]
>>> p = ncfdtr(2, dfd, 0.25, 15)
>>> p
array([ 0.8097138 , 0.93020416, 0.96787852])
Compute the inverse. We recover the values of `dfd`, as expected:
>>> ncfdtridfd(2, p, 0.25, 15)
array([ 1., 2., 3.])
""")
add_newdoc("ncfdtridfn",
"""
ncfdtridfn(p, dfd, nc, f, out=None)
Calculate degrees of freedom (numerator) for the noncentral F-distribution.
This is the inverse with respect to `dfn` of `ncfdtr`.
See `ncfdtr` for more details.
Parameters
----------
p : array_like
Value of the cumulative distribution function. Must be in the
range [0, 1].
dfd : array_like
Degrees of freedom of the denominator sum of squares. Range (0, inf).
nc : array_like
Noncentrality parameter. Should be in range (0, 1e4).
f : float
Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
Optional output array for the function results
Returns
-------
dfn : scalar or ndarray
Degrees of freedom of the numerator sum of squares.
See Also
--------
ncfdtr : CDF of the non-central F distribution.
ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.
ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.
ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.
Notes
-----
The value of the cumulative noncentral F distribution is not necessarily
monotone in either degrees of freedom. There thus may be two values that
provide a given CDF value. This routine assumes monotonicity and will
find an arbitrary one of the two values.
Examples
--------
>>> from scipy.special import ncfdtr, ncfdtridfn
Compute the CDF for several values of `dfn`:
>>> dfn = [1, 2, 3]
>>> p = ncfdtr(dfn, 2, 0.25, 15)
>>> p
array([ 0.92562363, 0.93020416, 0.93188394])
Compute the inverse. We recover the values of `dfn`, as expected:
>>> ncfdtridfn(p, 2, 0.25, 15)
array([ 1., 2., 3.])
""")
add_newdoc("ncfdtrinc",
"""
ncfdtrinc(dfn, dfd, p, f, out=None)
Calculate non-centrality parameter for non-central F distribution.
This is the inverse with respect to `nc` of `ncfdtr`.
See `ncfdtr` for more details.
Parameters
----------
dfn : array_like
Degrees of freedom of the numerator sum of squares. Range (0, inf).
dfd : array_like
Degrees of freedom of the denominator sum of squares. Range (0, inf).
p : array_like
Value of the cumulative distribution function. Must be in the
range [0, 1].
f : array_like
Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
Optional output array for the function results
Returns
-------
nc : scalar or ndarray
Noncentrality parameter.
See Also
--------
ncfdtr : CDF of the non-central F distribution.
ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.
ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.
ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.
Examples
--------
>>> from scipy.special import ncfdtr, ncfdtrinc
Compute the CDF for several values of `nc`:
>>> nc = [0.5, 1.5, 2.0]
>>> p = ncfdtr(2, 3, nc, 15)
>>> p
array([ 0.96309246, 0.94327955, 0.93304098])
Compute the inverse. We recover the values of `nc`, as expected:
>>> ncfdtrinc(2, 3, p, 15)
array([ 0.5, 1.5, 2. ])
""")
add_newdoc("nctdtr",
"""
nctdtr(df, nc, t, out=None)
Cumulative distribution function of the non-central `t` distribution.
Parameters
----------
df : array_like
Degrees of freedom of the distribution. Should be in range (0, inf).
nc : array_like
Noncentrality parameter. Should be in range (-1e6, 1e6).
t : array_like
Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
Optional output array for the function results
Returns
-------
cdf : scalar or ndarray
The calculated CDF. If all inputs are scalar, the return will be a
float. Otherwise, it will be an array.
See Also
--------
nctdtrit : Inverse CDF (iCDF) of the non-central t distribution.
nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.
nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
Plot the CDF of the non-central t distribution, for nc=0. Compare with the
t-distribution from scipy.stats:
>>> x = np.linspace(-5, 5, num=500)
>>> df = 3
>>> nct_stats = stats.t.cdf(x, df)
>>> nct_special = special.nctdtr(df, 0, x)
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, nct_stats, 'b-', lw=3)
>>> ax.plot(x, nct_special, 'r-')
>>> plt.show()
""")
add_newdoc("nctdtridf",
"""
nctdtridf(p, nc, t, out=None)
Calculate degrees of freedom for non-central t distribution.
See `nctdtr` for more details.
Parameters
----------
p : array_like
CDF values, in range (0, 1].
nc : array_like
Noncentrality parameter. Should be in range (-1e6, 1e6).
t : array_like
Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
Optional output array for the function results
Returns
-------
df : scalar or ndarray
The degrees of freedom. If all inputs are scalar, the return will be a
float. Otherwise, it will be an array.
See Also
--------
nctdtr : CDF of the non-central `t` distribution.
nctdtrit : Inverse CDF (iCDF) of the non-central t distribution.
nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.
Examples
--------
>>> from scipy.special import nctdtr, nctdtridf
Compute the CDF for several values of `df`:
>>> df = [1, 2, 3]
>>> p = nctdtr(df, 0.25, 1)
>>> p
array([0.67491974, 0.716464 , 0.73349456])
Compute the inverse. We recover the values of `df`, as expected:
>>> nctdtridf(p, 0.25, 1)
array([1., 2., 3.])
""")
add_newdoc("nctdtrinc",
"""
nctdtrinc(df, p, t, out=None)
Calculate non-centrality parameter for non-central t distribution.
See `nctdtr` for more details.
Parameters
----------
df : array_like
Degrees of freedom of the distribution. Should be in range (0, inf).
p : array_like
CDF values, in range (0, 1].
t : array_like
Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
Optional output array for the function results
Returns
-------
nc : scalar or ndarray
Noncentrality parameter
See Also
--------
nctdtr : CDF of the non-central `t` distribution.
nctdtrit : Inverse CDF (iCDF) of the non-central t distribution.
nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.
Examples
--------
>>> from scipy.special import nctdtr, nctdtrinc
Compute the CDF for several values of `nc`:
>>> nc = [0.5, 1.5, 2.5]
>>> p = nctdtr(3, nc, 1.5)
>>> p
array([0.77569497, 0.45524533, 0.1668691 ])
Compute the inverse. We recover the values of `nc`, as expected:
>>> nctdtrinc(3, p, 1.5)
array([0.5, 1.5, 2.5])
""")
add_newdoc("nctdtrit",
"""
nctdtrit(df, nc, p, out=None)
Inverse cumulative distribution function of the non-central t distribution.
See `nctdtr` for more details.
Parameters
----------
df : array_like
Degrees of freedom of the distribution. Should be in range (0, inf).
nc : array_like
Noncentrality parameter. Should be in range (-1e6, 1e6).
p : array_like
CDF values, in range (0, 1].
out : ndarray, optional
Optional output array for the function results
Returns
-------
t : scalar or ndarray
Quantiles
See Also
--------
nctdtr : CDF of the non-central `t` distribution.
nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.
nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.
Examples
--------
>>> from scipy.special import nctdtr, nctdtrit
Compute the CDF for several values of `t`:
>>> t = [0.5, 1, 1.5]
>>> p = nctdtr(3, 1, t)
>>> p
array([0.29811049, 0.46922687, 0.6257559 ])
Compute the inverse. We recover the values of `t`, as expected:
>>> nctdtrit(3, 1, p)
array([0.5, 1. , 1.5])
""")
add_newdoc("ndtr",
r"""
ndtr(x, out=None)
Cumulative distribution of the standard normal distribution.
Returns the area under the standard Gaussian probability
density function, integrated from minus infinity to `x`
.. math::
\frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp(-t^2/2) dt
Parameters
----------
x : array_like, real or complex
Argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The value of the normal CDF evaluated at `x`
See Also
--------
log_ndtr : Logarithm of ndtr
ndtri : Inverse of ndtr, standard normal percentile function
erf : Error function
erfc : 1 - erf
scipy.stats.norm : Normal distribution
Examples
--------
Evaluate `ndtr` at one point.
>>> import numpy as np
>>> from scipy.special import ndtr
>>> ndtr(0.5)
0.6914624612740131
Evaluate the function at several points by providing a NumPy array
or list for `x`.
>>> ndtr([0, 0.5, 2])
array([0.5 , 0.69146246, 0.97724987])
Plot the function.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-5, 5, 100)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, ndtr(x))
>>> ax.set_title(r"Standard normal cumulative distribution function $\Phi$")
>>> plt.show()
""")
add_newdoc("nrdtrimn",
"""
nrdtrimn(p, std, x, out=None)
Calculate mean of normal distribution given other params.
Parameters
----------
p : array_like
CDF values, in range (0, 1].
std : array_like
Standard deviation.
x : array_like
Quantiles, i.e. the upper limit of integration.
out : ndarray, optional
Optional output array for the function results
Returns
-------
mn : scalar or ndarray
The mean of the normal distribution.
See Also
--------
scipy.stats.norm : Normal distribution
ndtr : Standard normal cumulative probability distribution
ndtri : Inverse of standard normal CDF with respect to quantile
nrdtrisd : Inverse of normal distribution CDF with respect to
standard deviation
Examples
--------
`nrdtrimn` can be used to recover the mean of a normal distribution
if we know the CDF value `p` for a given quantile `x` and the
standard deviation `std`. First, we calculate
the normal distribution CDF for an exemplary parameter set.
>>> from scipy.stats import norm
>>> mean = 3.
>>> std = 2.
>>> x = 6.
>>> p = norm.cdf(x, loc=mean, scale=std)
>>> p
0.9331927987311419
Verify that `nrdtrimn` returns the original value for `mean`.
>>> from scipy.special import nrdtrimn
>>> nrdtrimn(p, std, x)
3.0000000000000004
""")
add_newdoc("nrdtrisd",
"""
nrdtrisd(mn, p, x, out=None)
Calculate standard deviation of normal distribution given other params.
Parameters
----------
mn : scalar or ndarray
The mean of the normal distribution.
p : array_like
CDF values, in range (0, 1].
x : array_like
Quantiles, i.e. the upper limit of integration.
out : ndarray, optional
Optional output array for the function results
Returns
-------
std : scalar or ndarray
Standard deviation.
See Also
--------
scipy.stats.norm : Normal distribution
ndtr : Standard normal cumulative probability distribution
ndtri : Inverse of standard normal CDF with respect to quantile
nrdtrimn : Inverse of normal distribution CDF with respect to
mean
Examples
--------
`nrdtrisd` can be used to recover the standard deviation of a normal
distribution if we know the CDF value `p` for a given quantile `x` and
the mean `mn`. First, we calculate the normal distribution CDF for an
exemplary parameter set.
>>> from scipy.stats import norm
>>> mean = 3.
>>> std = 2.
>>> x = 6.
>>> p = norm.cdf(x, loc=mean, scale=std)
>>> p
0.9331927987311419
Verify that `nrdtrisd` returns the original value for `std`.
>>> from scipy.special import nrdtrisd
>>> nrdtrisd(mean, p, x)
2.0000000000000004
""")
add_newdoc("log_ndtr",
"""
log_ndtr(x, out=None)
Logarithm of Gaussian cumulative distribution function.
Returns the log of the area under the standard Gaussian probability
density function, integrated from minus infinity to `x`::
log(1/sqrt(2*pi) * integral(exp(-t**2 / 2), t=-inf..x))
Parameters
----------
x : array_like, real or complex
Argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The value of the log of the normal CDF evaluated at `x`
See Also
--------
erf
erfc
scipy.stats.norm
ndtr
Examples
--------
>>> import numpy as np
>>> from scipy.special import log_ndtr, ndtr
The benefit of ``log_ndtr(x)`` over the naive implementation
``np.log(ndtr(x))`` is most evident with moderate to large positive
values of ``x``:
>>> x = np.array([6, 7, 9, 12, 15, 25])
>>> log_ndtr(x)
array([-9.86587646e-010, -1.27981254e-012, -1.12858841e-019,
-1.77648211e-033, -3.67096620e-051, -3.05669671e-138])
The results of the naive calculation for the moderate ``x`` values
have only 5 or 6 correct significant digits. For values of ``x``
greater than approximately 8.3, the naive expression returns 0:
>>> np.log(ndtr(x))
array([-9.86587701e-10, -1.27986510e-12, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00])
""")
add_newdoc("ndtri",
"""
ndtri(y, out=None)
Inverse of `ndtr` vs x
Returns the argument x for which the area under the standard normal
probability density function (integrated from minus infinity to `x`)
is equal to y.
Parameters
----------
p : array_like
Probability
out : ndarray, optional
Optional output array for the function results
Returns
-------
x : scalar or ndarray
Value of x such that ``ndtr(x) == p``.
See Also
--------
ndtr : Standard normal cumulative probability distribution
ndtri_exp : Inverse of log_ndtr
Examples
--------
`ndtri` is the percentile function of the standard normal distribution.
This means it returns the inverse of the cumulative density `ndtr`. First,
let us compute a cumulative density value.
>>> import numpy as np
>>> from scipy.special import ndtri, ndtr
>>> cdf_val = ndtr(2)
>>> cdf_val
0.9772498680518208
Verify that `ndtri` yields the original value for `x` up to floating point
errors.
>>> ndtri(cdf_val)
2.0000000000000004
Plot the function. For that purpose, we provide a NumPy array as argument.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0.01, 1, 200)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, ndtri(x))
>>> ax.set_title("Standard normal percentile function")
>>> plt.show()
""")
add_newdoc("obl_ang1",
"""
obl_ang1(m, n, c, x, out=None)
Oblate spheroidal angular function of the first kind and its derivative
Computes the oblate spheroidal angular function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.
Parameters
----------
m : array_like
Mode parameter m (nonnegative)
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
x : array_like
Parameter x (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
See Also
--------
obl_ang1_cv
""")
add_newdoc("obl_ang1_cv",
"""
obl_ang1_cv(m, n, c, cv, x, out=None)
Oblate spheroidal angular function obl_ang1 for precomputed characteristic value
Computes the oblate spheroidal angular function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.
Parameters
----------
m : array_like
Mode parameter m (nonnegative)
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
cv : array_like
Characteristic value
x : array_like
Parameter x (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
See Also
--------
obl_ang1
""")
add_newdoc("obl_cv",
"""
obl_cv(m, n, c, out=None)
Characteristic value of oblate spheroidal function
Computes the characteristic value of oblate spheroidal wave
functions of order `m`, `n` (n>=m) and spheroidal parameter `c`.
Parameters
----------
m : array_like
Mode parameter m (nonnegative)
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
out : ndarray, optional
Optional output array for the function results
Returns
-------
cv : scalar or ndarray
Characteristic value
""")
add_newdoc("obl_rad1",
"""
obl_rad1(m, n, c, x, out=None)
Oblate spheroidal radial function of the first kind and its derivative
Computes the oblate spheroidal radial function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.
Parameters
----------
m : array_like
Mode parameter m (nonnegative)
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
x : array_like
Parameter x (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
See Also
--------
obl_rad1_cv
""")
add_newdoc("obl_rad1_cv",
"""
obl_rad1_cv(m, n, c, cv, x, out=None)
Oblate spheroidal radial function obl_rad1 for precomputed characteristic value
Computes the oblate spheroidal radial function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.
Parameters
----------
m : array_like
Mode parameter m (nonnegative)
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
cv : array_like
Characteristic value
x : array_like
Parameter x (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
See Also
--------
obl_rad1
""")
add_newdoc("obl_rad2",
"""
obl_rad2(m, n, c, x, out=None)
Oblate spheroidal radial function of the second kind and its derivative.
Computes the oblate spheroidal radial function of the second kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.
Parameters
----------
m : array_like
Mode parameter m (nonnegative)
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
x : array_like
Parameter x (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
See Also
--------
obl_rad2_cv
""")
add_newdoc("obl_rad2_cv",
"""
obl_rad2_cv(m, n, c, cv, x, out=None)
Oblate spheroidal radial function obl_rad2 for precomputed characteristic value
Computes the oblate spheroidal radial function of the second kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.
Parameters
----------
m : array_like
Mode parameter m (nonnegative)
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
cv : array_like
Characteristic value
x : array_like
Parameter x (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
See Also
--------
obl_rad2
""")
add_newdoc("pbdv",
"""
pbdv(v, x, out=None)
Parabolic cylinder function D
Returns (d, dp) the parabolic cylinder function Dv(x) in d and the
derivative, Dv'(x) in dp.
Parameters
----------
v : array_like
Real parameter
x : array_like
Real argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
d : scalar or ndarray
Value of the function
dp : scalar or ndarray
Value of the derivative vs x
""")
add_newdoc("pbvv",
"""
pbvv(v, x, out=None)
Parabolic cylinder function V
Returns the parabolic cylinder function Vv(x) in v and the
derivative, Vv'(x) in vp.
Parameters
----------
v : array_like
Real parameter
x : array_like
Real argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
v : scalar or ndarray
Value of the function
vp : scalar or ndarray
Value of the derivative vs x
""")
add_newdoc("pbwa",
r"""
pbwa(a, x, out=None)
Parabolic cylinder function W.
The function is a particular solution to the differential equation
.. math::
y'' + \left(\frac{1}{4}x^2 - a\right)y = 0,
for a full definition see section 12.14 in [1]_.
Parameters
----------
a : array_like
Real parameter
x : array_like
Real argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
w : scalar or ndarray
Value of the function
wp : scalar or ndarray
Value of the derivative in x
Notes
-----
The function is a wrapper for a Fortran routine by Zhang and Jin
[2]_. The implementation is accurate only for ``|a|, |x| < 5`` and
returns NaN outside that range.
References
----------
.. [1] Digital Library of Mathematical Functions, 14.30.
https://dlmf.nist.gov/14.30
.. [2] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html
""")
add_newdoc("pdtr",
r"""
pdtr(k, m, out=None)
Poisson cumulative distribution function.
Defined as the probability that a Poisson-distributed random
variable with event rate :math:`m` is less than or equal to
:math:`k`. More concretely, this works out to be [1]_
.. math::
\exp(-m) \sum_{j = 0}^{\lfloor{k}\rfloor} \frac{m^j}{j!}.
Parameters
----------
k : array_like
Number of occurrences (nonnegative, real)
m : array_like
Shape parameter (nonnegative, real)
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the Poisson cumulative distribution function
See Also
--------
pdtrc : Poisson survival function
pdtrik : inverse of `pdtr` with respect to `k`
pdtri : inverse of `pdtr` with respect to `m`
References
----------
.. [1] https://en.wikipedia.org/wiki/Poisson_distribution
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is a cumulative distribution function, so it converges to 1
monotonically as `k` goes to infinity.
>>> sc.pdtr([1, 10, 100, np.inf], 1)
array([0.73575888, 0.99999999, 1. , 1. ])
It is discontinuous at integers and constant between integers.
>>> sc.pdtr([1, 1.5, 1.9, 2], 1)
array([0.73575888, 0.73575888, 0.73575888, 0.9196986 ])
""")
add_newdoc("pdtrc",
"""
pdtrc(k, m, out=None)
Poisson survival function
Returns the sum of the terms from k+1 to infinity of the Poisson
distribution: sum(exp(-m) * m**j / j!, j=k+1..inf) = gammainc(
k+1, m). Arguments must both be non-negative doubles.
Parameters
----------
k : array_like
Number of occurrences (nonnegative, real)
m : array_like
Shape parameter (nonnegative, real)
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the Poisson survival function
See Also
--------
pdtr : Poisson cumulative distribution function
pdtrik : inverse of `pdtr` with respect to `k`
pdtri : inverse of `pdtr` with respect to `m`
""")
add_newdoc("pdtri",
"""
pdtri(k, y, out=None)
Inverse to `pdtr` vs m
Returns the Poisson variable `m` such that the sum from 0 to `k` of
the Poisson density is equal to the given probability `y`:
calculated by ``gammaincinv(k + 1, y)``. `k` must be a nonnegative
integer and `y` between 0 and 1.
Parameters
----------
k : array_like
Number of occurrences (nonnegative, real)
y : array_like
Probability
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of the shape parameter `m` such that ``pdtr(k, m) = p``
See Also
--------
pdtr : Poisson cumulative distribution function
pdtrc : Poisson survival function
pdtrik : inverse of `pdtr` with respect to `k`
""")
add_newdoc("pdtrik",
"""
pdtrik(p, m, out=None)
Inverse to `pdtr` vs `m`.
Parameters
----------
m : array_like
Shape parameter (nonnegative, real)
p : array_like
Probability
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The number of occurrences `k` such that ``pdtr(k, m) = p``
See Also
--------
pdtr : Poisson cumulative distribution function
pdtrc : Poisson survival function
pdtri : inverse of `pdtr` with respect to `m`
""")
add_newdoc("poch",
r"""
poch(z, m, out=None)
Pochhammer symbol.
The Pochhammer symbol (rising factorial) is defined as
.. math::
(z)_m = \frac{\Gamma(z + m)}{\Gamma(z)}
For positive integer `m` it reads
.. math::
(z)_m = z (z + 1) ... (z + m - 1)
See [dlmf]_ for more details.
Parameters
----------
z, m : array_like
Real-valued arguments.
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The value of the function.
References
----------
.. [dlmf] Nist, Digital Library of Mathematical Functions
https://dlmf.nist.gov/5.2#iii
Examples
--------
>>> import scipy.special as sc
It is 1 when m is 0.
>>> sc.poch([1, 2, 3, 4], 0)
array([1., 1., 1., 1.])
For z equal to 1 it reduces to the factorial function.
>>> sc.poch(1, 5)
120.0
>>> 1 * 2 * 3 * 4 * 5
120
It can be expressed in terms of the gamma function.
>>> z, m = 3.7, 2.1
>>> sc.poch(z, m)
20.529581933776953
>>> sc.gamma(z + m) / sc.gamma(z)
20.52958193377696
""")
add_newdoc("powm1", """
powm1(x, y, out=None)
Computes ``x**y - 1``.
This function is useful when `y` is near 0, or when `x` is near 1.
The function is implemented for real types only (unlike ``numpy.power``,
which accepts complex inputs).
Parameters
----------
x : array_like
The base. Must be a real type (i.e. integer or float, not complex).
y : array_like
The exponent. Must be a real type (i.e. integer or float, not complex).
Returns
-------
array_like
Result of the calculation
Notes
-----
.. versionadded:: 1.10.0
The underlying code is implemented for single precision and double
precision floats only. Unlike `numpy.power`, integer inputs to
`powm1` are converted to floating point, and complex inputs are
not accepted.
Note the following edge cases:
* ``powm1(x, 0)`` returns 0 for any ``x``, including 0, ``inf``
and ``nan``.
* ``powm1(1, y)`` returns 0 for any ``y``, including ``nan``
and ``inf``.
Examples
--------
>>> import numpy as np
>>> from scipy.special import powm1
>>> x = np.array([1.2, 10.0, 0.9999999975])
>>> y = np.array([1e-9, 1e-11, 0.1875])
>>> powm1(x, y)
array([ 1.82321557e-10, 2.30258509e-11, -4.68749998e-10])
It can be verified that the relative errors in those results
are less than 2.5e-16.
Compare that to the result of ``x**y - 1``, where the
relative errors are all larger than 8e-8:
>>> x**y - 1
array([ 1.82321491e-10, 2.30258035e-11, -4.68750039e-10])
""")
add_newdoc("pro_ang1",
"""
pro_ang1(m, n, c, x, out=None)
Prolate spheroidal angular function of the first kind and its derivative
Computes the prolate spheroidal angular function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.
Parameters
----------
m : array_like
Nonnegative mode parameter m
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
x : array_like
Real parameter (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
""")
add_newdoc("pro_ang1_cv",
"""
pro_ang1_cv(m, n, c, cv, x, out=None)
Prolate spheroidal angular function pro_ang1 for precomputed characteristic value
Computes the prolate spheroidal angular function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.
Parameters
----------
m : array_like
Nonnegative mode parameter m
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
cv : array_like
Characteristic value
x : array_like
Real parameter (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
""")
add_newdoc("pro_cv",
"""
pro_cv(m, n, c, out=None)
Characteristic value of prolate spheroidal function
Computes the characteristic value of prolate spheroidal wave
functions of order `m`, `n` (n>=m) and spheroidal parameter `c`.
Parameters
----------
m : array_like
Nonnegative mode parameter m
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
out : ndarray, optional
Optional output array for the function results
Returns
-------
cv : scalar or ndarray
Characteristic value
""")
add_newdoc("pro_rad1",
"""
pro_rad1(m, n, c, x, out=None)
Prolate spheroidal radial function of the first kind and its derivative
Computes the prolate spheroidal radial function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.
Parameters
----------
m : array_like
Nonnegative mode parameter m
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
x : array_like
Real parameter (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
""")
add_newdoc("pro_rad1_cv",
"""
pro_rad1_cv(m, n, c, cv, x, out=None)
Prolate spheroidal radial function pro_rad1 for precomputed characteristic value
Computes the prolate spheroidal radial function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.
Parameters
----------
m : array_like
Nonnegative mode parameter m
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
cv : array_like
Characteristic value
x : array_like
Real parameter (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
""")
add_newdoc("pro_rad2",
"""
pro_rad2(m, n, c, x, out=None)
Prolate spheroidal radial function of the second kind and its derivative
Computes the prolate spheroidal radial function of the second kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.
Parameters
----------
m : array_like
Nonnegative mode parameter m
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
cv : array_like
Characteristic value
x : array_like
Real parameter (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
""")
add_newdoc("pro_rad2_cv",
"""
pro_rad2_cv(m, n, c, cv, x, out=None)
Prolate spheroidal radial function pro_rad2 for precomputed characteristic value
Computes the prolate spheroidal radial function of the second kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.
Parameters
----------
m : array_like
Nonnegative mode parameter m
n : array_like
Mode parameter n (>= m)
c : array_like
Spheroidal parameter
cv : array_like
Characteristic value
x : array_like
Real parameter (``|x| < 1.0``)
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Value of the function
sp : scalar or ndarray
Value of the derivative vs x
""")
add_newdoc("pseudo_huber",
r"""
pseudo_huber(delta, r, out=None)
Pseudo-Huber loss function.
.. math:: \mathrm{pseudo\_huber}(\delta, r) =
\delta^2 \left( \sqrt{ 1 + \left( \frac{r}{\delta} \right)^2 } - 1 \right)
Parameters
----------
delta : array_like
Input array, indicating the soft quadratic vs. linear loss changepoint.
r : array_like
Input array, possibly representing residuals.
out : ndarray, optional
Optional output array for the function results
Returns
-------
res : scalar or ndarray
The computed Pseudo-Huber loss function values.
See Also
--------
huber: Similar function which this function approximates
Notes
-----
Like `huber`, `pseudo_huber` often serves as a robust loss function
in statistics or machine learning to reduce the influence of outliers.
Unlike `huber`, `pseudo_huber` is smooth.
Typically, `r` represents residuals, the difference
between a model prediction and data. Then, for :math:`|r|\leq\delta`,
`pseudo_huber` resembles the squared error and for :math:`|r|>\delta` the
absolute error. This way, the Pseudo-Huber loss often achieves
a fast convergence in model fitting for small residuals like the squared
error loss function and still reduces the influence of outliers
(:math:`|r|>\delta`) like the absolute error loss. As :math:`\delta` is
the cutoff between squared and absolute error regimes, it has
to be tuned carefully for each problem. `pseudo_huber` is also
convex, making it suitable for gradient based optimization. [1]_ [2]_
.. versionadded:: 0.15.0
References
----------
.. [1] Hartley, Zisserman, "Multiple View Geometry in Computer Vision".
2003. Cambridge University Press. p. 619
.. [2] Charbonnier et al. "Deterministic edge-preserving regularization
in computed imaging". 1997. IEEE Trans. Image Processing.
6 (2): 298 - 311.
Examples
--------
Import all necessary modules.
>>> import numpy as np
>>> from scipy.special import pseudo_huber, huber
>>> import matplotlib.pyplot as plt
Calculate the function for ``delta=1`` at ``r=2``.
>>> pseudo_huber(1., 2.)
1.2360679774997898
Calculate the function at ``r=2`` for different `delta` by providing
a list or NumPy array for `delta`.
>>> pseudo_huber([1., 2., 4.], 3.)
array([2.16227766, 3.21110255, 4. ])
Calculate the function for ``delta=1`` at several points by providing
a list or NumPy array for `r`.
>>> pseudo_huber(2., np.array([1., 1.5, 3., 4.]))
array([0.47213595, 1. , 3.21110255, 4.94427191])
The function can be calculated for different `delta` and `r` by
providing arrays for both with compatible shapes for broadcasting.
>>> r = np.array([1., 2.5, 8., 10.])
>>> deltas = np.array([[1.], [5.], [9.]])
>>> print(r.shape, deltas.shape)
(4,) (3, 1)
>>> pseudo_huber(deltas, r)
array([[ 0.41421356, 1.6925824 , 7.06225775, 9.04987562],
[ 0.49509757, 2.95084972, 22.16990566, 30.90169944],
[ 0.49846624, 3.06693762, 27.37435121, 40.08261642]])
Plot the function for different `delta`.
>>> x = np.linspace(-4, 4, 500)
>>> deltas = [1, 2, 3]
>>> linestyles = ["dashed", "dotted", "dashdot"]
>>> fig, ax = plt.subplots()
>>> combined_plot_parameters = list(zip(deltas, linestyles))
>>> for delta, style in combined_plot_parameters:
... ax.plot(x, pseudo_huber(delta, x), label=rf"$\delta={delta}$",
... ls=style)
>>> ax.legend(loc="upper center")
>>> ax.set_xlabel("$x$")
>>> ax.set_title(r"Pseudo-Huber loss function $h_{\delta}(x)$")
>>> ax.set_xlim(-4, 4)
>>> ax.set_ylim(0, 8)
>>> plt.show()
Finally, illustrate the difference between `huber` and `pseudo_huber` by
plotting them and their gradients with respect to `r`. The plot shows
that `pseudo_huber` is continuously differentiable while `huber` is not
at the points :math:`\pm\delta`.
>>> def huber_grad(delta, x):
... grad = np.copy(x)
... linear_area = np.argwhere(np.abs(x) > delta)
... grad[linear_area]=delta*np.sign(x[linear_area])
... return grad
>>> def pseudo_huber_grad(delta, x):
... return x* (1+(x/delta)**2)**(-0.5)
>>> x=np.linspace(-3, 3, 500)
>>> delta = 1.
>>> fig, ax = plt.subplots(figsize=(7, 7))
>>> ax.plot(x, huber(delta, x), label="Huber", ls="dashed")
>>> ax.plot(x, huber_grad(delta, x), label="Huber Gradient", ls="dashdot")
>>> ax.plot(x, pseudo_huber(delta, x), label="Pseudo-Huber", ls="dotted")
>>> ax.plot(x, pseudo_huber_grad(delta, x), label="Pseudo-Huber Gradient",
... ls="solid")
>>> ax.legend(loc="upper center")
>>> plt.show()
""")
add_newdoc("psi",
"""
psi(z, out=None)
The digamma function.
The logarithmic derivative of the gamma function evaluated at ``z``.
Parameters
----------
z : array_like
Real or complex argument.
out : ndarray, optional
Array for the computed values of ``psi``.
Returns
-------
digamma : scalar or ndarray
Computed values of ``psi``.
Notes
-----
For large values not close to the negative real axis, ``psi`` is
computed using the asymptotic series (5.11.2) from [1]_. For small
arguments not close to the negative real axis, the recurrence
relation (5.5.2) from [1]_ is used until the argument is large
enough to use the asymptotic series. For values close to the
negative real axis, the reflection formula (5.5.4) from [1]_ is
used first. Note that ``psi`` has a family of zeros on the
negative real axis which occur between the poles at nonpositive
integers. Around the zeros the reflection formula suffers from
cancellation and the implementation loses precision. The sole
positive zero and the first negative zero, however, are handled
separately by precomputing series expansions using [2]_, so the
function should maintain full accuracy around the origin.
References
----------
.. [1] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/5
.. [2] Fredrik Johansson and others.
"mpmath: a Python library for arbitrary-precision floating-point arithmetic"
(Version 0.19) http://mpmath.org/
Examples
--------
>>> from scipy.special import psi
>>> z = 3 + 4j
>>> psi(z)
(1.55035981733341+1.0105022091860445j)
Verify psi(z) = psi(z + 1) - 1/z:
>>> psi(z + 1) - 1/z
(1.55035981733341+1.0105022091860445j)
""")
add_newdoc("radian",
"""
radian(d, m, s, out=None)
Convert from degrees to radians.
Returns the angle given in (d)egrees, (m)inutes, and (s)econds in
radians.
Parameters
----------
d : array_like
Degrees, can be real-valued.
m : array_like
Minutes, can be real-valued.
s : array_like
Seconds, can be real-valued.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Values of the inputs in radians.
Examples
--------
>>> import scipy.special as sc
There are many ways to specify an angle.
>>> sc.radian(90, 0, 0)
1.5707963267948966
>>> sc.radian(0, 60 * 90, 0)
1.5707963267948966
>>> sc.radian(0, 0, 60**2 * 90)
1.5707963267948966
The inputs can be real-valued.
>>> sc.radian(1.5, 0, 0)
0.02617993877991494
>>> sc.radian(1, 30, 0)
0.02617993877991494
""")
add_newdoc("rel_entr",
r"""
rel_entr(x, y, out=None)
Elementwise function for computing relative entropy.
.. math::
\mathrm{rel\_entr}(x, y) =
\begin{cases}
x \log(x / y) & x > 0, y > 0 \\
0 & x = 0, y \ge 0 \\
\infty & \text{otherwise}
\end{cases}
Parameters
----------
x, y : array_like
Input arrays
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Relative entropy of the inputs
See Also
--------
entr, kl_div, scipy.stats.entropy
Notes
-----
.. versionadded:: 0.15.0
This function is jointly convex in x and y.
The origin of this function is in convex programming; see
[1]_. Given two discrete probability distributions :math:`p_1,
\ldots, p_n` and :math:`q_1, \ldots, q_n`, the definition of relative
entropy in the context of *information theory* is
.. math::
\sum_{i = 1}^n \mathrm{rel\_entr}(p_i, q_i).
To compute the latter quantity, use `scipy.stats.entropy`.
See [2]_ for details.
References
----------
.. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*.
Cambridge University Press, 2004.
:doi:`https://doi.org/10.1017/CBO9780511804441`
.. [2] Kullback-Leibler divergence,
https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence
""")
add_newdoc("rgamma",
r"""
rgamma(z, out=None)
Reciprocal of the gamma function.
Defined as :math:`1 / \Gamma(z)`, where :math:`\Gamma` is the
gamma function. For more on the gamma function see `gamma`.
Parameters
----------
z : array_like
Real or complex valued input
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Function results
See Also
--------
gamma, gammaln, loggamma
Notes
-----
The gamma function has no zeros and has simple poles at
nonpositive integers, so `rgamma` is an entire function with zeros
at the nonpositive integers. See the discussion in [dlmf]_ for
more details.
References
----------
.. [dlmf] Nist, Digital Library of Mathematical functions,
https://dlmf.nist.gov/5.2#i
Examples
--------
>>> import scipy.special as sc
It is the reciprocal of the gamma function.
>>> sc.rgamma([1, 2, 3, 4])
array([1. , 1. , 0.5 , 0.16666667])
>>> 1 / sc.gamma([1, 2, 3, 4])
array([1. , 1. , 0.5 , 0.16666667])
It is zero at nonpositive integers.
>>> sc.rgamma([0, -1, -2, -3])
array([0., 0., 0., 0.])
It rapidly underflows to zero along the positive real axis.
>>> sc.rgamma([10, 100, 179])
array([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])
""")
add_newdoc("round",
"""
round(x, out=None)
Round to the nearest integer.
Returns the nearest integer to `x`. If `x` ends in 0.5 exactly,
the nearest even integer is chosen.
Parameters
----------
x : array_like
Real valued input.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
The nearest integers to the elements of `x`. The result is of
floating type, not integer type.
Examples
--------
>>> import scipy.special as sc
It rounds to even.
>>> sc.round([0.5, 1.5])
array([0., 2.])
""")
add_newdoc("shichi",
r"""
shichi(x, out=None)
Hyperbolic sine and cosine integrals.
The hyperbolic sine integral is
.. math::
\int_0^x \frac{\sinh{t}}{t}dt
and the hyperbolic cosine integral is
.. math::
\gamma + \log(x) + \int_0^x \frac{\cosh{t} - 1}{t} dt
where :math:`\gamma` is Euler's constant and :math:`\log` is the
principal branch of the logarithm [1]_.
Parameters
----------
x : array_like
Real or complex points at which to compute the hyperbolic sine
and cosine integrals.
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
si : scalar or ndarray
Hyperbolic sine integral at ``x``
ci : scalar or ndarray
Hyperbolic cosine integral at ``x``
See Also
--------
sici : Sine and cosine integrals.
exp1 : Exponential integral E1.
expi : Exponential integral Ei.
Notes
-----
For real arguments with ``x < 0``, ``chi`` is the real part of the
hyperbolic cosine integral. For such points ``chi(x)`` and ``chi(x
+ 0j)`` differ by a factor of ``1j*pi``.
For real arguments the function is computed by calling Cephes'
[2]_ *shichi* routine. For complex arguments the algorithm is based
on Mpmath's [3]_ *shi* and *chi* routines.
References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
(See Section 5.2.)
.. [2] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
.. [3] Fredrik Johansson and others.
"mpmath: a Python library for arbitrary-precision floating-point
arithmetic" (Version 0.19) http://mpmath.org/
Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import shichi, sici
`shichi` accepts real or complex input:
>>> shichi(0.5)
(0.5069967498196671, -0.05277684495649357)
>>> shichi(0.5 + 2.5j)
((0.11772029666668238+1.831091777729851j),
(0.29912435887648825+1.7395351121166562j))
The hyperbolic sine and cosine integrals Shi(z) and Chi(z) are
related to the sine and cosine integrals Si(z) and Ci(z) by
* Shi(z) = -i*Si(i*z)
* Chi(z) = Ci(-i*z) + i*pi/2
>>> z = 0.25 + 5j
>>> shi, chi = shichi(z)
>>> shi, -1j*sici(1j*z)[0] # Should be the same.
((-0.04834719325101729+1.5469354086921228j),
(-0.04834719325101729+1.5469354086921228j))
>>> chi, sici(-1j*z)[1] + 1j*np.pi/2 # Should be the same.
((-0.19568708973868087+1.556276312103824j),
(-0.19568708973868087+1.556276312103824j))
Plot the functions evaluated on the real axis:
>>> xp = np.geomspace(1e-8, 4.0, 250)
>>> x = np.concatenate((-xp[::-1], xp))
>>> shi, chi = shichi(x)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, shi, label='Shi(x)')
>>> ax.plot(x, chi, '--', label='Chi(x)')
>>> ax.set_xlabel('x')
>>> ax.set_title('Hyperbolic Sine and Cosine Integrals')
>>> ax.legend(shadow=True, framealpha=1, loc='lower right')
>>> ax.grid(True)
>>> plt.show()
""")
add_newdoc("sici",
r"""
sici(x, out=None)
Sine and cosine integrals.
The sine integral is
.. math::
\int_0^x \frac{\sin{t}}{t}dt
and the cosine integral is
.. math::
\gamma + \log(x) + \int_0^x \frac{\cos{t} - 1}{t}dt
where :math:`\gamma` is Euler's constant and :math:`\log` is the
principal branch of the logarithm [1]_.
Parameters
----------
x : array_like
Real or complex points at which to compute the sine and cosine
integrals.
out : tuple of ndarray, optional
Optional output arrays for the function results
Returns
-------
si : scalar or ndarray
Sine integral at ``x``
ci : scalar or ndarray
Cosine integral at ``x``
See Also
--------
shichi : Hyperbolic sine and cosine integrals.
exp1 : Exponential integral E1.
expi : Exponential integral Ei.
Notes
-----
For real arguments with ``x < 0``, ``ci`` is the real part of the
cosine integral. For such points ``ci(x)`` and ``ci(x + 0j)``
differ by a factor of ``1j*pi``.
For real arguments the function is computed by calling Cephes'
[2]_ *sici* routine. For complex arguments the algorithm is based
on Mpmath's [3]_ *si* and *ci* routines.
References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
(See Section 5.2.)
.. [2] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
.. [3] Fredrik Johansson and others.
"mpmath: a Python library for arbitrary-precision floating-point
arithmetic" (Version 0.19) http://mpmath.org/
Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import sici, exp1
`sici` accepts real or complex input:
>>> sici(2.5)
(1.7785201734438267, 0.2858711963653835)
>>> sici(2.5 + 3j)
((4.505735874563953+0.06863305018999577j),
(0.0793644206906966-2.935510262937543j))
For z in the right half plane, the sine and cosine integrals are
related to the exponential integral E1 (implemented in SciPy as
`scipy.special.exp1`) by
* Si(z) = (E1(i*z) - E1(-i*z))/2i + pi/2
* Ci(z) = -(E1(i*z) + E1(-i*z))/2
See [1]_ (equations 5.2.21 and 5.2.23).
We can verify these relations:
>>> z = 2 - 3j
>>> sici(z)
((4.54751388956229-1.3991965806460565j),
(1.408292501520851+2.9836177420296055j))
>>> (exp1(1j*z) - exp1(-1j*z))/2j + np.pi/2 # Same as sine integral
(4.54751388956229-1.3991965806460565j)
>>> -(exp1(1j*z) + exp1(-1j*z))/2 # Same as cosine integral
(1.408292501520851+2.9836177420296055j)
Plot the functions evaluated on the real axis; the dotted horizontal
lines are at pi/2 and -pi/2:
>>> x = np.linspace(-16, 16, 150)
>>> si, ci = sici(x)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, si, label='Si(x)')
>>> ax.plot(x, ci, '--', label='Ci(x)')
>>> ax.legend(shadow=True, framealpha=1, loc='upper left')
>>> ax.set_xlabel('x')
>>> ax.set_title('Sine and Cosine Integrals')
>>> ax.axhline(np.pi/2, linestyle=':', alpha=0.5, color='k')
>>> ax.axhline(-np.pi/2, linestyle=':', alpha=0.5, color='k')
>>> ax.grid(True)
>>> plt.show()
""")
add_newdoc("sindg",
"""
sindg(x, out=None)
Sine of the angle `x` given in degrees.
Parameters
----------
x : array_like
Angle, given in degrees.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Sine at the input.
See Also
--------
cosdg, tandg, cotdg
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is more accurate than using sine directly.
>>> x = 180 * np.arange(3)
>>> sc.sindg(x)
array([ 0., -0., 0.])
>>> np.sin(x * np.pi / 180)
array([ 0.0000000e+00, 1.2246468e-16, -2.4492936e-16])
""")
add_newdoc("smirnov",
r"""
smirnov(n, d, out=None)
Kolmogorov-Smirnov complementary cumulative distribution function
Returns the exact Kolmogorov-Smirnov complementary cumulative
distribution function,(aka the Survival Function) of Dn+ (or Dn-)
for a one-sided test of equality between an empirical and a
theoretical distribution. It is equal to the probability that the
maximum difference between a theoretical distribution and an empirical
one based on `n` samples is greater than d.
Parameters
----------
n : int
Number of samples
d : float array_like
Deviation between the Empirical CDF (ECDF) and the target CDF.
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The value(s) of smirnov(n, d), Prob(Dn+ >= d) (Also Prob(Dn- >= d))
See Also
--------
smirnovi : The Inverse Survival Function for the distribution
scipy.stats.ksone : Provides the functionality as a continuous distribution
kolmogorov, kolmogi : Functions for the two-sided distribution
Notes
-----
`smirnov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this
function is exposed in `scpy.special`, but the recommended way to achieve
the most accurate CDF/SF/PDF/PPF/ISF computations is to use the
`stats.ksone` distribution.
Examples
--------
>>> import numpy as np
>>> from scipy.special import smirnov
>>> from scipy.stats import norm
Show the probability of a gap at least as big as 0, 0.5 and 1.0 for a
sample of size 5.
>>> smirnov(5, [0, 0.5, 1.0])
array([ 1. , 0.056, 0. ])
Compare a sample of size 5 against N(0, 1), the standard normal
distribution with mean 0 and standard deviation 1.
`x` is the sample.
>>> x = np.array([-1.392, -0.135, 0.114, 0.190, 1.82])
>>> target = norm(0, 1)
>>> cdfs = target.cdf(x)
>>> cdfs
array([0.0819612 , 0.44630594, 0.5453811 , 0.57534543, 0.9656205 ])
Construct the empirical CDF and the K-S statistics (Dn+, Dn-, Dn).
>>> n = len(x)
>>> ecdfs = np.arange(n+1, dtype=float)/n
>>> cols = np.column_stack([x, ecdfs[1:], cdfs, cdfs - ecdfs[:n],
... ecdfs[1:] - cdfs])
>>> with np.printoptions(precision=3):
... print(cols)
[[-1.392 0.2 0.082 0.082 0.118]
[-0.135 0.4 0.446 0.246 -0.046]
[ 0.114 0.6 0.545 0.145 0.055]
[ 0.19 0.8 0.575 -0.025 0.225]
[ 1.82 1. 0.966 0.166 0.034]]
>>> gaps = cols[:, -2:]
>>> Dnpm = np.max(gaps, axis=0)
>>> print(f'Dn-={Dnpm[0]:f}, Dn+={Dnpm[1]:f}')
Dn-=0.246306, Dn+=0.224655
>>> probs = smirnov(n, Dnpm)
>>> print(f'For a sample of size {n} drawn from N(0, 1):',
... f' Smirnov n={n}: Prob(Dn- >= {Dnpm[0]:f}) = {probs[0]:.4f}',
... f' Smirnov n={n}: Prob(Dn+ >= {Dnpm[1]:f}) = {probs[1]:.4f}',
... sep='\n')
For a sample of size 5 drawn from N(0, 1):
Smirnov n=5: Prob(Dn- >= 0.246306) = 0.4711
Smirnov n=5: Prob(Dn+ >= 0.224655) = 0.5245
Plot the empirical CDF and the standard normal CDF.
>>> import matplotlib.pyplot as plt
>>> plt.step(np.concatenate(([-2.5], x, [2.5])),
... np.concatenate((ecdfs, [1])),
... where='post', label='Empirical CDF')
>>> xx = np.linspace(-2.5, 2.5, 100)
>>> plt.plot(xx, target.cdf(xx), '--', label='CDF for N(0, 1)')
Add vertical lines marking Dn+ and Dn-.
>>> iminus, iplus = np.argmax(gaps, axis=0)
>>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r',
... alpha=0.5, lw=4)
>>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='m',
... alpha=0.5, lw=4)
>>> plt.grid(True)
>>> plt.legend(framealpha=1, shadow=True)
>>> plt.show()
""")
add_newdoc("smirnovi",
"""
smirnovi(n, p, out=None)
Inverse to `smirnov`
Returns `d` such that ``smirnov(n, d) == p``, the critical value
corresponding to `p`.
Parameters
----------
n : int
Number of samples
p : float array_like
Probability
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
The value(s) of smirnovi(n, p), the critical values.
See Also
--------
smirnov : The Survival Function (SF) for the distribution
scipy.stats.ksone : Provides the functionality as a continuous distribution
kolmogorov, kolmogi : Functions for the two-sided distribution
scipy.stats.kstwobign : Two-sided Kolmogorov-Smirnov distribution, large n
Notes
-----
`smirnov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this
function is exposed in `scpy.special`, but the recommended way to achieve
the most accurate CDF/SF/PDF/PPF/ISF computations is to use the
`stats.ksone` distribution.
Examples
--------
>>> from scipy.special import smirnovi, smirnov
>>> n = 24
>>> deviations = [0.1, 0.2, 0.3]
Use `smirnov` to compute the complementary CDF of the Smirnov
distribution for the given number of samples and deviations.
>>> p = smirnov(n, deviations)
>>> p
array([0.58105083, 0.12826832, 0.01032231])
The inverse function ``smirnovi(n, p)`` returns ``deviations``.
>>> smirnovi(n, p)
array([0.1, 0.2, 0.3])
""")
add_newdoc("_smirnovc",
"""
_smirnovc(n, d)
Internal function, do not use.
""")
add_newdoc("_smirnovci",
"""
Internal function, do not use.
""")
add_newdoc("_smirnovp",
"""
_smirnovp(n, p)
Internal function, do not use.
""")
add_newdoc("spence",
r"""
spence(z, out=None)
Spence's function, also known as the dilogarithm.
It is defined to be
.. math::
\int_1^z \frac{\log(t)}{1 - t}dt
for complex :math:`z`, where the contour of integration is taken
to avoid the branch cut of the logarithm. Spence's function is
analytic everywhere except the negative real axis where it has a
branch cut.
Parameters
----------
z : array_like
Points at which to evaluate Spence's function
out : ndarray, optional
Optional output array for the function results
Returns
-------
s : scalar or ndarray
Computed values of Spence's function
Notes
-----
There is a different convention which defines Spence's function by
the integral
.. math::
-\int_0^z \frac{\log(1 - t)}{t}dt;
this is our ``spence(1 - z)``.
Examples
--------
>>> import numpy as np
>>> from scipy.special import spence
>>> import matplotlib.pyplot as plt
The function is defined for complex inputs:
>>> spence([1-1j, 1.5+2j, 3j, -10-5j])
array([-0.20561676+0.91596559j, -0.86766909-1.39560134j,
-0.59422064-2.49129918j, -1.14044398+6.80075924j])
For complex inputs on the branch cut, which is the negative real axis,
the function returns the limit for ``z`` with positive imaginary part.
For example, in the following, note the sign change of the imaginary
part of the output for ``z = -2`` and ``z = -2 - 1e-8j``:
>>> spence([-2 + 1e-8j, -2, -2 - 1e-8j])
array([2.32018041-3.45139229j, 2.32018042-3.4513923j ,
2.32018041+3.45139229j])
The function returns ``nan`` for real inputs on the branch cut:
>>> spence(-1.5)
nan
Verify some particular values: ``spence(0) = pi**2/6``,
``spence(1) = 0`` and ``spence(2) = -pi**2/12``.
>>> spence([0, 1, 2])
array([ 1.64493407, 0. , -0.82246703])
>>> np.pi**2/6, -np.pi**2/12
(1.6449340668482264, -0.8224670334241132)
Verify the identity::
spence(z) + spence(1 - z) = pi**2/6 - log(z)*log(1 - z)
>>> z = 3 + 4j
>>> spence(z) + spence(1 - z)
(-2.6523186143876067+1.8853470951513935j)
>>> np.pi**2/6 - np.log(z)*np.log(1 - z)
(-2.652318614387606+1.885347095151394j)
Plot the function for positive real input.
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0, 6, 400)
>>> ax.plot(x, spence(x))
>>> ax.grid()
>>> ax.set_xlabel('x')
>>> ax.set_title('spence(x)')
>>> plt.show()
""")
add_newdoc(
"stdtr",
r"""
stdtr(df, t, out=None)
Student t distribution cumulative distribution function
Returns the integral:
.. math::
\frac{\Gamma((df+1)/2)}{\sqrt{\pi df} \Gamma(df/2)}
\int_{-\infty}^t (1+x^2/df)^{-(df+1)/2}\, dx
Parameters
----------
df : array_like
Degrees of freedom
t : array_like
Upper bound of the integral
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Value of the Student t CDF at t
See Also
--------
stdtridf : inverse of stdtr with respect to `df`
stdtrit : inverse of stdtr with respect to `t`
scipy.stats.t : student t distribution
Notes
-----
The student t distribution is also available as `scipy.stats.t`.
Calling `stdtr` directly can improve performance compared to the
``cdf`` method of `scipy.stats.t` (see last example below).
Examples
--------
Calculate the function for ``df=3`` at ``t=1``.
>>> import numpy as np
>>> from scipy.special import stdtr
>>> import matplotlib.pyplot as plt
>>> stdtr(3, 1)
0.8044988905221148
Plot the function for three different degrees of freedom.
>>> x = np.linspace(-10, 10, 1000)
>>> fig, ax = plt.subplots()
>>> parameters = [(1, "solid"), (3, "dashed"), (10, "dotted")]
>>> for (df, linestyle) in parameters:
... ax.plot(x, stdtr(df, x), ls=linestyle, label=f"$df={df}$")
>>> ax.legend()
>>> ax.set_title("Student t distribution cumulative distribution function")
>>> plt.show()
The function can be computed for several degrees of freedom at the same
time by providing a NumPy array or list for `df`:
>>> stdtr([1, 2, 3], 1)
array([0.75 , 0.78867513, 0.80449889])
It is possible to calculate the function at several points for several
different degrees of freedom simultaneously by providing arrays for `df`
and `t` with shapes compatible for broadcasting. Compute `stdtr` at
4 points for 3 degrees of freedom resulting in an array of shape 3x4.
>>> dfs = np.array([[1], [2], [3]])
>>> t = np.array([2, 4, 6, 8])
>>> dfs.shape, t.shape
((3, 1), (4,))
>>> stdtr(dfs, t)
array([[0.85241638, 0.92202087, 0.94743154, 0.96041658],
[0.90824829, 0.97140452, 0.98666426, 0.99236596],
[0.93033702, 0.98599577, 0.99536364, 0.99796171]])
The t distribution is also available as `scipy.stats.t`. Calling `stdtr`
directly can be much faster than calling the ``cdf`` method of
`scipy.stats.t`. To get the same results, one must use the following
parametrization: ``scipy.stats.t(df).cdf(x) = stdtr(df, x)``.
>>> from scipy.stats import t
>>> df, x = 3, 1
>>> stdtr_result = stdtr(df, x) # this can be faster than below
>>> stats_result = t(df).cdf(x)
>>> stats_result == stdtr_result # test that results are equal
True
""")
add_newdoc("stdtridf",
"""
stdtridf(p, t, out=None)
Inverse of `stdtr` vs df
Returns the argument df such that stdtr(df, t) is equal to `p`.
Parameters
----------
p : array_like
Probability
t : array_like
Upper bound of the integral
out : ndarray, optional
Optional output array for the function results
Returns
-------
df : scalar or ndarray
Value of `df` such that ``stdtr(df, t) == p``
See Also
--------
stdtr : Student t CDF
stdtrit : inverse of stdtr with respect to `t`
scipy.stats.t : Student t distribution
Examples
--------
Compute the student t cumulative distribution function for one
parameter set.
>>> from scipy.special import stdtr, stdtridf
>>> df, x = 5, 2
>>> cdf_value = stdtr(df, x)
>>> cdf_value
0.9490302605850709
Verify that `stdtridf` recovers the original value for `df` given
the CDF value and `x`.
>>> stdtridf(cdf_value, x)
5.0
""")
add_newdoc("stdtrit",
"""
stdtrit(df, p, out=None)
The `p`-th quantile of the student t distribution.
This function is the inverse of the student t distribution cumulative
distribution function (CDF), returning `t` such that `stdtr(df, t) = p`.
Returns the argument `t` such that stdtr(df, t) is equal to `p`.
Parameters
----------
df : array_like
Degrees of freedom
p : array_like
Probability
out : ndarray, optional
Optional output array for the function results
Returns
-------
t : scalar or ndarray
Value of `t` such that ``stdtr(df, t) == p``
See Also
--------
stdtr : Student t CDF
stdtridf : inverse of stdtr with respect to `df`
scipy.stats.t : Student t distribution
Notes
-----
The student t distribution is also available as `scipy.stats.t`. Calling
`stdtrit` directly can improve performance compared to the ``ppf``
method of `scipy.stats.t` (see last example below).
Examples
--------
`stdtrit` represents the inverse of the student t distribution CDF which
is available as `stdtr`. Here, we calculate the CDF for ``df`` at
``x=1``. `stdtrit` then returns ``1`` up to floating point errors
given the same value for `df` and the computed CDF value.
>>> import numpy as np
>>> from scipy.special import stdtr, stdtrit
>>> import matplotlib.pyplot as plt
>>> df = 3
>>> x = 1
>>> cdf_value = stdtr(df, x)
>>> stdtrit(df, cdf_value)
0.9999999994418539
Plot the function for three different degrees of freedom.
>>> x = np.linspace(0, 1, 1000)
>>> parameters = [(1, "solid"), (2, "dashed"), (5, "dotted")]
>>> fig, ax = plt.subplots()
>>> for (df, linestyle) in parameters:
... ax.plot(x, stdtrit(df, x), ls=linestyle, label=f"$df={df}$")
>>> ax.legend()
>>> ax.set_ylim(-10, 10)
>>> ax.set_title("Student t distribution quantile function")
>>> plt.show()
The function can be computed for several degrees of freedom at the same
time by providing a NumPy array or list for `df`:
>>> stdtrit([1, 2, 3], 0.7)
array([0.72654253, 0.6172134 , 0.58438973])
It is possible to calculate the function at several points for several
different degrees of freedom simultaneously by providing arrays for `df`
and `p` with shapes compatible for broadcasting. Compute `stdtrit` at
4 points for 3 degrees of freedom resulting in an array of shape 3x4.
>>> dfs = np.array([[1], [2], [3]])
>>> p = np.array([0.2, 0.4, 0.7, 0.8])
>>> dfs.shape, p.shape
((3, 1), (4,))
>>> stdtrit(dfs, p)
array([[-1.37638192, -0.3249197 , 0.72654253, 1.37638192],
[-1.06066017, -0.28867513, 0.6172134 , 1.06066017],
[-0.97847231, -0.27667066, 0.58438973, 0.97847231]])
The t distribution is also available as `scipy.stats.t`. Calling `stdtrit`
directly can be much faster than calling the ``ppf`` method of
`scipy.stats.t`. To get the same results, one must use the following
parametrization: ``scipy.stats.t(df).ppf(x) = stdtrit(df, x)``.
>>> from scipy.stats import t
>>> df, x = 3, 0.5
>>> stdtrit_result = stdtrit(df, x) # this can be faster than below
>>> stats_result = t(df).ppf(x)
>>> stats_result == stdtrit_result # test that results are equal
True
""")
add_newdoc("struve",
r"""
struve(v, x, out=None)
Struve function.
Return the value of the Struve function of order `v` at `x`. The Struve
function is defined as,
.. math::
H_v(x) = (z/2)^{v + 1} \sum_{n=0}^\infty
\frac{(-1)^n (z/2)^{2n}}{\Gamma(n + \frac{3}{2}) \Gamma(n + v + \frac{3}{2})},
where :math:`\Gamma` is the gamma function.
Parameters
----------
v : array_like
Order of the Struve function (float).
x : array_like
Argument of the Struve function (float; must be positive unless `v` is
an integer).
out : ndarray, optional
Optional output array for the function results
Returns
-------
H : scalar or ndarray
Value of the Struve function of order `v` at `x`.
See Also
--------
modstruve: Modified Struve function
Notes
-----
Three methods discussed in [1]_ are used to evaluate the Struve function:
- power series
- expansion in Bessel functions (if :math:`|z| < |v| + 20`)
- asymptotic large-z expansion (if :math:`z \geq 0.7v + 12`)
Rounding errors are estimated based on the largest terms in the sums, and
the result associated with the smallest error is returned.
References
----------
.. [1] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/11
Examples
--------
Calculate the Struve function of order 1 at 2.
>>> import numpy as np
>>> from scipy.special import struve
>>> import matplotlib.pyplot as plt
>>> struve(1, 2.)
0.6467637282835622
Calculate the Struve function at 2 for orders 1, 2 and 3 by providing
a list for the order parameter `v`.
>>> struve([1, 2, 3], 2.)
array([0.64676373, 0.28031806, 0.08363767])
Calculate the Struve function of order 1 for several points by providing
an array for `x`.
>>> points = np.array([2., 5., 8.])
>>> struve(1, points)
array([0.64676373, 0.80781195, 0.48811605])
Compute the Struve function for several orders at several points by
providing arrays for `v` and `z`. The arrays have to be broadcastable
to the correct shapes.
>>> orders = np.array([[1], [2], [3]])
>>> points.shape, orders.shape
((3,), (3, 1))
>>> struve(orders, points)
array([[0.64676373, 0.80781195, 0.48811605],
[0.28031806, 1.56937455, 1.51769363],
[0.08363767, 1.50872065, 2.98697513]])
Plot the Struve functions of order 0 to 3 from -10 to 10.
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> for i in range(4):
... ax.plot(x, struve(i, x), label=f'$H_{i!r}$')
>>> ax.legend(ncol=2)
>>> ax.set_xlim(-10, 10)
>>> ax.set_title(r"Struve functions $H_{\nu}$")
>>> plt.show()
""")
add_newdoc("tandg",
"""
tandg(x, out=None)
Tangent of angle `x` given in degrees.
Parameters
----------
x : array_like
Angle, given in degrees.
out : ndarray, optional
Optional output array for the function results.
Returns
-------
scalar or ndarray
Tangent at the input.
See Also
--------
sindg, cosdg, cotdg
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
It is more accurate than using tangent directly.
>>> x = 180 * np.arange(3)
>>> sc.tandg(x)
array([0., 0., 0.])
>>> np.tan(x * np.pi / 180)
array([ 0.0000000e+00, -1.2246468e-16, -2.4492936e-16])
""")
add_newdoc(
"tklmbda",
r"""
tklmbda(x, lmbda, out=None)
Cumulative distribution function of the Tukey lambda distribution.
Parameters
----------
x, lmbda : array_like
Parameters
out : ndarray, optional
Optional output array for the function results
Returns
-------
cdf : scalar or ndarray
Value of the Tukey lambda CDF
See Also
--------
scipy.stats.tukeylambda : Tukey lambda distribution
Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import tklmbda, expit
Compute the cumulative distribution function (CDF) of the Tukey lambda
distribution at several ``x`` values for `lmbda` = -1.5.
>>> x = np.linspace(-2, 2, 9)
>>> x
array([-2. , -1.5, -1. , -0.5, 0. , 0.5, 1. , 1.5, 2. ])
>>> tklmbda(x, -1.5)
array([0.34688734, 0.3786554 , 0.41528805, 0.45629737, 0.5 ,
0.54370263, 0.58471195, 0.6213446 , 0.65311266])
When `lmbda` is 0, the function is the logistic sigmoid function,
which is implemented in `scipy.special` as `expit`.
>>> tklmbda(x, 0)
array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5 ,
0.62245933, 0.73105858, 0.81757448, 0.88079708])
>>> expit(x)
array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5 ,
0.62245933, 0.73105858, 0.81757448, 0.88079708])
When `lmbda` is 1, the Tukey lambda distribution is uniform on the
interval [-1, 1], so the CDF increases linearly.
>>> t = np.linspace(-1, 1, 9)
>>> tklmbda(t, 1)
array([0. , 0.125, 0.25 , 0.375, 0.5 , 0.625, 0.75 , 0.875, 1. ])
In the following, we generate plots for several values of `lmbda`.
The first figure shows graphs for `lmbda` <= 0.
>>> styles = ['-', '-.', '--', ':']
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-12, 12, 500)
>>> for k, lmbda in enumerate([-1.0, -0.5, 0.0]):
... y = tklmbda(x, lmbda)
... ax.plot(x, y, styles[k], label=rf'$\lambda$ = {lmbda:-4.1f}')
>>> ax.set_title(r'tklmbda(x, $\lambda$)')
>>> ax.set_label('x')
>>> ax.legend(framealpha=1, shadow=True)
>>> ax.grid(True)
The second figure shows graphs for `lmbda` > 0. The dots in the
graphs show the bounds of the support of the distribution.
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-4.2, 4.2, 500)
>>> lmbdas = [0.25, 0.5, 1.0, 1.5]
>>> for k, lmbda in enumerate(lmbdas):
... y = tklmbda(x, lmbda)
... ax.plot(x, y, styles[k], label=fr'$\lambda$ = {lmbda}')
>>> ax.set_prop_cycle(None)
>>> for lmbda in lmbdas:
... ax.plot([-1/lmbda, 1/lmbda], [0, 1], '.', ms=8)
>>> ax.set_title(r'tklmbda(x, $\lambda$)')
>>> ax.set_xlabel('x')
>>> ax.legend(framealpha=1, shadow=True)
>>> ax.grid(True)
>>> plt.tight_layout()
>>> plt.show()
The CDF of the Tukey lambda distribution is also implemented as the
``cdf`` method of `scipy.stats.tukeylambda`. In the following,
``tukeylambda.cdf(x, -0.5)`` and ``tklmbda(x, -0.5)`` compute the
same values:
>>> from scipy.stats import tukeylambda
>>> x = np.linspace(-2, 2, 9)
>>> tukeylambda.cdf(x, -0.5)
array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5 ,
0.58671839, 0.66458323, 0.72906142, 0.78004843])
>>> tklmbda(x, -0.5)
array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5 ,
0.58671839, 0.66458323, 0.72906142, 0.78004843])
The implementation in ``tukeylambda`` also provides location and scale
parameters, and other methods such as ``pdf()`` (the probability
density function) and ``ppf()`` (the inverse of the CDF), so for
working with the Tukey lambda distribution, ``tukeylambda`` is more
generally useful. The primary advantage of ``tklmbda`` is that it is
significantly faster than ``tukeylambda.cdf``.
""")
add_newdoc("wofz",
"""
wofz(z, out=None)
Faddeeva function
Returns the value of the Faddeeva function for complex argument::
exp(-z**2) * erfc(-i*z)
Parameters
----------
z : array_like
complex argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Value of the Faddeeva function
See Also
--------
dawsn, erf, erfc, erfcx, erfi
References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
http://ab-initio.mit.edu/Faddeeva
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-3, 3)
>>> z = special.wofz(x)
>>> plt.plot(x, z.real, label='wofz(x).real')
>>> plt.plot(x, z.imag, label='wofz(x).imag')
>>> plt.xlabel('$x$')
>>> plt.legend(framealpha=1, shadow=True)
>>> plt.grid(alpha=0.25)
>>> plt.show()
""")
add_newdoc("xlogy",
"""
xlogy(x, y, out=None)
Compute ``x*log(y)`` so that the result is 0 if ``x = 0``.
Parameters
----------
x : array_like
Multiplier
y : array_like
Argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
z : scalar or ndarray
Computed x*log(y)
Notes
-----
The log function used in the computation is the natural log.
.. versionadded:: 0.13.0
Examples
--------
We can use this function to calculate the binary logistic loss also
known as the binary cross entropy. This loss function is used for
binary classification problems and is defined as:
.. math::
L = 1/n * \\sum_{i=0}^n -(y_i*log(y\\_pred_i) + (1-y_i)*log(1-y\\_pred_i))
We can define the parameters `x` and `y` as y and y_pred respectively.
y is the array of the actual labels which over here can be either 0 or 1.
y_pred is the array of the predicted probabilities with respect to
the positive class (1).
>>> import numpy as np
>>> from scipy.special import xlogy
>>> y = np.array([0, 1, 0, 1, 1, 0])
>>> y_pred = np.array([0.3, 0.8, 0.4, 0.7, 0.9, 0.2])
>>> n = len(y)
>>> loss = -(xlogy(y, y_pred) + xlogy(1 - y, 1 - y_pred)).sum()
>>> loss /= n
>>> loss
0.29597052165495025
A lower loss is usually better as it indicates that the predictions are
similar to the actual labels. In this example since our predicted
probabilities are close to the actual labels, we get an overall loss
that is reasonably low and appropriate.
""")
add_newdoc("xlog1py",
"""
xlog1py(x, y, out=None)
Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``.
Parameters
----------
x : array_like
Multiplier
y : array_like
Argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
z : scalar or ndarray
Computed x*log1p(y)
Notes
-----
.. versionadded:: 0.13.0
Examples
--------
This example shows how the function can be used to calculate the log of
the probability mass function for a geometric discrete random variable.
The probability mass function of the geometric distribution is defined
as follows:
.. math:: f(k) = (1-p)^{k-1} p
where :math:`p` is the probability of a single success
and :math:`1-p` is the probability of a single failure
and :math:`k` is the number of trials to get the first success.
>>> import numpy as np
>>> from scipy.special import xlog1py
>>> p = 0.5
>>> k = 100
>>> _pmf = np.power(1 - p, k - 1) * p
>>> _pmf
7.888609052210118e-31
If we take k as a relatively large number the value of the probability
mass function can become very low. In such cases taking the log of the
pmf would be more suitable as the log function can change the values
to a scale that is more appropriate to work with.
>>> _log_pmf = xlog1py(k - 1, -p) + np.log(p)
>>> _log_pmf
-69.31471805599453
We can confirm that we get a value close to the original pmf value by
taking the exponential of the log pmf.
>>> _orig_pmf = np.exp(_log_pmf)
>>> np.isclose(_pmf, _orig_pmf)
True
""")
add_newdoc("y0",
r"""
y0(x, out=None)
Bessel function of the second kind of order 0.
Parameters
----------
x : array_like
Argument (float).
out : ndarray, optional
Optional output array for the function results
Returns
-------
Y : scalar or ndarray
Value of the Bessel function of the second kind of order 0 at `x`.
See Also
--------
j0: Bessel function of the first kind of order 0
yv: Bessel function of the first kind
Notes
-----
The domain is divided into the intervals [0, 5] and (5, infinity). In the
first interval a rational approximation :math:`R(x)` is employed to
compute,
.. math::
Y_0(x) = R(x) + \frac{2 \log(x) J_0(x)}{\pi},
where :math:`J_0` is the Bessel function of the first kind of order 0.
In the second interval, the Hankel asymptotic expansion is employed with
two rational functions of degree 6/6 and 7/7.
This function is a wrapper for the Cephes [1]_ routine `y0`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function at one point:
>>> from scipy.special import y0
>>> y0(1.)
0.08825696421567697
Calculate at several points:
>>> import numpy as np
>>> y0(np.array([0.5, 2., 3.]))
array([-0.44451873, 0.51037567, 0.37685001])
Plot the function from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = y0(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("y1",
"""
y1(x, out=None)
Bessel function of the second kind of order 1.
Parameters
----------
x : array_like
Argument (float).
out : ndarray, optional
Optional output array for the function results
Returns
-------
Y : scalar or ndarray
Value of the Bessel function of the second kind of order 1 at `x`.
See Also
--------
j1: Bessel function of the first kind of order 1
yn: Bessel function of the second kind
yv: Bessel function of the second kind
Notes
-----
The domain is divided into the intervals [0, 8] and (8, infinity). In the
first interval a 25 term Chebyshev expansion is used, and computing
:math:`J_1` (the Bessel function of the first kind) is required. In the
second, the asymptotic trigonometric representation is employed using two
rational functions of degree 5/5.
This function is a wrapper for the Cephes [1]_ routine `y1`.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Calculate the function at one point:
>>> from scipy.special import y1
>>> y1(1.)
-0.7812128213002888
Calculate at several points:
>>> import numpy as np
>>> y1(np.array([0.5, 2., 3.]))
array([-1.47147239, -0.10703243, 0.32467442])
Plot the function from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = y1(x)
>>> ax.plot(x, y)
>>> plt.show()
""")
add_newdoc("yn",
r"""
yn(n, x, out=None)
Bessel function of the second kind of integer order and real argument.
Parameters
----------
n : array_like
Order (integer).
x : array_like
Argument (float).
out : ndarray, optional
Optional output array for the function results
Returns
-------
Y : scalar or ndarray
Value of the Bessel function, :math:`Y_n(x)`.
See Also
--------
yv : For real order and real or complex argument.
y0: faster implementation of this function for order 0
y1: faster implementation of this function for order 1
Notes
-----
Wrapper for the Cephes [1]_ routine `yn`.
The function is evaluated by forward recurrence on `n`, starting with
values computed by the Cephes routines `y0` and `y1`. If `n = 0` or 1,
the routine for `y0` or `y1` is called directly.
References
----------
.. [1] Cephes Mathematical Functions Library,
http://www.netlib.org/cephes/
Examples
--------
Evaluate the function of order 0 at one point.
>>> from scipy.special import yn
>>> yn(0, 1.)
0.08825696421567697
Evaluate the function at one point for different orders.
>>> yn(0, 1.), yn(1, 1.), yn(2, 1.)
(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)
The evaluation for different orders can be carried out in one call by
providing a list or NumPy array as argument for the `v` parameter:
>>> yn([0, 1, 2], 1.)
array([ 0.08825696, -0.78121282, -1.65068261])
Evaluate the function at several points for order 0 by providing an
array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 3., 8.])
>>> yn(0, points)
array([-0.44451873, 0.37685001, 0.22352149])
If `z` is an array, the order parameter `v` must be broadcastable to
the correct shape if different orders shall be computed in one call.
To calculate the orders 0 and 1 for an 1D array:
>>> orders = np.array([[0], [1]])
>>> orders.shape
(2, 1)
>>> yn(orders, points)
array([[-0.44451873, 0.37685001, 0.22352149],
[-1.47147239, 0.32467442, -0.15806046]])
Plot the functions of order 0 to 3 from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> for i in range(4):
... ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')
>>> ax.set_ylim(-3, 1)
>>> ax.legend()
>>> plt.show()
""")
add_newdoc("yv",
r"""
yv(v, z, out=None)
Bessel function of the second kind of real order and complex argument.
Parameters
----------
v : array_like
Order (float).
z : array_like
Argument (float or complex).
out : ndarray, optional
Optional output array for the function results
Returns
-------
Y : scalar or ndarray
Value of the Bessel function of the second kind, :math:`Y_v(x)`.
See Also
--------
yve : :math:`Y_v` with leading exponential behavior stripped off.
y0: faster implementation of this function for order 0
y1: faster implementation of this function for order 1
Notes
-----
For positive `v` values, the computation is carried out using the
AMOS [1]_ `zbesy` routine, which exploits the connection to the Hankel
Bessel functions :math:`H_v^{(1)}` and :math:`H_v^{(2)}`,
.. math:: Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}).
For negative `v` values the formula,
.. math:: Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v)
is used, where :math:`J_v(z)` is the Bessel function of the first kind,
computed using the AMOS routine `zbesj`. Note that the second term is
exactly zero for integer `v`; to improve accuracy the second term is
explicitly omitted for `v` values such that `v = floor(v)`.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
Examples
--------
Evaluate the function of order 0 at one point.
>>> from scipy.special import yv
>>> yv(0, 1.)
0.088256964215677
Evaluate the function at one point for different orders.
>>> yv(0, 1.), yv(1, 1.), yv(1.5, 1.)
(0.088256964215677, -0.7812128213002889, -1.102495575160179)
The evaluation for different orders can be carried out in one call by
providing a list or NumPy array as argument for the `v` parameter:
>>> yv([0, 1, 1.5], 1.)
array([ 0.08825696, -0.78121282, -1.10249558])
Evaluate the function at several points for order 0 by providing an
array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 3., 8.])
>>> yv(0, points)
array([-0.44451873, 0.37685001, 0.22352149])
If `z` is an array, the order parameter `v` must be broadcastable to
the correct shape if different orders shall be computed in one call.
To calculate the orders 0 and 1 for an 1D array:
>>> orders = np.array([[0], [1]])
>>> orders.shape
(2, 1)
>>> yv(orders, points)
array([[-0.44451873, 0.37685001, 0.22352149],
[-1.47147239, 0.32467442, -0.15806046]])
Plot the functions of order 0 to 3 from 0 to 10.
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> for i in range(4):
... ax.plot(x, yv(i, x), label=f'$Y_{i!r}$')
>>> ax.set_ylim(-3, 1)
>>> ax.legend()
>>> plt.show()
""")
add_newdoc("yve",
r"""
yve(v, z, out=None)
Exponentially scaled Bessel function of the second kind of real order.
Returns the exponentially scaled Bessel function of the second
kind of real order `v` at complex `z`::
yve(v, z) = yv(v, z) * exp(-abs(z.imag))
Parameters
----------
v : array_like
Order (float).
z : array_like
Argument (float or complex).
out : ndarray, optional
Optional output array for the function results
Returns
-------
Y : scalar or ndarray
Value of the exponentially scaled Bessel function.
See Also
--------
yv: Unscaled Bessel function of the second kind of real order.
Notes
-----
For positive `v` values, the computation is carried out using the
AMOS [1]_ `zbesy` routine, which exploits the connection to the Hankel
Bessel functions :math:`H_v^{(1)}` and :math:`H_v^{(2)}`,
.. math:: Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}).
For negative `v` values the formula,
.. math:: Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v)
is used, where :math:`J_v(z)` is the Bessel function of the first kind,
computed using the AMOS routine `zbesj`. Note that the second term is
exactly zero for integer `v`; to improve accuracy the second term is
explicitly omitted for `v` values such that `v = floor(v)`.
Exponentially scaled Bessel functions are useful for large `z`:
for these, the unscaled Bessel functions can easily under-or overflow.
References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
of a Complex Argument and Nonnegative Order",
http://netlib.org/amos/
Examples
--------
Compare the output of `yv` and `yve` for large complex arguments for `z`
by computing their values for order ``v=1`` at ``z=1000j``. We see that
`yv` returns nan but `yve` returns a finite number:
>>> import numpy as np
>>> from scipy.special import yv, yve
>>> v = 1
>>> z = 1000j
>>> yv(v, z), yve(v, z)
((nan+nanj), (-0.012610930256928629+7.721967686709076e-19j))
For real arguments for `z`, `yve` returns the same as `yv` up to
floating point errors.
>>> v, z = 1, 1000
>>> yv(v, z), yve(v, z)
(-0.02478433129235178, -0.02478433129235179)
The function can be evaluated for several orders at the same time by
providing a list or NumPy array for `v`:
>>> yve([1, 2, 3], 1j)
array([-0.20791042+0.14096627j, 0.38053618-0.04993878j,
0.00815531-1.66311097j])
In the same way, the function can be evaluated at several points in one
call by providing a list or NumPy array for `z`:
>>> yve(1, np.array([1j, 2j, 3j]))
array([-0.20791042+0.14096627j, -0.21526929+0.01205044j,
-0.19682671+0.00127278j])
It is also possible to evaluate several orders at several points
at the same time by providing arrays for `v` and `z` with
broadcasting compatible shapes. Compute `yve` for two different orders
`v` and three points `z` resulting in a 2x3 array.
>>> v = np.array([[1], [2]])
>>> z = np.array([3j, 4j, 5j])
>>> v.shape, z.shape
((2, 1), (3,))
>>> yve(v, z)
array([[-1.96826713e-01+1.27277544e-03j, -1.78750840e-01+1.45558819e-04j,
-1.63972267e-01+1.73494110e-05j],
[1.94960056e-03-1.11782545e-01j, 2.02902325e-04-1.17626501e-01j,
2.27727687e-05-1.17951906e-01j]])
""")
add_newdoc("_zeta",
"""
_zeta(x, q)
Internal function, Hurwitz zeta.
""")
add_newdoc("zetac",
"""
zetac(x, out=None)
Riemann zeta function minus 1.
This function is defined as
.. math:: \\zeta(x) = \\sum_{k=2}^{\\infty} 1 / k^x,
where ``x > 1``. For ``x < 1`` the analytic continuation is
computed. For more information on the Riemann zeta function, see
[dlmf]_.
Parameters
----------
x : array_like of float
Values at which to compute zeta(x) - 1 (must be real).
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Values of zeta(x) - 1.
See Also
--------
zeta
References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/25
Examples
--------
>>> import numpy as np
>>> from scipy.special import zetac, zeta
Some special values:
>>> zetac(2), np.pi**2/6 - 1
(0.64493406684822641, 0.6449340668482264)
>>> zetac(-1), -1.0/12 - 1
(-1.0833333333333333, -1.0833333333333333)
Compare ``zetac(x)`` to ``zeta(x) - 1`` for large `x`:
>>> zetac(60), zeta(60) - 1
(8.673617380119933e-19, 0.0)
""")
add_newdoc("_riemann_zeta",
"""
Internal function, use `zeta` instead.
""")
add_newdoc("_struve_asymp_large_z",
"""
_struve_asymp_large_z(v, z, is_h)
Internal function for testing `struve` & `modstruve`
Evaluates using asymptotic expansion
Returns
-------
v, err
""")
add_newdoc("_struve_power_series",
"""
_struve_power_series(v, z, is_h)
Internal function for testing `struve` & `modstruve`
Evaluates using power series
Returns
-------
v, err
""")
add_newdoc("_struve_bessel_series",
"""
_struve_bessel_series(v, z, is_h)
Internal function for testing `struve` & `modstruve`
Evaluates using Bessel function series
Returns
-------
v, err
""")
add_newdoc("_spherical_jn",
"""
Internal function, use `spherical_jn` instead.
""")
add_newdoc("_spherical_jn_d",
"""
Internal function, use `spherical_jn` instead.
""")
add_newdoc("_spherical_yn",
"""
Internal function, use `spherical_yn` instead.
""")
add_newdoc("_spherical_yn_d",
"""
Internal function, use `spherical_yn` instead.
""")
add_newdoc("_spherical_in",
"""
Internal function, use `spherical_in` instead.
""")
add_newdoc("_spherical_in_d",
"""
Internal function, use `spherical_in` instead.
""")
add_newdoc("_spherical_kn",
"""
Internal function, use `spherical_kn` instead.
""")
add_newdoc("_spherical_kn_d",
"""
Internal function, use `spherical_kn` instead.
""")
add_newdoc("loggamma",
r"""
loggamma(z, out=None)
Principal branch of the logarithm of the gamma function.
Defined to be :math:`\log(\Gamma(x))` for :math:`x > 0` and
extended to the complex plane by analytic continuation. The
function has a single branch cut on the negative real axis.
.. versionadded:: 0.18.0
Parameters
----------
z : array_like
Values in the complex plane at which to compute ``loggamma``
out : ndarray, optional
Output array for computed values of ``loggamma``
Returns
-------
loggamma : scalar or ndarray
Values of ``loggamma`` at z.
See Also
--------
gammaln : logarithm of the absolute value of the gamma function
gammasgn : sign of the gamma function
Notes
-----
It is not generally true that :math:`\log\Gamma(z) =
\log(\Gamma(z))`, though the real parts of the functions do
agree. The benefit of not defining `loggamma` as
:math:`\log(\Gamma(z))` is that the latter function has a
complicated branch cut structure whereas `loggamma` is analytic
except for on the negative real axis.
The identities
.. math::
\exp(\log\Gamma(z)) &= \Gamma(z) \\
\log\Gamma(z + 1) &= \log(z) + \log\Gamma(z)
make `loggamma` useful for working in complex logspace.
On the real line `loggamma` is related to `gammaln` via
``exp(loggamma(x + 0j)) = gammasgn(x)*exp(gammaln(x))``, up to
rounding error.
The implementation here is based on [hare1997]_.
References
----------
.. [hare1997] D.E.G. Hare,
*Computing the Principal Branch of log-Gamma*,
Journal of Algorithms, Volume 25, Issue 2, November 1997, pages 221-236.
""")
add_newdoc("_sinpi",
"""
Internal function, do not use.
""")
add_newdoc("_cospi",
"""
Internal function, do not use.
""")
add_newdoc("owens_t",
"""
owens_t(h, a, out=None)
Owen's T Function.
The function T(h, a) gives the probability of the event
(X > h and 0 < Y < a * X) where X and Y are independent
standard normal random variables.
Parameters
----------
h: array_like
Input value.
a: array_like
Input value.
out : ndarray, optional
Optional output array for the function results
Returns
-------
t: scalar or ndarray
Probability of the event (X > h and 0 < Y < a * X),
where X and Y are independent standard normal random variables.
References
----------
.. [1] M. Patefield and D. Tandy, "Fast and accurate calculation of
Owen's T Function", Statistical Software vol. 5, pp. 1-25, 2000.
Examples
--------
>>> from scipy import special
>>> a = 3.5
>>> h = 0.78
>>> special.owens_t(h, a)
0.10877216734852274
""")
add_newdoc("_factorial",
"""
Internal function, do not use.
""")
add_newdoc("wright_bessel",
r"""
wright_bessel(a, b, x, out=None)
Wright's generalized Bessel function.
Wright's generalized Bessel function is an entire function and defined as
.. math:: \Phi(a, b; x) = \sum_{k=0}^\infty \frac{x^k}{k! \Gamma(a k + b)}
See Also [1].
Parameters
----------
a : array_like of float
a >= 0
b : array_like of float
b >= 0
x : array_like of float
x >= 0
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Value of the Wright's generalized Bessel function
Notes
-----
Due to the complexity of the function with its three parameters, only
non-negative arguments are implemented.
References
----------
.. [1] Digital Library of Mathematical Functions, 10.46.
https://dlmf.nist.gov/10.46.E1
Examples
--------
>>> from scipy.special import wright_bessel
>>> a, b, x = 1.5, 1.1, 2.5
>>> wright_bessel(a, b-1, x)
4.5314465939443025
Now, let us verify the relation
.. math:: \Phi(a, b-1; x) = a x \Phi(a, b+a; x) + (b-1) \Phi(a, b; x)
>>> a * x * wright_bessel(a, b+a, x) + (b-1) * wright_bessel(a, b, x)
4.5314465939443025
""")
add_newdoc("ndtri_exp",
r"""
ndtri_exp(y, out=None)
Inverse of `log_ndtr` vs x. Allows for greater precision than
`ndtri` composed with `numpy.exp` for very small values of y and for
y close to 0.
Parameters
----------
y : array_like of float
Function argument
out : ndarray, optional
Optional output array for the function results
Returns
-------
scalar or ndarray
Inverse of the log CDF of the standard normal distribution, evaluated
at y.
See Also
--------
log_ndtr : log of the standard normal cumulative distribution function
ndtr : standard normal cumulative distribution function
ndtri : standard normal percentile function
Examples
--------
>>> import numpy as np
>>> import scipy.special as sc
`ndtri_exp` agrees with the naive implementation when the latter does
not suffer from underflow.
>>> sc.ndtri_exp(-1)
-0.33747496376420244
>>> sc.ndtri(np.exp(-1))
-0.33747496376420244
For extreme values of y, the naive approach fails
>>> sc.ndtri(np.exp(-800))
-inf
>>> sc.ndtri(np.exp(-1e-20))
inf
whereas `ndtri_exp` is still able to compute the result to high precision.
>>> sc.ndtri_exp(-800)
-39.88469483825668
>>> sc.ndtri_exp(-1e-20)
9.262340089798409
""")
add_newdoc("_stirling2_inexact",
r"""
Internal function, do not use.
""")
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