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checkhand/venv/lib/python3.11/site-packages/scipy/special/_basic.py
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#
# Author: Travis Oliphant, 2002
#
import numpy as np
import math
import warnings
from collections import defaultdict
from heapq import heapify, heappop
from numpy import (pi, asarray, floor, isscalar, sqrt, where,
sin, place, issubdtype, extract, inexact, nan, zeros, sinc)
from . import _ufuncs
from ._ufuncs import (mathieu_a, mathieu_b, iv, jv, gamma, rgamma,
psi, hankel1, hankel2, yv, kv, poch, binom,
_stirling2_inexact)
from ._gufuncs import _lqn, _lqmn, _rctj, _rcty
from ._input_validation import _nonneg_int_or_fail
from . import _specfun
from ._comb import _comb_int
from ._multiufuncs import (assoc_legendre_p_all,
legendre_p_all)
from scipy._lib.deprecation import _deprecated
__all__ = [
'ai_zeros',
'assoc_laguerre',
'bei_zeros',
'beip_zeros',
'ber_zeros',
'bernoulli',
'berp_zeros',
'bi_zeros',
'clpmn',
'comb',
'digamma',
'diric',
'erf_zeros',
'euler',
'factorial',
'factorial2',
'factorialk',
'fresnel_zeros',
'fresnelc_zeros',
'fresnels_zeros',
'h1vp',
'h2vp',
'ivp',
'jn_zeros',
'jnjnp_zeros',
'jnp_zeros',
'jnyn_zeros',
'jvp',
'kei_zeros',
'keip_zeros',
'kelvin_zeros',
'ker_zeros',
'kerp_zeros',
'kvp',
'lmbda',
'lpmn',
'lpn',
'lqmn',
'lqn',
'mathieu_even_coef',
'mathieu_odd_coef',
'obl_cv_seq',
'pbdn_seq',
'pbdv_seq',
'pbvv_seq',
'perm',
'polygamma',
'pro_cv_seq',
'riccati_jn',
'riccati_yn',
'sinc',
'softplus',
'stirling2',
'y0_zeros',
'y1_zeros',
'y1p_zeros',
'yn_zeros',
'ynp_zeros',
'yvp',
'zeta'
]
__DEPRECATION_MSG_1_15 = (
"`scipy.special.{}` is deprecated as of SciPy 1.15.0 and will be "
"removed in SciPy 1.17.0. Please use `scipy.special.{}` instead."
)
# mapping k to last n such that factorialk(n, k) < np.iinfo(np.int64).max
_FACTORIALK_LIMITS_64BITS = {1: 20, 2: 33, 3: 44, 4: 54, 5: 65,
6: 74, 7: 84, 8: 93, 9: 101}
# mapping k to last n such that factorialk(n, k) < np.iinfo(np.int32).max
_FACTORIALK_LIMITS_32BITS = {1: 12, 2: 19, 3: 25, 4: 31, 5: 37,
6: 43, 7: 47, 8: 51, 9: 56}
def diric(x, n):
"""Periodic sinc function, also called the Dirichlet function.
The Dirichlet function is defined as::
diric(x, n) = sin(x * n/2) / (n * sin(x / 2)),
where `n` is a positive integer.
Parameters
----------
x : array_like
Input data
n : int
Integer defining the periodicity.
Returns
-------
diric : ndarray
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-8*np.pi, 8*np.pi, num=201)
>>> plt.figure(figsize=(8, 8));
>>> for idx, n in enumerate([2, 3, 4, 9]):
... plt.subplot(2, 2, idx+1)
... plt.plot(x, special.diric(x, n))
... plt.title('diric, n={}'.format(n))
>>> plt.show()
The following example demonstrates that `diric` gives the magnitudes
(modulo the sign and scaling) of the Fourier coefficients of a
rectangular pulse.
Suppress output of values that are effectively 0:
>>> np.set_printoptions(suppress=True)
Create a signal `x` of length `m` with `k` ones:
>>> m = 8
>>> k = 3
>>> x = np.zeros(m)
>>> x[:k] = 1
Use the FFT to compute the Fourier transform of `x`, and
inspect the magnitudes of the coefficients:
>>> np.abs(np.fft.fft(x))
array([ 3. , 2.41421356, 1. , 0.41421356, 1. ,
0.41421356, 1. , 2.41421356])
Now find the same values (up to sign) using `diric`. We multiply
by `k` to account for the different scaling conventions of
`numpy.fft.fft` and `diric`:
>>> theta = np.linspace(0, 2*np.pi, m, endpoint=False)
>>> k * special.diric(theta, k)
array([ 3. , 2.41421356, 1. , -0.41421356, -1. ,
-0.41421356, 1. , 2.41421356])
"""
x, n = asarray(x), asarray(n)
n = asarray(n + (x-x))
x = asarray(x + (n-n))
if issubdtype(x.dtype, inexact):
ytype = x.dtype
else:
ytype = float
y = zeros(x.shape, ytype)
# empirical minval for 32, 64 or 128 bit float computations
# where sin(x/2) < minval, result is fixed at +1 or -1
if np.finfo(ytype).eps < 1e-18:
minval = 1e-11
elif np.finfo(ytype).eps < 1e-15:
minval = 1e-7
else:
minval = 1e-3
mask1 = (n <= 0) | (n != floor(n))
place(y, mask1, nan)
x = x / 2
denom = sin(x)
mask2 = (1-mask1) & (abs(denom) < minval)
xsub = extract(mask2, x)
nsub = extract(mask2, n)
zsub = xsub / pi
place(y, mask2, pow(-1, np.round(zsub)*(nsub-1)))
mask = (1-mask1) & (1-mask2)
xsub = extract(mask, x)
nsub = extract(mask, n)
dsub = extract(mask, denom)
place(y, mask, sin(nsub*xsub)/(nsub*dsub))
return y
def jnjnp_zeros(nt):
"""Compute zeros of integer-order Bessel functions Jn and Jn'.
Results are arranged in order of the magnitudes of the zeros.
Parameters
----------
nt : int
Number (<=1200) of zeros to compute
Returns
-------
zo[l-1] : ndarray
Value of the lth zero of Jn(x) and Jn'(x). Of length `nt`.
n[l-1] : ndarray
Order of the Jn(x) or Jn'(x) associated with lth zero. Of length `nt`.
m[l-1] : ndarray
Serial number of the zeros of Jn(x) or Jn'(x) associated
with lth zero. Of length `nt`.
t[l-1] : ndarray
0 if lth zero in zo is zero of Jn(x), 1 if it is a zero of Jn'(x). Of
length `nt`.
See Also
--------
jn_zeros, jnp_zeros : to get separated arrays of zeros.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt > 1200):
raise ValueError("Number must be integer <= 1200.")
nt = int(nt)
n, m, t, zo = _specfun.jdzo(nt)
return zo[1:nt+1], n[:nt], m[:nt], t[:nt]
def jnyn_zeros(n, nt):
"""Compute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x).
Returns 4 arrays of length `nt`, corresponding to the first `nt`
zeros of Jn(x), Jn'(x), Yn(x), and Yn'(x), respectively. The zeros
are returned in ascending order.
Parameters
----------
n : int
Order of the Bessel functions
nt : int
Number (<=1200) of zeros to compute
Returns
-------
Jn : ndarray
First `nt` zeros of Jn
Jnp : ndarray
First `nt` zeros of Jn'
Yn : ndarray
First `nt` zeros of Yn
Ynp : ndarray
First `nt` zeros of Yn'
See Also
--------
jn_zeros, jnp_zeros, yn_zeros, ynp_zeros
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first three roots of :math:`J_1`, :math:`J_1'`,
:math:`Y_1` and :math:`Y_1'`.
>>> from scipy.special import jnyn_zeros
>>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3)
>>> jn_roots, yn_roots
(array([ 3.83170597, 7.01558667, 10.17346814]),
array([2.19714133, 5.42968104, 8.59600587]))
Plot :math:`J_1`, :math:`J_1'`, :math:`Y_1`, :math:`Y_1'` and their roots.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import jnyn_zeros, jvp, jn, yvp, yn
>>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3)
>>> fig, ax = plt.subplots()
>>> xmax= 11
>>> x = np.linspace(0, xmax)
>>> x[0] += 1e-15
>>> ax.plot(x, jn(1, x), label=r"$J_1$", c='r')
>>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$", c='b')
>>> ax.plot(x, yn(1, x), label=r"$Y_1$", c='y')
>>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$", c='c')
>>> zeros = np.zeros((3, ))
>>> ax.scatter(jn_roots, zeros, s=30, c='r', zorder=5,
... label=r"$J_1$ roots")
>>> ax.scatter(jnp_roots, zeros, s=30, c='b', zorder=5,
... label=r"$J_1'$ roots")
>>> ax.scatter(yn_roots, zeros, s=30, c='y', zorder=5,
... label=r"$Y_1$ roots")
>>> ax.scatter(ynp_roots, zeros, s=30, c='c', zorder=5,
... label=r"$Y_1'$ roots")
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_ylim(-0.6, 0.6)
>>> ax.set_xlim(0, xmax)
>>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75))
>>> plt.tight_layout()
>>> plt.show()
"""
if not (isscalar(nt) and isscalar(n)):
raise ValueError("Arguments must be scalars.")
if (floor(n) != n) or (floor(nt) != nt):
raise ValueError("Arguments must be integers.")
if (nt <= 0):
raise ValueError("nt > 0")
return _specfun.jyzo(abs(n), nt)
def jn_zeros(n, nt):
r"""Compute zeros of integer-order Bessel functions Jn.
Compute `nt` zeros of the Bessel functions :math:`J_n(x)` on the
interval :math:`(0, \infty)`. The zeros are returned in ascending
order. Note that this interval excludes the zero at :math:`x = 0`
that exists for :math:`n > 0`.
Parameters
----------
n : int
Order of Bessel function
nt : int
Number of zeros to return
Returns
-------
ndarray
First `nt` zeros of the Bessel function.
See Also
--------
jv: Real-order Bessel functions of the first kind
jnp_zeros: Zeros of :math:`Jn'`
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four positive roots of :math:`J_3`.
>>> from scipy.special import jn_zeros
>>> jn_zeros(3, 4)
array([ 6.3801619 , 9.76102313, 13.01520072, 16.22346616])
Plot :math:`J_3` and its first four positive roots. Note
that the root located at 0 is not returned by `jn_zeros`.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import jn, jn_zeros
>>> j3_roots = jn_zeros(3, 4)
>>> xmax = 18
>>> xmin = -1
>>> x = np.linspace(xmin, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, jn(3, x), label=r'$J_3$')
>>> ax.scatter(j3_roots, np.zeros((4, )), s=30, c='r',
... label=r"$J_3$_Zeros", zorder=5)
>>> ax.scatter(0, 0, s=30, c='k',
... label=r"Root at 0", zorder=5)
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend()
>>> plt.show()
"""
return jnyn_zeros(n, nt)[0]
def jnp_zeros(n, nt):
r"""Compute zeros of integer-order Bessel function derivatives Jn'.
Compute `nt` zeros of the functions :math:`J_n'(x)` on the
interval :math:`(0, \infty)`. The zeros are returned in ascending
order. Note that this interval excludes the zero at :math:`x = 0`
that exists for :math:`n > 1`.
Parameters
----------
n : int
Order of Bessel function
nt : int
Number of zeros to return
Returns
-------
ndarray
First `nt` zeros of the Bessel function.
See Also
--------
jvp: Derivatives of integer-order Bessel functions of the first kind
jv: Float-order Bessel functions of the first kind
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four roots of :math:`J_2'`.
>>> from scipy.special import jnp_zeros
>>> jnp_zeros(2, 4)
array([ 3.05423693, 6.70613319, 9.96946782, 13.17037086])
As `jnp_zeros` yields the roots of :math:`J_n'`, it can be used to
compute the locations of the peaks of :math:`J_n`. Plot
:math:`J_2`, :math:`J_2'` and the locations of the roots of :math:`J_2'`.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import jn, jnp_zeros, jvp
>>> j2_roots = jnp_zeros(2, 4)
>>> xmax = 15
>>> x = np.linspace(0, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, jn(2, x), label=r'$J_2$')
>>> ax.plot(x, jvp(2, x, 1), label=r"$J_2'$")
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.scatter(j2_roots, np.zeros((4, )), s=30, c='r',
... label=r"Roots of $J_2'$", zorder=5)
>>> ax.set_ylim(-0.4, 0.8)
>>> ax.set_xlim(0, xmax)
>>> plt.legend()
>>> plt.show()
"""
return jnyn_zeros(n, nt)[1]
def yn_zeros(n, nt):
r"""Compute zeros of integer-order Bessel function Yn(x).
Compute `nt` zeros of the functions :math:`Y_n(x)` on the interval
:math:`(0, \infty)`. The zeros are returned in ascending order.
Parameters
----------
n : int
Order of Bessel function
nt : int
Number of zeros to return
Returns
-------
ndarray
First `nt` zeros of the Bessel function.
See Also
--------
yn: Bessel function of the second kind for integer order
yv: Bessel function of the second kind for real order
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four roots of :math:`Y_2`.
>>> from scipy.special import yn_zeros
>>> yn_zeros(2, 4)
array([ 3.38424177, 6.79380751, 10.02347798, 13.20998671])
Plot :math:`Y_2` and its first four roots.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import yn, yn_zeros
>>> xmin = 2
>>> xmax = 15
>>> x = np.linspace(xmin, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.hlines(0, xmin, xmax, color='k')
>>> ax.plot(x, yn(2, x), label=r'$Y_2$')
>>> ax.scatter(yn_zeros(2, 4), np.zeros((4, )), s=30, c='r',
... label='Roots', zorder=5)
>>> ax.set_ylim(-0.4, 0.4)
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend()
>>> plt.show()
"""
return jnyn_zeros(n, nt)[2]
def ynp_zeros(n, nt):
r"""Compute zeros of integer-order Bessel function derivatives Yn'(x).
Compute `nt` zeros of the functions :math:`Y_n'(x)` on the
interval :math:`(0, \infty)`. The zeros are returned in ascending
order.
Parameters
----------
n : int
Order of Bessel function
nt : int
Number of zeros to return
Returns
-------
ndarray
First `nt` zeros of the Bessel derivative function.
See Also
--------
yvp
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four roots of the first derivative of the
Bessel function of second kind for order 0 :math:`Y_0'`.
>>> from scipy.special import ynp_zeros
>>> ynp_zeros(0, 4)
array([ 2.19714133, 5.42968104, 8.59600587, 11.74915483])
Plot :math:`Y_0`, :math:`Y_0'` and confirm visually that the roots of
:math:`Y_0'` are located at local extrema of :math:`Y_0`.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import yn, ynp_zeros, yvp
>>> zeros = ynp_zeros(0, 4)
>>> xmax = 13
>>> x = np.linspace(0, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, yn(0, x), label=r'$Y_0$')
>>> ax.plot(x, yvp(0, x, 1), label=r"$Y_0'$")
>>> ax.scatter(zeros, np.zeros((4, )), s=30, c='r',
... label=r"Roots of $Y_0'$", zorder=5)
>>> for root in zeros:
... y0_extremum = yn(0, root)
... lower = min(0, y0_extremum)
... upper = max(0, y0_extremum)
... ax.vlines(root, lower, upper, color='r')
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_ylim(-0.6, 0.6)
>>> ax.set_xlim(0, xmax)
>>> plt.legend()
>>> plt.show()
"""
return jnyn_zeros(n, nt)[3]
def y0_zeros(nt, complex=False):
"""Compute nt zeros of Bessel function Y0(z), and derivative at each zero.
The derivatives are given by Y0'(z0) = -Y1(z0) at each zero z0.
Parameters
----------
nt : int
Number of zeros to return
complex : bool, default False
Set to False to return only the real zeros; set to True to return only
the complex zeros with negative real part and positive imaginary part.
Note that the complex conjugates of the latter are also zeros of the
function, but are not returned by this routine.
Returns
-------
z0n : ndarray
Location of nth zero of Y0(z)
y0pz0n : ndarray
Value of derivative Y0'(z0) for nth zero
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first 4 real roots and the derivatives at the roots of
:math:`Y_0`:
>>> import numpy as np
>>> from scipy.special import y0_zeros
>>> zeros, grads = y0_zeros(4)
>>> with np.printoptions(precision=5):
... print(f"Roots: {zeros}")
... print(f"Gradients: {grads}")
Roots: [ 0.89358+0.j 3.95768+0.j 7.08605+0.j 10.22235+0.j]
Gradients: [-0.87942+0.j 0.40254+0.j -0.3001 +0.j 0.2497 +0.j]
Plot the real part of :math:`Y_0` and the first four computed roots.
>>> import matplotlib.pyplot as plt
>>> from scipy.special import y0
>>> xmin = 0
>>> xmax = 11
>>> x = np.linspace(xmin, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.hlines(0, xmin, xmax, color='k')
>>> ax.plot(x, y0(x), label=r'$Y_0$')
>>> zeros, grads = y0_zeros(4)
>>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r',
... label=r'$Y_0$_zeros', zorder=5)
>>> ax.set_ylim(-0.5, 0.6)
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend(ncol=2)
>>> plt.show()
Compute the first 4 complex roots and the derivatives at the roots of
:math:`Y_0` by setting ``complex=True``:
>>> y0_zeros(4, True)
(array([ -2.40301663+0.53988231j, -5.5198767 +0.54718001j,
-8.6536724 +0.54841207j, -11.79151203+0.54881912j]),
array([ 0.10074769-0.88196771j, -0.02924642+0.5871695j ,
0.01490806-0.46945875j, -0.00937368+0.40230454j]))
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("Arguments must be scalar positive integer.")
kf = 0
kc = not complex
return _specfun.cyzo(nt, kf, kc)
def y1_zeros(nt, complex=False):
"""Compute nt zeros of Bessel function Y1(z), and derivative at each zero.
The derivatives are given by Y1'(z1) = Y0(z1) at each zero z1.
Parameters
----------
nt : int
Number of zeros to return
complex : bool, default False
Set to False to return only the real zeros; set to True to return only
the complex zeros with negative real part and positive imaginary part.
Note that the complex conjugates of the latter are also zeros of the
function, but are not returned by this routine.
Returns
-------
z1n : ndarray
Location of nth zero of Y1(z)
y1pz1n : ndarray
Value of derivative Y1'(z1) for nth zero
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first 4 real roots and the derivatives at the roots of
:math:`Y_1`:
>>> import numpy as np
>>> from scipy.special import y1_zeros
>>> zeros, grads = y1_zeros(4)
>>> with np.printoptions(precision=5):
... print(f"Roots: {zeros}")
... print(f"Gradients: {grads}")
Roots: [ 2.19714+0.j 5.42968+0.j 8.59601+0.j 11.74915+0.j]
Gradients: [ 0.52079+0.j -0.34032+0.j 0.27146+0.j -0.23246+0.j]
Extract the real parts:
>>> realzeros = zeros.real
>>> realzeros
array([ 2.19714133, 5.42968104, 8.59600587, 11.74915483])
Plot :math:`Y_1` and the first four computed roots.
>>> import matplotlib.pyplot as plt
>>> from scipy.special import y1
>>> xmin = 0
>>> xmax = 13
>>> x = np.linspace(xmin, xmax, 500)
>>> zeros, grads = y1_zeros(4)
>>> fig, ax = plt.subplots()
>>> ax.hlines(0, xmin, xmax, color='k')
>>> ax.plot(x, y1(x), label=r'$Y_1$')
>>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r',
... label=r'$Y_1$_zeros', zorder=5)
>>> ax.set_ylim(-0.5, 0.5)
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend()
>>> plt.show()
Compute the first 4 complex roots and the derivatives at the roots of
:math:`Y_1` by setting ``complex=True``:
>>> y1_zeros(4, True)
(array([ -0.50274327+0.78624371j, -3.83353519+0.56235654j,
-7.01590368+0.55339305j, -10.17357383+0.55127339j]),
array([-0.45952768+1.31710194j, 0.04830191-0.69251288j,
-0.02012695+0.51864253j, 0.011614 -0.43203296j]))
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("Arguments must be scalar positive integer.")
kf = 1
kc = not complex
return _specfun.cyzo(nt, kf, kc)
def y1p_zeros(nt, complex=False):
"""Compute nt zeros of Bessel derivative Y1'(z), and value at each zero.
The values are given by Y1(z1) at each z1 where Y1'(z1)=0.
Parameters
----------
nt : int
Number of zeros to return
complex : bool, default False
Set to False to return only the real zeros; set to True to return only
the complex zeros with negative real part and positive imaginary part.
Note that the complex conjugates of the latter are also zeros of the
function, but are not returned by this routine.
Returns
-------
z1pn : ndarray
Location of nth zero of Y1'(z)
y1z1pn : ndarray
Value of derivative Y1(z1) for nth zero
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four roots of :math:`Y_1'` and the values of
:math:`Y_1` at these roots.
>>> import numpy as np
>>> from scipy.special import y1p_zeros
>>> y1grad_roots, y1_values = y1p_zeros(4)
>>> with np.printoptions(precision=5):
... print(f"Y1' Roots: {y1grad_roots.real}")
... print(f"Y1 values: {y1_values.real}")
Y1' Roots: [ 3.68302 6.9415 10.1234 13.28576]
Y1 values: [ 0.41673 -0.30317 0.25091 -0.21897]
`y1p_zeros` can be used to calculate the extremal points of :math:`Y_1`
directly. Here we plot :math:`Y_1` and the first four extrema.
>>> import matplotlib.pyplot as plt
>>> from scipy.special import y1, yvp
>>> y1_roots, y1_values_at_roots = y1p_zeros(4)
>>> real_roots = y1_roots.real
>>> xmax = 15
>>> x = np.linspace(0, xmax, 500)
>>> x[0] += 1e-15
>>> fig, ax = plt.subplots()
>>> ax.plot(x, y1(x), label=r'$Y_1$')
>>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$")
>>> ax.scatter(real_roots, np.zeros((4, )), s=30, c='r',
... label=r"Roots of $Y_1'$", zorder=5)
>>> ax.scatter(real_roots, y1_values_at_roots.real, s=30, c='k',
... label=r"Extrema of $Y_1$", zorder=5)
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_ylim(-0.5, 0.5)
>>> ax.set_xlim(0, xmax)
>>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75))
>>> plt.tight_layout()
>>> plt.show()
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("Arguments must be scalar positive integer.")
kf = 2
kc = not complex
return _specfun.cyzo(nt, kf, kc)
def _bessel_diff_formula(v, z, n, L, phase):
# from AMS55.
# L(v, z) = J(v, z), Y(v, z), H1(v, z), H2(v, z), phase = -1
# L(v, z) = I(v, z) or exp(v*pi*i)K(v, z), phase = 1
# For K, you can pull out the exp((v-k)*pi*i) into the caller
v = asarray(v)
p = 1.0
s = L(v-n, z)
for i in range(1, n+1):
p = phase * (p * (n-i+1)) / i # = choose(k, i)
s += p*L(v-n + i*2, z)
return s / (2.**n)
def jvp(v, z, n=1):
"""Compute derivatives of Bessel functions of the first kind.
Compute the nth derivative of the Bessel function `Jv` with
respect to `z`.
Parameters
----------
v : array_like or float
Order of Bessel function
z : complex
Argument at which to evaluate the derivative; can be real or
complex.
n : int, default 1
Order of derivative. For 0 returns the Bessel function `jv` itself.
Returns
-------
scalar or ndarray
Values of the derivative of the Bessel function.
Notes
-----
The derivative is computed using the relation DLFM 10.6.7 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.6.E7
Examples
--------
Compute the Bessel function of the first kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import jvp
>>> jvp(0, 1, 0), jvp(0, 1, 1), jvp(0, 1, 2)
(0.7651976865579666, -0.44005058574493355, -0.3251471008130331)
Compute the first derivative of the Bessel function of the first
kind for several orders at 1 by providing an array for `v`.
>>> jvp([0, 1, 2], 1, 1)
array([-0.44005059, 0.3251471 , 0.21024362])
Compute the first derivative of the Bessel function of the first
kind of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0., 1.5, 3.])
>>> jvp(0, points, 1)
array([-0. , -0.55793651, -0.33905896])
Plot the Bessel function of the first kind of order 1 and its
first three derivatives.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-10, 10, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, jvp(1, x, 0), label=r"$J_1$")
>>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$")
>>> ax.plot(x, jvp(1, x, 2), label=r"$J_1''$")
>>> ax.plot(x, jvp(1, x, 3), label=r"$J_1'''$")
>>> plt.legend()
>>> plt.show()
"""
n = _nonneg_int_or_fail(n, 'n')
if n == 0:
return jv(v, z)
else:
return _bessel_diff_formula(v, z, n, jv, -1)
def yvp(v, z, n=1):
"""Compute derivatives of Bessel functions of the second kind.
Compute the nth derivative of the Bessel function `Yv` with
respect to `z`.
Parameters
----------
v : array_like of float
Order of Bessel function
z : complex
Argument at which to evaluate the derivative
n : int, default 1
Order of derivative. For 0 returns the BEssel function `yv`
Returns
-------
scalar or ndarray
nth derivative of the Bessel function.
See Also
--------
yv : Bessel functions of the second kind
Notes
-----
The derivative is computed using the relation DLFM 10.6.7 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.6.E7
Examples
--------
Compute the Bessel function of the second kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import yvp
>>> yvp(0, 1, 0), yvp(0, 1, 1), yvp(0, 1, 2)
(0.088256964215677, 0.7812128213002889, -0.8694697855159659)
Compute the first derivative of the Bessel function of the second
kind for several orders at 1 by providing an array for `v`.
>>> yvp([0, 1, 2], 1, 1)
array([0.78121282, 0.86946979, 2.52015239])
Compute the first derivative of the Bessel function of the
second kind of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> yvp(0, points, 1)
array([ 1.47147239, 0.41230863, -0.32467442])
Plot the Bessel function of the second kind of order 1 and its
first three derivatives.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> x[0] += 1e-15
>>> fig, ax = plt.subplots()
>>> ax.plot(x, yvp(1, x, 0), label=r"$Y_1$")
>>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$")
>>> ax.plot(x, yvp(1, x, 2), label=r"$Y_1''$")
>>> ax.plot(x, yvp(1, x, 3), label=r"$Y_1'''$")
>>> ax.set_ylim(-10, 10)
>>> plt.legend()
>>> plt.show()
"""
n = _nonneg_int_or_fail(n, 'n')
if n == 0:
return yv(v, z)
else:
return _bessel_diff_formula(v, z, n, yv, -1)
def kvp(v, z, n=1):
"""Compute derivatives of real-order modified Bessel function Kv(z)
Kv(z) is the modified Bessel function of the second kind.
Derivative is calculated with respect to `z`.
Parameters
----------
v : array_like of float
Order of Bessel function
z : array_like of complex
Argument at which to evaluate the derivative
n : int, default 1
Order of derivative. For 0 returns the Bessel function `kv` itself.
Returns
-------
out : ndarray
The results
See Also
--------
kv
Notes
-----
The derivative is computed using the relation DLFM 10.29.5 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 6.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.29.E5
Examples
--------
Compute the modified bessel function of the second kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import kvp
>>> kvp(0, 1, 0), kvp(0, 1, 1), kvp(0, 1, 2)
(0.42102443824070834, -0.6019072301972346, 1.0229316684379428)
Compute the first derivative of the modified Bessel function of the second
kind for several orders at 1 by providing an array for `v`.
>>> kvp([0, 1, 2], 1, 1)
array([-0.60190723, -1.02293167, -3.85158503])
Compute the first derivative of the modified Bessel function of the
second kind of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> kvp(0, points, 1)
array([-1.65644112, -0.2773878 , -0.04015643])
Plot the modified bessel function of the second kind and its
first three derivatives.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, kvp(1, x, 0), label=r"$K_1$")
>>> ax.plot(x, kvp(1, x, 1), label=r"$K_1'$")
>>> ax.plot(x, kvp(1, x, 2), label=r"$K_1''$")
>>> ax.plot(x, kvp(1, x, 3), label=r"$K_1'''$")
>>> ax.set_ylim(-2.5, 2.5)
>>> plt.legend()
>>> plt.show()
"""
n = _nonneg_int_or_fail(n, 'n')
if n == 0:
return kv(v, z)
else:
return (-1)**n * _bessel_diff_formula(v, z, n, kv, 1)
def ivp(v, z, n=1):
"""Compute derivatives of modified Bessel functions of the first kind.
Compute the nth derivative of the modified Bessel function `Iv`
with respect to `z`.
Parameters
----------
v : array_like or float
Order of Bessel function
z : array_like
Argument at which to evaluate the derivative; can be real or
complex.
n : int, default 1
Order of derivative. For 0, returns the Bessel function `iv` itself.
Returns
-------
scalar or ndarray
nth derivative of the modified Bessel function.
See Also
--------
iv
Notes
-----
The derivative is computed using the relation DLFM 10.29.5 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 6.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.29.E5
Examples
--------
Compute the modified Bessel function of the first kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import ivp
>>> ivp(0, 1, 0), ivp(0, 1, 1), ivp(0, 1, 2)
(1.2660658777520084, 0.565159103992485, 0.7009067737595233)
Compute the first derivative of the modified Bessel function of the first
kind for several orders at 1 by providing an array for `v`.
>>> ivp([0, 1, 2], 1, 1)
array([0.5651591 , 0.70090677, 0.29366376])
Compute the first derivative of the modified Bessel function of the
first kind of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0., 1.5, 3.])
>>> ivp(0, points, 1)
array([0. , 0.98166643, 3.95337022])
Plot the modified Bessel function of the first kind of order 1 and its
first three derivatives.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-5, 5, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, ivp(1, x, 0), label=r"$I_1$")
>>> ax.plot(x, ivp(1, x, 1), label=r"$I_1'$")
>>> ax.plot(x, ivp(1, x, 2), label=r"$I_1''$")
>>> ax.plot(x, ivp(1, x, 3), label=r"$I_1'''$")
>>> plt.legend()
>>> plt.show()
"""
n = _nonneg_int_or_fail(n, 'n')
if n == 0:
return iv(v, z)
else:
return _bessel_diff_formula(v, z, n, iv, 1)
def h1vp(v, z, n=1):
"""Compute derivatives of Hankel function H1v(z) with respect to `z`.
Parameters
----------
v : array_like
Order of Hankel function
z : array_like
Argument at which to evaluate the derivative. Can be real or
complex.
n : int, default 1
Order of derivative. For 0 returns the Hankel function `h1v` itself.
Returns
-------
scalar or ndarray
Values of the derivative of the Hankel function.
See Also
--------
hankel1
Notes
-----
The derivative is computed using the relation DLFM 10.6.7 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.6.E7
Examples
--------
Compute the Hankel function of the first kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import h1vp
>>> h1vp(0, 1, 0), h1vp(0, 1, 1), h1vp(0, 1, 2)
((0.7651976865579664+0.088256964215677j),
(-0.44005058574493355+0.7812128213002889j),
(-0.3251471008130329-0.8694697855159659j))
Compute the first derivative of the Hankel function of the first kind
for several orders at 1 by providing an array for `v`.
>>> h1vp([0, 1, 2], 1, 1)
array([-0.44005059+0.78121282j, 0.3251471 +0.86946979j,
0.21024362+2.52015239j])
Compute the first derivative of the Hankel function of the first kind
of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> h1vp(0, points, 1)
array([-0.24226846+1.47147239j, -0.55793651+0.41230863j,
-0.33905896-0.32467442j])
"""
n = _nonneg_int_or_fail(n, 'n')
if n == 0:
return hankel1(v, z)
else:
return _bessel_diff_formula(v, z, n, hankel1, -1)
def h2vp(v, z, n=1):
"""Compute derivatives of Hankel function H2v(z) with respect to `z`.
Parameters
----------
v : array_like
Order of Hankel function
z : array_like
Argument at which to evaluate the derivative. Can be real or
complex.
n : int, default 1
Order of derivative. For 0 returns the Hankel function `h2v` itself.
Returns
-------
scalar or ndarray
Values of the derivative of the Hankel function.
See Also
--------
hankel2
Notes
-----
The derivative is computed using the relation DLFM 10.6.7 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.6.E7
Examples
--------
Compute the Hankel function of the second kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import h2vp
>>> h2vp(0, 1, 0), h2vp(0, 1, 1), h2vp(0, 1, 2)
((0.7651976865579664-0.088256964215677j),
(-0.44005058574493355-0.7812128213002889j),
(-0.3251471008130329+0.8694697855159659j))
Compute the first derivative of the Hankel function of the second kind
for several orders at 1 by providing an array for `v`.
>>> h2vp([0, 1, 2], 1, 1)
array([-0.44005059-0.78121282j, 0.3251471 -0.86946979j,
0.21024362-2.52015239j])
Compute the first derivative of the Hankel function of the second kind
of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> h2vp(0, points, 1)
array([-0.24226846-1.47147239j, -0.55793651-0.41230863j,
-0.33905896+0.32467442j])
"""
n = _nonneg_int_or_fail(n, 'n')
if n == 0:
return hankel2(v, z)
else:
return _bessel_diff_formula(v, z, n, hankel2, -1)
def riccati_jn(n, x):
r"""Compute Ricatti-Bessel function of the first kind and its derivative.
The Ricatti-Bessel function of the first kind is defined as :math:`x
j_n(x)`, where :math:`j_n` is the spherical Bessel function of the first
kind of order :math:`n`.
This function computes the value and first derivative of the
Ricatti-Bessel function for all orders up to and including `n`.
Parameters
----------
n : int
Maximum order of function to compute
x : float
Argument at which to evaluate
Returns
-------
jn : ndarray
Value of j0(x), ..., jn(x)
jnp : ndarray
First derivative j0'(x), ..., jn'(x)
Notes
-----
The computation is carried out via backward recurrence, using the
relation DLMF 10.51.1 [2]_.
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/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.51.E1
"""
if not (isscalar(n) and isscalar(x)):
raise ValueError("arguments must be scalars.")
n = _nonneg_int_or_fail(n, 'n', strict=False)
if (n == 0):
n1 = 1
else:
n1 = n
jn = np.empty((n1 + 1,), dtype=np.float64)
jnp = np.empty_like(jn)
_rctj(x, out=(jn, jnp))
return jn[:(n+1)], jnp[:(n+1)]
def riccati_yn(n, x):
"""Compute Ricatti-Bessel function of the second kind and its derivative.
The Ricatti-Bessel function of the second kind is defined here as :math:`+x
y_n(x)`, where :math:`y_n` is the spherical Bessel function of the second
kind of order :math:`n`. *Note that this is in contrast to a common convention
that includes a minus sign in the definition.*
This function computes the value and first derivative of the function for
all orders up to and including `n`.
Parameters
----------
n : int
Maximum order of function to compute
x : float
Argument at which to evaluate
Returns
-------
yn : ndarray
Value of y0(x), ..., yn(x)
ynp : ndarray
First derivative y0'(x), ..., yn'(x)
Notes
-----
The computation is carried out via ascending recurrence, using the
relation DLMF 10.51.1 [2]_.
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/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.51.E1
"""
if not (isscalar(n) and isscalar(x)):
raise ValueError("arguments must be scalars.")
n = _nonneg_int_or_fail(n, 'n', strict=False)
if (n == 0):
n1 = 1
else:
n1 = n
yn = np.empty((n1 + 1,), dtype=np.float64)
ynp = np.empty_like(yn)
_rcty(x, out=(yn, ynp))
return yn[:(n+1)], ynp[:(n+1)]
def erf_zeros(nt):
"""Compute the first nt zero in the first quadrant, ordered by absolute value.
Zeros in the other quadrants can be obtained by using the symmetries
erf(-z) = erf(z) and erf(conj(z)) = conj(erf(z)).
Parameters
----------
nt : int
The number of zeros to compute
Returns
-------
The locations of the zeros of erf : ndarray (complex)
Complex values at which zeros of erf(z)
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
>>> from scipy import special
>>> special.erf_zeros(1)
array([1.45061616+1.880943j])
Check that erf is (close to) zero for the value returned by erf_zeros
>>> special.erf(special.erf_zeros(1))
array([4.95159469e-14-1.16407394e-16j])
"""
if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt):
raise ValueError("Argument must be positive scalar integer.")
return _specfun.cerzo(nt)
def fresnelc_zeros(nt):
"""Compute nt complex zeros of cosine Fresnel integral C(z).
Parameters
----------
nt : int
Number of zeros to compute
Returns
-------
fresnelc_zeros: ndarray
Zeros of the cosine Fresnel integral
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt):
raise ValueError("Argument must be positive scalar integer.")
return _specfun.fcszo(1, nt)
def fresnels_zeros(nt):
"""Compute nt complex zeros of sine Fresnel integral S(z).
Parameters
----------
nt : int
Number of zeros to compute
Returns
-------
fresnels_zeros: ndarray
Zeros of the sine Fresnel integral
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt):
raise ValueError("Argument must be positive scalar integer.")
return _specfun.fcszo(2, nt)
def fresnel_zeros(nt):
"""Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z).
Parameters
----------
nt : int
Number of zeros to compute
Returns
-------
zeros_sine: ndarray
Zeros of the sine Fresnel integral
zeros_cosine : ndarray
Zeros of the cosine Fresnel integral
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt):
raise ValueError("Argument must be positive scalar integer.")
return _specfun.fcszo(2, nt), _specfun.fcszo(1, nt)
def assoc_laguerre(x, n, k=0.0):
"""Compute the generalized (associated) Laguerre polynomial of degree n and order k.
The polynomial :math:`L^{(k)}_n(x)` is orthogonal over ``[0, inf)``,
with weighting function ``exp(-x) * x**k`` with ``k > -1``.
Parameters
----------
x : float or ndarray
Points where to evaluate the Laguerre polynomial
n : int
Degree of the Laguerre polynomial
k : int
Order of the Laguerre polynomial
Returns
-------
assoc_laguerre: float or ndarray
Associated laguerre polynomial values
Notes
-----
`assoc_laguerre` is a simple wrapper around `eval_genlaguerre`, with
reversed argument order ``(x, n, k=0.0) --> (n, k, x)``.
"""
return _ufuncs.eval_genlaguerre(n, k, x)
digamma = psi
def polygamma(n, x):
r"""Polygamma functions.
Defined as :math:`\psi^{(n)}(x)` where :math:`\psi` is the
`digamma` function. See [dlmf]_ for details.
Parameters
----------
n : array_like
The order of the derivative of the digamma function; must be
integral
x : array_like
Real valued input
Returns
-------
ndarray
Function results
See Also
--------
digamma
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/5.15
Examples
--------
>>> from scipy import special
>>> x = [2, 3, 25.5]
>>> special.polygamma(1, x)
array([ 0.64493407, 0.39493407, 0.03999467])
>>> special.polygamma(0, x) == special.psi(x)
array([ True, True, True], dtype=bool)
"""
n, x = asarray(n), asarray(x)
fac2 = (-1.0)**(n+1) * gamma(n+1.0) * zeta(n+1, x)
return where(n == 0, psi(x), fac2)
def mathieu_even_coef(m, q):
r"""Fourier coefficients for even Mathieu and modified Mathieu functions.
The Fourier series of the even solutions of the Mathieu differential
equation are of the form
.. math:: \mathrm{ce}_{2n}(z, q) = \sum_{k=0}^{\infty} A_{(2n)}^{(2k)} \cos 2kz
.. math:: \mathrm{ce}_{2n+1}(z, q) =
\sum_{k=0}^{\infty} A_{(2n+1)}^{(2k+1)} \cos (2k+1)z
This function returns the coefficients :math:`A_{(2n)}^{(2k)}` for even
input m=2n, and the coefficients :math:`A_{(2n+1)}^{(2k+1)}` for odd input
m=2n+1.
Parameters
----------
m : int
Order of Mathieu functions. Must be non-negative.
q : float (>=0)
Parameter of Mathieu functions. Must be non-negative.
Returns
-------
Ak : ndarray
Even or odd Fourier coefficients, corresponding to even or odd m.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/28.4#i
"""
if not (isscalar(m) and isscalar(q)):
raise ValueError("m and q must be scalars.")
if (q < 0):
raise ValueError("q >=0")
if (m != floor(m)) or (m < 0):
raise ValueError("m must be an integer >=0.")
if (q <= 1):
qm = 7.5 + 56.1*sqrt(q) - 134.7*q + 90.7*sqrt(q)*q
else:
qm = 17.0 + 3.1*sqrt(q) - .126*q + .0037*sqrt(q)*q
km = int(qm + 0.5*m)
if km > 251:
warnings.warn("Too many predicted coefficients.", RuntimeWarning, stacklevel=2)
kd = 1
m = int(floor(m))
if m % 2:
kd = 2
a = mathieu_a(m, q)
fc = _specfun.fcoef(kd, m, q, a)
return fc[:km]
def mathieu_odd_coef(m, q):
r"""Fourier coefficients for even Mathieu and modified Mathieu functions.
The Fourier series of the odd solutions of the Mathieu differential
equation are of the form
.. math:: \mathrm{se}_{2n+1}(z, q) =
\sum_{k=0}^{\infty} B_{(2n+1)}^{(2k+1)} \sin (2k+1)z
.. math:: \mathrm{se}_{2n+2}(z, q) =
\sum_{k=0}^{\infty} B_{(2n+2)}^{(2k+2)} \sin (2k+2)z
This function returns the coefficients :math:`B_{(2n+2)}^{(2k+2)}` for even
input m=2n+2, and the coefficients :math:`B_{(2n+1)}^{(2k+1)}` for odd
input m=2n+1.
Parameters
----------
m : int
Order of Mathieu functions. Must be non-negative.
q : float (>=0)
Parameter of Mathieu functions. Must be non-negative.
Returns
-------
Bk : ndarray
Even or odd Fourier coefficients, corresponding to even or odd m.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not (isscalar(m) and isscalar(q)):
raise ValueError("m and q must be scalars.")
if (q < 0):
raise ValueError("q >=0")
if (m != floor(m)) or (m <= 0):
raise ValueError("m must be an integer > 0")
if (q <= 1):
qm = 7.5 + 56.1*sqrt(q) - 134.7*q + 90.7*sqrt(q)*q
else:
qm = 17.0 + 3.1*sqrt(q) - .126*q + .0037*sqrt(q)*q
km = int(qm + 0.5*m)
if km > 251:
warnings.warn("Too many predicted coefficients.", RuntimeWarning, stacklevel=2)
kd = 4
m = int(floor(m))
if m % 2:
kd = 3
b = mathieu_b(m, q)
fc = _specfun.fcoef(kd, m, q, b)
return fc[:km]
@_deprecated(__DEPRECATION_MSG_1_15.format("lpmn", "assoc_legendre_p_all"))
def lpmn(m, n, z):
"""Sequence of associated Legendre functions of the first kind.
Computes the associated Legendre function of the first kind of order m and
degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``.
Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and
``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``.
This function takes a real argument ``z``. For complex arguments ``z``
use clpmn instead.
.. deprecated:: 1.15.0
This function is deprecated and will be removed in SciPy 1.17.0.
Please `scipy.special.assoc_legendre_p_all` instead.
Parameters
----------
m : int
``|m| <= n``; the order of the Legendre function.
n : int
where ``n >= 0``; the degree of the Legendre function. Often
called ``l`` (lower case L) in descriptions of the associated
Legendre function
z : array_like
Input value.
Returns
-------
Pmn_z : (m+1, n+1) array
Values for all orders 0..m and degrees 0..n
Pmn_d_z : (m+1, n+1) array
Derivatives for all orders 0..m and degrees 0..n
See Also
--------
clpmn: associated Legendre functions of the first kind for complex z
Notes
-----
In the interval (-1, 1), Ferrer's function of the first kind is
returned. The phase convention used for the intervals (1, inf)
and (-inf, -1) is such that the result is always real.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/14.3
"""
n = _nonneg_int_or_fail(n, 'n', strict=False)
if (abs(m) > n):
raise ValueError("m must be <= n.")
if np.iscomplexobj(z):
raise ValueError("Argument must be real. Use clpmn instead.")
m, n = int(m), int(n) # Convert to int to maintain backwards compatibility.
branch_cut = np.where(np.abs(z) <= 1, 2, 3)
p, pd = assoc_legendre_p_all(n, abs(m), z, branch_cut=branch_cut, diff_n=1)
p = np.swapaxes(p, 0, 1)
pd = np.swapaxes(pd, 0, 1)
if (m >= 0):
p = p[:(m + 1)]
pd = pd[:(m + 1)]
else:
p = np.insert(p[:(m - 1):-1], 0, p[0], axis=0)
pd = np.insert(pd[:(m - 1):-1], 0, pd[0], axis=0)
return p, pd
@_deprecated(__DEPRECATION_MSG_1_15.format("clpmn", "assoc_legendre_p_all"))
def clpmn(m, n, z, type=3):
"""Associated Legendre function of the first kind for complex arguments.
Computes the associated Legendre function of the first kind of order m and
degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``.
Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and
``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``.
.. deprecated:: 1.15.0
This function is deprecated and will be removed in SciPy 1.17.0.
Please use `scipy.special.assoc_legendre_p_all` instead.
Parameters
----------
m : int
``|m| <= n``; the order of the Legendre function.
n : int
where ``n >= 0``; the degree of the Legendre function. Often
called ``l`` (lower case L) in descriptions of the associated
Legendre function
z : array_like, float or complex
Input value.
type : int, optional
takes values 2 or 3
2: cut on the real axis ``|x| > 1``
3: cut on the real axis ``-1 < x < 1`` (default)
Returns
-------
Pmn_z : (m+1, n+1) array
Values for all orders ``0..m`` and degrees ``0..n``
Pmn_d_z : (m+1, n+1) array
Derivatives for all orders ``0..m`` and degrees ``0..n``
See Also
--------
lpmn: associated Legendre functions of the first kind for real z
Notes
-----
By default, i.e. for ``type=3``, phase conventions are chosen according
to [1]_ such that the function is analytic. The cut lies on the interval
(-1, 1). Approaching the cut from above or below in general yields a phase
factor with respect to Ferrer's function of the first kind
(cf. `lpmn`).
For ``type=2`` a cut at ``|x| > 1`` is chosen. Approaching the real values
on the interval (-1, 1) in the complex plane yields Ferrer's function
of the first kind.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/14.21
"""
if (abs(m) > n):
raise ValueError("m must be <= n.")
if not (type == 2 or type == 3):
raise ValueError("type must be either 2 or 3.")
m, n = int(m), int(n) # Convert to int to maintain backwards compatibility.
if not np.iscomplexobj(z):
z = np.asarray(z, dtype=complex)
out, out_jac = assoc_legendre_p_all(n, abs(m), z, branch_cut=type, diff_n=1)
out = np.swapaxes(out, 0, 1)
out_jac = np.swapaxes(out_jac, 0, 1)
if (m >= 0):
out = out[:(m + 1)]
out_jac = out_jac[:(m + 1)]
else:
out = np.insert(out[:(m - 1):-1], 0, out[0], axis=0)
out_jac = np.insert(out_jac[:(m - 1):-1], 0, out_jac[0], axis=0)
return out, out_jac
def lqmn(m, n, z):
"""Sequence of associated Legendre functions of the second kind.
Computes the associated Legendre function of the second kind of order m and
degree n, ``Qmn(z)`` = :math:`Q_n^m(z)`, and its derivative, ``Qmn'(z)``.
Returns two arrays of size ``(m+1, n+1)`` containing ``Qmn(z)`` and
``Qmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``.
Parameters
----------
m : int
``|m| <= n``; the order of the Legendre function.
n : int
where ``n >= 0``; the degree of the Legendre function. Often
called ``l`` (lower case L) in descriptions of the associated
Legendre function
z : array_like, complex
Input value.
Returns
-------
Qmn_z : (m+1, n+1) array
Values for all orders 0..m and degrees 0..n
Qmn_d_z : (m+1, n+1) array
Derivatives for all orders 0..m and degrees 0..n
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(m) or (m < 0):
raise ValueError("m must be a non-negative integer.")
if not isscalar(n) or (n < 0):
raise ValueError("n must be a non-negative integer.")
m, n = int(m), int(n) # Convert to int to maintain backwards compatibility.
# Ensure neither m nor n == 0
mm = max(1, m)
nn = max(1, n)
z = np.asarray(z)
if (not np.issubdtype(z.dtype, np.inexact)):
z = z.astype(np.float64)
if np.iscomplexobj(z):
q = np.empty((mm + 1, nn + 1) + z.shape, dtype=np.complex128)
else:
q = np.empty((mm + 1, nn + 1) + z.shape, dtype=np.float64)
qd = np.empty_like(q)
if (z.ndim == 0):
_lqmn(z, out=(q, qd))
else:
# new axes must be last for the ufunc
_lqmn(z,
out=(np.moveaxis(q, (0, 1), (-2, -1)),
np.moveaxis(qd, (0, 1), (-2, -1))))
return q[:(m+1), :(n+1)], qd[:(m+1), :(n+1)]
def bernoulli(n):
"""Bernoulli numbers B0..Bn (inclusive).
Parameters
----------
n : int
Indicated the number of terms in the Bernoulli series to generate.
Returns
-------
ndarray
The Bernoulli numbers ``[B(0), B(1), ..., B(n)]``.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] "Bernoulli number", Wikipedia, https://en.wikipedia.org/wiki/Bernoulli_number
Examples
--------
>>> import numpy as np
>>> from scipy.special import bernoulli, zeta
>>> bernoulli(4)
array([ 1. , -0.5 , 0.16666667, 0. , -0.03333333])
The Wikipedia article ([2]_) points out the relationship between the
Bernoulli numbers and the zeta function, ``B_n^+ = -n * zeta(1 - n)``
for ``n > 0``:
>>> n = np.arange(1, 5)
>>> -n * zeta(1 - n)
array([ 0.5 , 0.16666667, -0. , -0.03333333])
Note that, in the notation used in the wikipedia article,
`bernoulli` computes ``B_n^-`` (i.e. it used the convention that
``B_1`` is -1/2). The relation given above is for ``B_n^+``, so the
sign of 0.5 does not match the output of ``bernoulli(4)``.
"""
if not isscalar(n) or (n < 0):
raise ValueError("n must be a non-negative integer.")
n = int(n)
if (n < 2):
n1 = 2
else:
n1 = n
return _specfun.bernob(int(n1))[:(n+1)]
def euler(n):
"""Euler numbers E(0), E(1), ..., E(n).
The Euler numbers [1]_ are also known as the secant numbers.
Because ``euler(n)`` returns floating point values, it does not give
exact values for large `n`. The first inexact value is E(22).
Parameters
----------
n : int
The highest index of the Euler number to be returned.
Returns
-------
ndarray
The Euler numbers [E(0), E(1), ..., E(n)].
The odd Euler numbers, which are all zero, are included.
References
----------
.. [1] Sequence A122045, The On-Line Encyclopedia of Integer Sequences,
https://oeis.org/A122045
.. [2] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
>>> import numpy as np
>>> from scipy.special import euler
>>> euler(6)
array([ 1., 0., -1., 0., 5., 0., -61.])
>>> euler(13).astype(np.int64)
array([ 1, 0, -1, 0, 5, 0, -61,
0, 1385, 0, -50521, 0, 2702765, 0])
>>> euler(22)[-1] # Exact value of E(22) is -69348874393137901.
-69348874393137976.0
"""
if not isscalar(n) or (n < 0):
raise ValueError("n must be a non-negative integer.")
n = int(n)
if (n < 2):
n1 = 2
else:
n1 = n
return _specfun.eulerb(n1)[:(n+1)]
@_deprecated(__DEPRECATION_MSG_1_15.format("lpn", "legendre_p_all"))
def lpn(n, z):
"""Legendre function of the first kind.
Compute sequence of Legendre functions of the first kind (polynomials),
Pn(z) and derivatives for all degrees from 0 to n (inclusive).
See also special.legendre for polynomial class.
.. deprecated:: 1.15.0
This function is deprecated and will be removed in SciPy 1.17.0.
Please use `scipy.special.legendre_p_all` instead.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
return legendre_p_all(n, z, diff_n=1)
def lqn(n, z):
"""Legendre function of the second kind.
Compute sequence of Legendre functions of the second kind, Qn(z) and
derivatives for all degrees from 0 to n (inclusive).
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
n = _nonneg_int_or_fail(n, 'n', strict=False)
if (n < 1):
n1 = 1
else:
n1 = n
z = np.asarray(z)
if (not np.issubdtype(z.dtype, np.inexact)):
z = z.astype(float)
if np.iscomplexobj(z):
qn = np.empty((n1 + 1,) + z.shape, dtype=np.complex128)
else:
qn = np.empty((n1 + 1,) + z.shape, dtype=np.float64)
qd = np.empty_like(qn)
if (z.ndim == 0):
_lqn(z, out=(qn, qd))
else:
# new axes must be last for the ufunc
_lqn(z,
out=(np.moveaxis(qn, 0, -1),
np.moveaxis(qd, 0, -1)))
return qn[:(n+1)], qd[:(n+1)]
def ai_zeros(nt):
"""
Compute `nt` zeros and values of the Airy function Ai and its derivative.
Computes the first `nt` zeros, `a`, of the Airy function Ai(x);
first `nt` zeros, `ap`, of the derivative of the Airy function Ai'(x);
the corresponding values Ai(a');
and the corresponding values Ai'(a).
Parameters
----------
nt : int
Number of zeros to compute
Returns
-------
a : ndarray
First `nt` zeros of Ai(x)
ap : ndarray
First `nt` zeros of Ai'(x)
ai : ndarray
Values of Ai(x) evaluated at first `nt` zeros of Ai'(x)
aip : ndarray
Values of Ai'(x) evaluated at first `nt` zeros of Ai(x)
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
>>> from scipy import special
>>> a, ap, ai, aip = special.ai_zeros(3)
>>> a
array([-2.33810741, -4.08794944, -5.52055983])
>>> ap
array([-1.01879297, -3.24819758, -4.82009921])
>>> ai
array([ 0.53565666, -0.41901548, 0.38040647])
>>> aip
array([ 0.70121082, -0.80311137, 0.86520403])
"""
kf = 1
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be a positive integer scalar.")
return _specfun.airyzo(nt, kf)
def bi_zeros(nt):
"""
Compute `nt` zeros and values of the Airy function Bi and its derivative.
Computes the first `nt` zeros, b, of the Airy function Bi(x);
first `nt` zeros, b', of the derivative of the Airy function Bi'(x);
the corresponding values Bi(b');
and the corresponding values Bi'(b).
Parameters
----------
nt : int
Number of zeros to compute
Returns
-------
b : ndarray
First `nt` zeros of Bi(x)
bp : ndarray
First `nt` zeros of Bi'(x)
bi : ndarray
Values of Bi(x) evaluated at first `nt` zeros of Bi'(x)
bip : ndarray
Values of Bi'(x) evaluated at first `nt` zeros of Bi(x)
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
>>> from scipy import special
>>> b, bp, bi, bip = special.bi_zeros(3)
>>> b
array([-1.17371322, -3.2710933 , -4.83073784])
>>> bp
array([-2.29443968, -4.07315509, -5.51239573])
>>> bi
array([-0.45494438, 0.39652284, -0.36796916])
>>> bip
array([ 0.60195789, -0.76031014, 0.83699101])
"""
kf = 2
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be a positive integer scalar.")
return _specfun.airyzo(nt, kf)
def lmbda(v, x):
r"""Jahnke-Emden Lambda function, Lambdav(x).
This function is defined as [2]_,
.. math:: \Lambda_v(x) = \Gamma(v+1) \frac{J_v(x)}{(x/2)^v},
where :math:`\Gamma` is the gamma function and :math:`J_v` is the
Bessel function of the first kind.
Parameters
----------
v : float
Order of the Lambda function
x : float
Value at which to evaluate the function and derivatives
Returns
-------
vl : ndarray
Values of Lambda_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
dl : ndarray
Derivatives Lambda_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] Jahnke, E. and Emde, F. "Tables of Functions with Formulae and
Curves" (4th ed.), Dover, 1945
"""
if not (isscalar(v) and isscalar(x)):
raise ValueError("arguments must be scalars.")
if (v < 0):
raise ValueError("argument must be > 0.")
n = int(v)
v0 = v - n
if (n < 1):
n1 = 1
else:
n1 = n
v1 = n1 + v0
if (v != floor(v)):
vm, vl, dl = _specfun.lamv(v1, x)
else:
vm, vl, dl = _specfun.lamn(v1, x)
return vl[:(n+1)], dl[:(n+1)]
def pbdv_seq(v, x):
"""Parabolic cylinder functions Dv(x) and derivatives.
Parameters
----------
v : float
Order of the parabolic cylinder function
x : float
Value at which to evaluate the function and derivatives
Returns
-------
dv : ndarray
Values of D_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
dp : ndarray
Derivatives D_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 13.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not (isscalar(v) and isscalar(x)):
raise ValueError("arguments must be scalars.")
n = int(v)
v0 = v-n
if (n < 1):
n1 = 1
else:
n1 = n
v1 = n1 + v0
dv, dp, pdf, pdd = _specfun.pbdv(v1, x)
return dv[:n1+1], dp[:n1+1]
def pbvv_seq(v, x):
"""Parabolic cylinder functions Vv(x) and derivatives.
Parameters
----------
v : float
Order of the parabolic cylinder function
x : float
Value at which to evaluate the function and derivatives
Returns
-------
dv : ndarray
Values of V_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
dp : ndarray
Derivatives V_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 13.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not (isscalar(v) and isscalar(x)):
raise ValueError("arguments must be scalars.")
n = int(v)
v0 = v-n
if (n <= 1):
n1 = 1
else:
n1 = n
v1 = n1 + v0
dv, dp, pdf, pdd = _specfun.pbvv(v1, x)
return dv[:n1+1], dp[:n1+1]
def pbdn_seq(n, z):
"""Parabolic cylinder functions Dn(z) and derivatives.
Parameters
----------
n : int
Order of the parabolic cylinder function
z : complex
Value at which to evaluate the function and derivatives
Returns
-------
dv : ndarray
Values of D_i(z), for i=0, ..., i=n.
dp : ndarray
Derivatives D_i'(z), for i=0, ..., i=n.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 13.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not (isscalar(n) and isscalar(z)):
raise ValueError("arguments must be scalars.")
if (floor(n) != n):
raise ValueError("n must be an integer.")
if (abs(n) <= 1):
n1 = 1
else:
n1 = n
cpb, cpd = _specfun.cpbdn(n1, z)
return cpb[:n1+1], cpd[:n1+1]
def ber_zeros(nt):
"""Compute nt zeros of the Kelvin function ber.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the Kelvin function.
See Also
--------
ber
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be positive integer scalar.")
return _specfun.klvnzo(nt, 1)
def bei_zeros(nt):
"""Compute nt zeros of the Kelvin function bei.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the Kelvin function.
See Also
--------
bei
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be positive integer scalar.")
return _specfun.klvnzo(nt, 2)
def ker_zeros(nt):
"""Compute nt zeros of the Kelvin function ker.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the Kelvin function.
See Also
--------
ker
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be positive integer scalar.")
return _specfun.klvnzo(nt, 3)
def kei_zeros(nt):
"""Compute nt zeros of the Kelvin function kei.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the Kelvin function.
See Also
--------
kei
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be positive integer scalar.")
return _specfun.klvnzo(nt, 4)
def berp_zeros(nt):
"""Compute nt zeros of the derivative of the Kelvin function ber.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the derivative of the Kelvin function.
See Also
--------
ber, berp
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first 5 zeros of the derivative of the Kelvin function.
>>> from scipy.special import berp_zeros
>>> berp_zeros(5)
array([ 6.03871081, 10.51364251, 14.96844542, 19.41757493, 23.86430432])
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be positive integer scalar.")
return _specfun.klvnzo(nt, 5)
def beip_zeros(nt):
"""Compute nt zeros of the derivative of the Kelvin function bei.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the derivative of the Kelvin function.
See Also
--------
bei, beip
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be positive integer scalar.")
return _specfun.klvnzo(nt, 6)
def kerp_zeros(nt):
"""Compute nt zeros of the derivative of the Kelvin function ker.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the derivative of the Kelvin function.
See Also
--------
ker, kerp
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be positive integer scalar.")
return _specfun.klvnzo(nt, 7)
def keip_zeros(nt):
"""Compute nt zeros of the derivative of the Kelvin function kei.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the derivative of the Kelvin function.
See Also
--------
kei, keip
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be positive integer scalar.")
return _specfun.klvnzo(nt, 8)
def kelvin_zeros(nt):
"""Compute nt zeros of all Kelvin functions.
Returned in a length-8 tuple of arrays of length nt. The tuple contains
the arrays of zeros of (ber, bei, ker, kei, ber', bei', ker', kei').
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
raise ValueError("nt must be positive integer scalar.")
return (_specfun.klvnzo(nt, 1),
_specfun.klvnzo(nt, 2),
_specfun.klvnzo(nt, 3),
_specfun.klvnzo(nt, 4),
_specfun.klvnzo(nt, 5),
_specfun.klvnzo(nt, 6),
_specfun.klvnzo(nt, 7),
_specfun.klvnzo(nt, 8))
def pro_cv_seq(m, n, c):
"""Characteristic values for prolate spheroidal wave functions.
Compute a sequence of characteristic values for the prolate
spheroidal wave functions for mode m and n'=m..n and spheroidal
parameter c.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not (isscalar(m) and isscalar(n) and isscalar(c)):
raise ValueError("Arguments must be scalars.")
if (n != floor(n)) or (m != floor(m)):
raise ValueError("Modes must be integers.")
if (n-m > 199):
raise ValueError("Difference between n and m is too large.")
maxL = n-m+1
return _specfun.segv(m, n, c, 1)[1][:maxL]
def obl_cv_seq(m, n, c):
"""Characteristic values for oblate spheroidal wave functions.
Compute a sequence of characteristic values for the oblate
spheroidal wave functions for mode m and n'=m..n and spheroidal
parameter c.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
"""
if not (isscalar(m) and isscalar(n) and isscalar(c)):
raise ValueError("Arguments must be scalars.")
if (n != floor(n)) or (m != floor(m)):
raise ValueError("Modes must be integers.")
if (n-m > 199):
raise ValueError("Difference between n and m is too large.")
maxL = n-m+1
return _specfun.segv(m, n, c, -1)[1][:maxL]
def comb(N, k, *, exact=False, repetition=False):
"""The number of combinations of N things taken k at a time.
This is often expressed as "N choose k".
Parameters
----------
N : int, ndarray
Number of things.
k : int, ndarray
Number of elements taken.
exact : bool, optional
For integers, if `exact` is False, then floating point precision is
used, otherwise the result is computed exactly.
repetition : bool, optional
If `repetition` is True, then the number of combinations with
repetition is computed.
Returns
-------
val : int, float, ndarray
The total number of combinations.
See Also
--------
binom : Binomial coefficient considered as a function of two real
variables.
Notes
-----
- Array arguments accepted only for exact=False case.
- If N < 0, or k < 0, then 0 is returned.
- If k > N and repetition=False, then 0 is returned.
Examples
--------
>>> import numpy as np
>>> from scipy.special import comb
>>> k = np.array([3, 4])
>>> n = np.array([10, 10])
>>> comb(n, k, exact=False)
array([ 120., 210.])
>>> comb(10, 3, exact=True)
120
>>> comb(10, 3, exact=True, repetition=True)
220
"""
if repetition:
return comb(N + k - 1, k, exact=exact)
if exact:
if int(N) == N and int(k) == k:
# _comb_int casts inputs to integers, which is safe & intended here
return _comb_int(N, k)
else:
raise ValueError("Non-integer `N` and `k` with `exact=True` is not "
"supported.")
else:
k, N = asarray(k), asarray(N)
cond = (k <= N) & (N >= 0) & (k >= 0)
vals = binom(N, k)
if isinstance(vals, np.ndarray):
vals[~cond] = 0
elif not cond:
vals = np.float64(0)
return vals
def perm(N, k, exact=False):
"""Permutations of N things taken k at a time, i.e., k-permutations of N.
It's also known as "partial permutations".
Parameters
----------
N : int, ndarray
Number of things.
k : int, ndarray
Number of elements taken.
exact : bool, optional
If ``True``, calculate the answer exactly using long integer arithmetic (`N`
and `k` must be scalar integers). If ``False``, a floating point approximation
is calculated (more rapidly) using `poch`. Default is ``False``.
Returns
-------
val : int, ndarray
The number of k-permutations of N.
Notes
-----
- Array arguments accepted only for exact=False case.
- If k > N, N < 0, or k < 0, then a 0 is returned.
Examples
--------
>>> import numpy as np
>>> from scipy.special import perm
>>> k = np.array([3, 4])
>>> n = np.array([10, 10])
>>> perm(n, k)
array([ 720., 5040.])
>>> perm(10, 3, exact=True)
720
"""
if exact:
N = np.squeeze(N)[()] # for backward compatibility (accepted size 1 arrays)
k = np.squeeze(k)[()]
if not (isscalar(N) and isscalar(k)):
raise ValueError("`N` and `k` must be scalar integers with `exact=True`.")
floor_N, floor_k = int(N), int(k)
non_integral = not (floor_N == N and floor_k == k)
if non_integral:
raise ValueError("Non-integer `N` and `k` with `exact=True` is not "
"supported.")
if (k > N) or (N < 0) or (k < 0):
return 0
val = 1
for i in range(floor_N - floor_k + 1, floor_N + 1):
val *= i
return val
else:
k, N = asarray(k), asarray(N)
cond = (k <= N) & (N >= 0) & (k >= 0)
vals = poch(N - k + 1, k)
if isinstance(vals, np.ndarray):
vals[~cond] = 0
elif not cond:
vals = np.float64(0)
return vals
# https://stackoverflow.com/a/16327037
def _range_prod(lo, hi, k=1):
"""
Product of a range of numbers spaced k apart (from hi).
For k=1, this returns the product of
lo * (lo+1) * (lo+2) * ... * (hi-2) * (hi-1) * hi
= hi! / (lo-1)!
For k>1, it correspond to taking only every k'th number when
counting down from hi - e.g. 18!!!! = _range_prod(1, 18, 4).
Breaks into smaller products first for speed:
_range_prod(2, 9) = ((2*3)*(4*5))*((6*7)*(8*9))
"""
if lo == 1 and k == 1:
return math.factorial(hi)
if lo + k < hi:
mid = (hi + lo) // 2
if k > 1:
# make sure mid is a multiple of k away from hi
mid = mid - ((mid - hi) % k)
return _range_prod(lo, mid, k) * _range_prod(mid + k, hi, k)
elif lo + k == hi:
return lo * hi
else:
return hi
def _factorialx_array_exact(n, k=1):
"""
Exact computation of factorial for an array.
The factorials are computed in incremental fashion, by taking
the sorted unique values of n and multiplying the intervening
numbers between the different unique values.
In other words, the factorial for the largest input is only
computed once, with each other result computed in the process.
k > 1 corresponds to the multifactorial.
"""
un = np.unique(n)
# numpy changed nan-sorting behaviour with 1.21, see numpy/numpy#18070;
# to unify the behaviour, we remove the nan's here; the respective
# values will be set separately at the end
un = un[~np.isnan(un)]
# Convert to object array if np.int64 can't handle size
if np.isnan(n).any():
dt = float
elif k in _FACTORIALK_LIMITS_64BITS.keys():
if un[-1] > _FACTORIALK_LIMITS_64BITS[k]:
# e.g. k=1: 21! > np.iinfo(np.int64).max
dt = object
elif un[-1] > _FACTORIALK_LIMITS_32BITS[k]:
# e.g. k=3: 26!!! > np.iinfo(np.int32).max
dt = np.int64
else:
dt = np.dtype("long")
else:
# for k >= 10, we always use object
dt = object
out = np.empty_like(n, dtype=dt)
# Handle invalid/trivial values
un = un[un > 1]
out[n < 2] = 1
out[n < 0] = 0
# Calculate products of each range of numbers
# we can only multiply incrementally if the values are k apart;
# therefore we partition `un` into "lanes", i.e. its residues modulo k
for lane in range(0, k):
ul = un[(un % k) == lane] if k > 1 else un
if ul.size:
# after np.unique, un resp. ul are sorted, ul[0] is the smallest;
# cast to python ints to avoid overflow with np.int-types
val = _range_prod(1, int(ul[0]), k=k)
out[n == ul[0]] = val
for i in range(len(ul) - 1):
# by the filtering above, we have ensured that prev & current
# are a multiple of k apart
prev = ul[i]
current = ul[i + 1]
# we already multiplied all factors until prev; continue
# building the full factorial from the following (`prev + 1`);
# use int() for the same reason as above
val *= _range_prod(int(prev + 1), int(current), k=k)
out[n == current] = val
if np.isnan(n).any():
out = out.astype(np.float64)
out[np.isnan(n)] = np.nan
return out
def _factorialx_array_approx(n, k, extend):
"""
Calculate approximation to multifactorial for array n and integer k.
Ensure that values aren't calculated unnecessarily.
"""
if extend == "complex":
return _factorialx_approx_core(n, k=k, extend=extend)
# at this point we are guaranteed that extend='zero' and that k>0 is an integer
result = zeros(n.shape)
# keep nans as nans
place(result, np.isnan(n), np.nan)
# only compute where n >= 0 (excludes nans), everything else is 0
cond = (n >= 0)
n_to_compute = extract(cond, n)
place(result, cond, _factorialx_approx_core(n_to_compute, k=k, extend=extend))
return result
def _gamma1p(vals):
"""
returns gamma(n+1), though with NaN at -1 instead of inf, c.f. #21827
"""
res = gamma(vals + 1)
# replace infinities at -1 (from gamma function at 0) with nan
# gamma only returns inf for real inputs; can ignore complex case
if isinstance(res, np.ndarray):
if not _is_subdtype(vals.dtype, "c"):
res[vals == -1] = np.nan
elif np.isinf(res) and vals == -1:
res = np.float64("nan")
return res
def _factorialx_approx_core(n, k, extend):
"""
Core approximation to multifactorial for array n and integer k.
"""
if k == 1:
# shortcut for k=1; same for both extensions, because we assume the
# handling of extend == 'zero' happens in _factorialx_array_approx
result = _gamma1p(n)
if isinstance(n, np.ndarray):
# gamma does not maintain 0-dim arrays; fix it
result = np.array(result)
return result
if extend == "complex":
# see https://numpy.org/doc/stable/reference/generated/numpy.power.html
p_dtype = complex if (_is_subdtype(type(k), "c") or k < 0) else None
with warnings.catch_warnings():
# do not warn about 0 * inf, nan / nan etc.; the results are correct
warnings.simplefilter("ignore", RuntimeWarning)
# don't use `(n-1)/k` in np.power; underflows if 0 is of a uintX type
result = np.power(k, n / k, dtype=p_dtype) * _gamma1p(n / k)
result *= rgamma(1 / k + 1) / np.power(k, 1 / k, dtype=p_dtype)
if isinstance(n, np.ndarray):
# ensure we keep array-ness for 0-dim inputs; already n/k above loses it
result = np.array(result)
return result
# at this point we are guaranteed that extend='zero' and that k>0 is an integer
n_mod_k = n % k
# scalar case separately, unified handling would be inefficient for arrays;
# don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below
if not isinstance(n, np.ndarray):
with warnings.catch_warnings():
# large n cause overflow warnings, but infinity is fine
warnings.simplefilter("ignore", RuntimeWarning)
return (
np.power(k, (n - n_mod_k) / k)
* gamma(n / k + 1) / gamma(n_mod_k / k + 1)
* max(n_mod_k, 1)
)
# factor that's independent of the residue class (see factorialk docstring)
with warnings.catch_warnings():
# large n cause overflow warnings, but infinity is fine
warnings.simplefilter("ignore", RuntimeWarning)
result = np.power(k, n / k) * gamma(n / k + 1)
# factor dependent on residue r (for `r=0` it's 1, so we skip `r=0`
# below and thus also avoid evaluating `max(r, 1)`)
def corr(k, r): return np.power(k, -r / k) / gamma(r / k + 1) * r
for r in np.unique(n_mod_k):
if r == 0:
continue
# cast to int because uint types break on `-r`
result[n_mod_k == r] *= corr(k, int(r))
return result
def _is_subdtype(dtype, dtypes):
"""
Shorthand for calculating whether dtype is subtype of some dtypes.
Also allows specifying a list instead of just a single dtype.
Additionaly, the most important supertypes from
https://numpy.org/doc/stable/reference/arrays.scalars.html
can optionally be specified using abbreviations as follows:
"i": np.integer
"f": np.floating
"c": np.complexfloating
"n": np.number (contains the other three)
"""
dtypes = dtypes if isinstance(dtypes, list) else [dtypes]
# map single character abbreviations, if they are in dtypes
mapping = {
"i": np.integer,
"f": np.floating,
"c": np.complexfloating,
"n": np.number
}
dtypes = [mapping.get(x, x) for x in dtypes]
return any(np.issubdtype(dtype, dt) for dt in dtypes)
def _factorialx_wrapper(fname, n, k, exact, extend):
"""
Shared implementation for factorial, factorial2 & factorialk.
"""
if extend not in ("zero", "complex"):
raise ValueError(
f"argument `extend` must be either 'zero' or 'complex', received: {extend}"
)
if exact and extend == "complex":
raise ValueError("Incompatible options: `exact=True` and `extend='complex'`")
msg_unsup = (
"Unsupported data type for {vname} in {fname}: {dtype}\n"
)
if fname == "factorial":
msg_unsup += (
"Permitted data types are integers and floating point numbers, "
"as well as complex numbers if `extend='complex' is passed."
)
else:
msg_unsup += (
"Permitted data types are integers, as well as floating point "
"numbers and complex numbers if `extend='complex' is passed."
)
msg_exact_not_possible = (
"`exact=True` only supports integers, cannot use data type {dtype}"
)
msg_needs_complex = (
"In order to use non-integer arguments, you must opt into this by passing "
"`extend='complex'`. Note that this changes the result for all negative "
"arguments (which by default return 0)."
)
if fname == "factorial2":
msg_needs_complex += (" Additionally, it will rescale the values of the double"
" factorial at even integers by a factor of sqrt(2/pi).")
elif fname == "factorialk":
msg_needs_complex += (" Additionally, it will perturb the values of the"
" multifactorial at most positive integers `n`.")
# check type of k
if not _is_subdtype(type(k), ["i", "f", "c"]):
raise ValueError(msg_unsup.format(vname="`k`", fname=fname, dtype=type(k)))
elif _is_subdtype(type(k), ["f", "c"]) and extend != "complex":
raise ValueError(msg_needs_complex)
# check value of k
if extend == "zero" and k < 1:
msg = f"For `extend='zero'`, k must be a positive integer, received: {k}"
raise ValueError(msg)
elif k == 0:
raise ValueError("Parameter k cannot be zero!")
# factorial allows floats also for extend="zero"
types_requiring_complex = "c" if fname == "factorial" else ["f", "c"]
# don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below
if np.ndim(n) == 0 and not isinstance(n, np.ndarray):
# scalar cases
if not _is_subdtype(type(n), ["i", "f", "c", type(None)]):
raise ValueError(msg_unsup.format(vname="`n`", fname=fname, dtype=type(n)))
elif _is_subdtype(type(n), types_requiring_complex) and extend != "complex":
raise ValueError(msg_needs_complex)
elif n is None or np.isnan(n):
complexify = (extend == "complex") and _is_subdtype(type(n), "c")
return np.complex128("nan+nanj") if complexify else np.float64("nan")
elif extend == "zero" and n < 0:
return 0 if exact else np.float64(0)
elif n in {0, 1}:
return 1 if exact else np.float64(1)
elif exact and _is_subdtype(type(n), "i"):
# calculate with integers; cast away other int types (like unsigned)
return _range_prod(1, int(n), k=k)
elif exact:
# only relevant for factorial
raise ValueError(msg_exact_not_possible.format(dtype=type(n)))
# approximation
return _factorialx_approx_core(n, k=k, extend=extend)
# arrays & array-likes
n = asarray(n)
if not _is_subdtype(n.dtype, ["i", "f", "c"]):
raise ValueError(msg_unsup.format(vname="`n`", fname=fname, dtype=n.dtype))
elif _is_subdtype(n.dtype, types_requiring_complex) and extend != "complex":
raise ValueError(msg_needs_complex)
elif exact and _is_subdtype(n.dtype, ["f"]):
# only relevant for factorial
raise ValueError(msg_exact_not_possible.format(dtype=n.dtype))
if n.size == 0:
# return empty arrays unchanged
return n
elif exact:
# calculate with integers
return _factorialx_array_exact(n, k=k)
# approximation
return _factorialx_array_approx(n, k=k, extend=extend)
def factorial(n, exact=False, extend="zero"):
"""
The factorial of a number or array of numbers.
The factorial of non-negative integer `n` is the product of all
positive integers less than or equal to `n`::
n! = n * (n - 1) * (n - 2) * ... * 1
Parameters
----------
n : int or float or complex (or array_like thereof)
Input values for ``n!``. Complex values require ``extend='complex'``.
By default, the return value for ``n < 0`` is 0.
exact : bool, optional
If ``exact`` is set to True, calculate the answer exactly using
integer arithmetic, otherwise approximate using the gamma function
(faster, but yields floats instead of integers).
Default is False.
extend : string, optional
One of ``'zero'`` or ``'complex'``; this determines how values ``n<0``
are handled - by default they are 0, but it is possible to opt into the
complex extension of the factorial (see below).
Returns
-------
nf : int or float or complex or ndarray
Factorial of ``n``, as integer, float or complex (depending on ``exact``
and ``extend``). Array inputs are returned as arrays.
Notes
-----
For arrays with ``exact=True``, the factorial is computed only once, for
the largest input, with each other result computed in the process.
The output dtype is increased to ``int64`` or ``object`` if necessary.
With ``exact=False`` the factorial is approximated using the gamma
function (which is also the definition of the complex extension):
.. math:: n! = \\Gamma(n+1)
Examples
--------
>>> import numpy as np
>>> from scipy.special import factorial
>>> arr = np.array([3, 4, 5])
>>> factorial(arr, exact=False)
array([ 6., 24., 120.])
>>> factorial(arr, exact=True)
array([ 6, 24, 120])
>>> factorial(5, exact=True)
120
"""
return _factorialx_wrapper("factorial", n, k=1, exact=exact, extend=extend)
def factorial2(n, exact=False, extend="zero"):
"""Double factorial.
This is the factorial with every second value skipped. E.g., ``7!! = 7 * 5
* 3 * 1``. It can be approximated numerically as::
n!! = 2 ** (n / 2) * gamma(n / 2 + 1) * sqrt(2 / pi) n odd
= 2 ** (n / 2) * gamma(n / 2 + 1) n even
= 2 ** (n / 2) * (n / 2)! n even
The formula for odd ``n`` is the basis for the complex extension.
Parameters
----------
n : int or float or complex (or array_like thereof)
Input values for ``n!!``. Non-integer values require ``extend='complex'``.
By default, the return value for ``n < 0`` is 0.
exact : bool, optional
If ``exact`` is set to True, calculate the answer exactly using
integer arithmetic, otherwise use above approximation (faster,
but yields floats instead of integers).
Default is False.
extend : string, optional
One of ``'zero'`` or ``'complex'``; this determines how values ``n<0``
are handled - by default they are 0, but it is possible to opt into the
complex extension of the double factorial. This also enables passing
complex values to ``n``.
.. warning::
Using the ``'complex'`` extension also changes the values of the
double factorial for even integers, reducing them by a factor of
``sqrt(2/pi) ~= 0.79``, see [1].
Returns
-------
nf : int or float or complex or ndarray
Double factorial of ``n``, as integer, float or complex (depending on
``exact`` and ``extend``). Array inputs are returned as arrays.
Examples
--------
>>> from scipy.special import factorial2
>>> factorial2(7, exact=False)
array(105.00000000000001)
>>> factorial2(7, exact=True)
105
References
----------
.. [1] Complex extension to double factorial
https://en.wikipedia.org/wiki/Double_factorial#Complex_arguments
"""
return _factorialx_wrapper("factorial2", n, k=2, exact=exact, extend=extend)
def factorialk(n, k, exact=False, extend="zero"):
"""Multifactorial of n of order k, n(!!...!).
This is the multifactorial of n skipping k values. For example,
factorialk(17, 4) = 17!!!! = 17 * 13 * 9 * 5 * 1
In particular, for any integer ``n``, we have
factorialk(n, 1) = factorial(n)
factorialk(n, 2) = factorial2(n)
Parameters
----------
n : int or float or complex (or array_like thereof)
Input values for multifactorial. Non-integer values require
``extend='complex'``. By default, the return value for ``n < 0`` is 0.
n : int or float or complex (or array_like thereof)
Order of multifactorial. Non-integer values require ``extend='complex'``.
exact : bool, optional
If ``exact`` is set to True, calculate the answer exactly using
integer arithmetic, otherwise use an approximation (faster,
but yields floats instead of integers)
Default is False.
extend : string, optional
One of ``'zero'`` or ``'complex'``; this determines how values ``n<0`` are
handled - by default they are 0, but it is possible to opt into the complex
extension of the multifactorial. This enables passing complex values,
not only to ``n`` but also to ``k``.
.. warning::
Using the ``'complex'`` extension also changes the values of the
multifactorial at integers ``n != 1 (mod k)`` by a factor depending
on both ``k`` and ``n % k``, see below or [1].
Returns
-------
nf : int or float or complex or ndarray
Multifactorial (order ``k``) of ``n``, as integer, float or complex (depending
on ``exact`` and ``extend``). Array inputs are returned as arrays.
Examples
--------
>>> from scipy.special import factorialk
>>> factorialk(5, k=1, exact=True)
120
>>> factorialk(5, k=3, exact=True)
10
>>> factorialk([5, 7, 9], k=3, exact=True)
array([ 10, 28, 162])
>>> factorialk([5, 7, 9], k=3, exact=False)
array([ 10., 28., 162.])
Notes
-----
While less straight-forward than for the double-factorial, it's possible to
calculate a general approximation formula of n!(k) by studying ``n`` for a given
remainder ``r < k`` (thus ``n = m * k + r``, resp. ``r = n % k``), which can be
put together into something valid for all integer values ``n >= 0`` & ``k > 0``::
n!(k) = k ** ((n - r)/k) * gamma(n/k + 1) / gamma(r/k + 1) * max(r, 1)
This is the basis of the approximation when ``exact=False``.
In principle, any fixed choice of ``r`` (ignoring its relation ``r = n%k``
to ``n``) would provide a suitable analytic continuation from integer ``n``
to complex ``z`` (not only satisfying the functional equation but also
being logarithmically convex, c.f. Bohr-Mollerup theorem) -- in fact, the
choice of ``r`` above only changes the function by a constant factor. The
final constraint that determines the canonical continuation is ``f(1) = 1``,
which forces ``r = 1`` (see also [1]).::
z!(k) = k ** ((z - 1)/k) * gamma(z/k + 1) / gamma(1/k + 1)
References
----------
.. [1] Complex extension to multifactorial
https://en.wikipedia.org/wiki/Double_factorial#Alternative_extension_of_the_multifactorial
"""
return _factorialx_wrapper("factorialk", n, k=k, exact=exact, extend=extend)
def stirling2(N, K, *, exact=False):
r"""Generate Stirling number(s) of the second kind.
Stirling numbers of the second kind count the number of ways to
partition a set with N elements into K non-empty subsets.
The values this function returns are calculated using a dynamic
program which avoids redundant computation across the subproblems
in the solution. For array-like input, this implementation also
avoids redundant computation across the different Stirling number
calculations.
The numbers are sometimes denoted
.. math::
{N \brace{K}}
see [1]_ for details. This is often expressed-verbally-as
"N subset K".
Parameters
----------
N : int, ndarray
Number of things.
K : int, ndarray
Number of non-empty subsets taken.
exact : bool, optional
Uses dynamic programming (DP) with floating point
numbers for smaller arrays and uses a second order approximation due to
Temme for larger entries of `N` and `K` that allows trading speed for
accuracy. See [2]_ for a description. Temme approximation is used for
values ``n>50``. The max error from the DP has max relative error
``4.5*10^-16`` for ``n<=50`` and the max error from the Temme approximation
has max relative error ``5*10^-5`` for ``51 <= n < 70`` and
``9*10^-6`` for ``70 <= n < 101``. Note that these max relative errors will
decrease further as `n` increases.
Returns
-------
val : int, float, ndarray
The number of partitions.
See Also
--------
comb : The number of combinations of N things taken k at a time.
Notes
-----
- If N < 0, or K < 0, then 0 is returned.
- If K > N, then 0 is returned.
The output type will always be `int` or ndarray of `object`.
The input must contain either numpy or python integers otherwise a
TypeError is raised.
References
----------
.. [1] R. L. Graham, D. E. Knuth and O. Patashnik, "Concrete
Mathematics: A Foundation for Computer Science," Addison-Wesley
Publishing Company, Boston, 1989. Chapter 6, page 258.
.. [2] Temme, Nico M. "Asymptotic estimates of Stirling numbers."
Studies in Applied Mathematics 89.3 (1993): 233-243.
Examples
--------
>>> import numpy as np
>>> from scipy.special import stirling2
>>> k = np.array([3, -1, 3])
>>> n = np.array([10, 10, 9])
>>> stirling2(n, k)
array([9330.0, 0.0, 3025.0])
"""
output_is_scalar = np.isscalar(N) and np.isscalar(K)
# make a min-heap of unique (n,k) pairs
N, K = asarray(N), asarray(K)
if not np.issubdtype(N.dtype, np.integer):
raise TypeError("Argument `N` must contain only integers")
if not np.issubdtype(K.dtype, np.integer):
raise TypeError("Argument `K` must contain only integers")
if not exact:
# NOTE: here we allow np.uint via casting to double types prior to
# passing to private ufunc dispatcher. All dispatched functions
# take double type for (n,k) arguments and return double.
return _stirling2_inexact(N.astype(float), K.astype(float))
nk_pairs = list(
set([(n.take(0), k.take(0))
for n, k in np.nditer([N, K], ['refs_ok'])])
)
heapify(nk_pairs)
# base mapping for small values
snsk_vals = defaultdict(int)
for pair in [(0, 0), (1, 1), (2, 1), (2, 2)]:
snsk_vals[pair] = 1
# for each pair in the min-heap, calculate the value, store for later
n_old, n_row = 2, [0, 1, 1]
while nk_pairs:
n, k = heappop(nk_pairs)
if n < 2 or k > n or k <= 0:
continue
elif k == n or k == 1:
snsk_vals[(n, k)] = 1
continue
elif n != n_old:
num_iters = n - n_old
while num_iters > 0:
n_row.append(1)
# traverse from back to remove second row
for j in range(len(n_row)-2, 1, -1):
n_row[j] = n_row[j]*j + n_row[j-1]
num_iters -= 1
snsk_vals[(n, k)] = n_row[k]
else:
snsk_vals[(n, k)] = n_row[k]
n_old, n_row = n, n_row
out_types = [object, object, object] if exact else [float, float, float]
# for each pair in the map, fetch the value, and populate the array
it = np.nditer(
[N, K, None],
['buffered', 'refs_ok'],
[['readonly'], ['readonly'], ['writeonly', 'allocate']],
op_dtypes=out_types,
)
with it:
while not it.finished:
it[2] = snsk_vals[(int(it[0]), int(it[1]))]
it.iternext()
output = it.operands[2]
# If N and K were both scalars, convert output to scalar.
if output_is_scalar:
output = output.take(0)
return output
def zeta(x, q=None, out=None):
r"""
Riemann or Hurwitz zeta function.
Parameters
----------
x : array_like of float or complex.
Input data
q : array_like of float, optional
Input data, must be real. Defaults to Riemann zeta. When `q` is
``None``, complex inputs `x` are supported. If `q` is not ``None``,
then currently only real inputs `x` with ``x >= 1`` are supported,
even when ``q = 1.0`` (corresponding to the Riemann zeta function).
out : ndarray, optional
Output array for the computed values.
Returns
-------
out : array_like
Values of zeta(x).
See Also
--------
zetac
Notes
-----
The two-argument version is the Hurwitz zeta function
.. math::
\zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x};
see [dlmf]_ for details. The Riemann zeta function corresponds to
the case when ``q = 1``.
For complex inputs with ``q = None``, points with
``abs(z.imag) > 1e9`` and ``0 <= abs(z.real) < 2.5`` are currently not
supported due to slow convergence causing excessive runtime.
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/25.11#i
Examples
--------
>>> import numpy as np
>>> from scipy.special import zeta, polygamma, factorial
Some specific values:
>>> zeta(2), np.pi**2/6
(1.6449340668482266, 1.6449340668482264)
>>> zeta(4), np.pi**4/90
(1.0823232337111381, 1.082323233711138)
First nontrivial zero:
>>> zeta(0.5 + 14.134725141734695j)
0 + 0j
Relation to the `polygamma` function:
>>> m = 3
>>> x = 1.25
>>> polygamma(m, x)
array(2.782144009188397)
>>> (-1)**(m+1) * factorial(m) * zeta(m+1, x)
2.7821440091883969
"""
if q is None:
return _ufuncs._riemann_zeta(x, out)
else:
return _ufuncs._zeta(x, q, out)
def softplus(x, **kwargs):
r"""
Compute the softplus function element-wise.
The softplus function is defined as: ``softplus(x) = log(1 + exp(x))``.
It is a smooth approximation of the rectifier function (ReLU).
Parameters
----------
x : array_like
Input value.
**kwargs
For other keyword-only arguments, see the
`ufunc docs <https://numpy.org/doc/stable/reference/ufuncs.html>`_.
Returns
-------
softplus : ndarray
Logarithm of ``exp(0) + exp(x)``.
Examples
--------
>>> from scipy import special
>>> special.softplus(0)
0.6931471805599453
>>> special.softplus([-1, 0, 1])
array([0.31326169, 0.69314718, 1.31326169])
"""
return np.logaddexp(0, x, **kwargs)
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