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ai-content-maker/.venv/Lib/site-packages/torch/special/__init__.py

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2024-05-03 04:18:51 +03:00
import torch
from torch._C import _add_docstr, _special # type: ignore[attr-defined]
from torch._torch_docs import common_args, multi_dim_common
__all__ = [
'airy_ai',
'bessel_j0',
'bessel_j1',
'bessel_y0',
'bessel_y1',
'chebyshev_polynomial_t',
'chebyshev_polynomial_u',
'chebyshev_polynomial_v',
'chebyshev_polynomial_w',
'digamma',
'entr',
'erf',
'erfc',
'erfcx',
'erfinv',
'exp2',
'expit',
'expm1',
'gammainc',
'gammaincc',
'gammaln',
'hermite_polynomial_h',
'hermite_polynomial_he',
'i0',
'i0e',
'i1',
'i1e',
'laguerre_polynomial_l',
'legendre_polynomial_p',
'log1p',
'log_ndtr',
'log_softmax',
'logit',
'logsumexp',
'modified_bessel_i0',
'modified_bessel_i1',
'modified_bessel_k0',
'modified_bessel_k1',
'multigammaln',
'ndtr',
'ndtri',
'polygamma',
'psi',
'round',
'shifted_chebyshev_polynomial_t',
'shifted_chebyshev_polynomial_u',
'shifted_chebyshev_polynomial_v',
'shifted_chebyshev_polynomial_w',
'scaled_modified_bessel_k0',
'scaled_modified_bessel_k1',
'sinc',
'softmax',
'spherical_bessel_j0',
'xlog1py',
'xlogy',
'zeta',
]
Tensor = torch.Tensor
entr = _add_docstr(_special.special_entr,
r"""
entr(input, *, out=None) -> Tensor
Computes the entropy on :attr:`input` (as defined below), elementwise.
.. math::
\begin{align}
\text{entr(x)} = \begin{cases}
-x * \ln(x) & x > 0 \\
0 & x = 0.0 \\
-\infty & x < 0
\end{cases}
\end{align}
""" + """
Args:
input (Tensor): the input tensor.
Keyword args:
out (Tensor, optional): the output tensor.
Example::
>>> a = torch.arange(-0.5, 1, 0.5)
>>> a
tensor([-0.5000, 0.0000, 0.5000])
>>> torch.special.entr(a)
tensor([ -inf, 0.0000, 0.3466])
""")
psi = _add_docstr(_special.special_psi,
r"""
psi(input, *, out=None) -> Tensor
Alias for :func:`torch.special.digamma`.
""")
digamma = _add_docstr(_special.special_digamma,
r"""
digamma(input, *, out=None) -> Tensor
Computes the logarithmic derivative of the gamma function on `input`.
.. math::
\digamma(x) = \frac{d}{dx} \ln\left(\Gamma\left(x\right)\right) = \frac{\Gamma'(x)}{\Gamma(x)}
""" + r"""
Args:
input (Tensor): the tensor to compute the digamma function on
Keyword args:
{out}
.. note:: This function is similar to SciPy's `scipy.special.digamma`.
.. note:: From PyTorch 1.8 onwards, the digamma function returns `-Inf` for `0`.
Previously it returned `NaN` for `0`.
Example::
>>> a = torch.tensor([1, 0.5])
>>> torch.special.digamma(a)
tensor([-0.5772, -1.9635])
""".format(**common_args))
gammaln = _add_docstr(_special.special_gammaln,
r"""
gammaln(input, *, out=None) -> Tensor
Computes the natural logarithm of the absolute value of the gamma function on :attr:`input`.
.. math::
\text{out}_{i} = \ln \Gamma(|\text{input}_{i}|)
""" + """
Args:
{input}
Keyword args:
{out}
Example::
>>> a = torch.arange(0.5, 2, 0.5)
>>> torch.special.gammaln(a)
tensor([ 0.5724, 0.0000, -0.1208])
""".format(**common_args))
polygamma = _add_docstr(_special.special_polygamma,
r"""
polygamma(n, input, *, out=None) -> Tensor
Computes the :math:`n^{th}` derivative of the digamma function on :attr:`input`.
:math:`n \geq 0` is called the order of the polygamma function.
.. math::
\psi^{(n)}(x) = \frac{d^{(n)}}{dx^{(n)}} \psi(x)
.. note::
This function is implemented only for nonnegative integers :math:`n \geq 0`.
""" + """
Args:
n (int): the order of the polygamma function
{input}
Keyword args:
{out}
Example::
>>> a = torch.tensor([1, 0.5])
>>> torch.special.polygamma(1, a)
tensor([1.64493, 4.9348])
>>> torch.special.polygamma(2, a)
tensor([ -2.4041, -16.8288])
>>> torch.special.polygamma(3, a)
tensor([ 6.4939, 97.4091])
>>> torch.special.polygamma(4, a)
tensor([ -24.8863, -771.4742])
""".format(**common_args))
erf = _add_docstr(_special.special_erf,
r"""
erf(input, *, out=None) -> Tensor
Computes the error function of :attr:`input`. The error function is defined as follows:
.. math::
\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.erf(torch.tensor([0, -1., 10.]))
tensor([ 0.0000, -0.8427, 1.0000])
""".format(**common_args))
erfc = _add_docstr(_special.special_erfc,
r"""
erfc(input, *, out=None) -> Tensor
Computes the complementary error function of :attr:`input`.
The complementary error function is defined as follows:
.. math::
\mathrm{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.erfc(torch.tensor([0, -1., 10.]))
tensor([ 1.0000, 1.8427, 0.0000])
""".format(**common_args))
erfcx = _add_docstr(_special.special_erfcx,
r"""
erfcx(input, *, out=None) -> Tensor
Computes the scaled complementary error function for each element of :attr:`input`.
The scaled complementary error function is defined as follows:
.. math::
\mathrm{erfcx}(x) = e^{x^2} \mathrm{erfc}(x)
""" + r"""
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.erfcx(torch.tensor([0, -1., 10.]))
tensor([ 1.0000, 5.0090, 0.0561])
""".format(**common_args))
erfinv = _add_docstr(_special.special_erfinv,
r"""
erfinv(input, *, out=None) -> Tensor
Computes the inverse error function of :attr:`input`.
The inverse error function is defined in the range :math:`(-1, 1)` as:
.. math::
\mathrm{erfinv}(\mathrm{erf}(x)) = x
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.erfinv(torch.tensor([0, 0.5, -1.]))
tensor([ 0.0000, 0.4769, -inf])
""".format(**common_args))
logit = _add_docstr(_special.special_logit,
r"""
logit(input, eps=None, *, out=None) -> Tensor
Returns a new tensor with the logit of the elements of :attr:`input`.
:attr:`input` is clamped to [eps, 1 - eps] when eps is not None.
When eps is None and :attr:`input` < 0 or :attr:`input` > 1, the function will yields NaN.
.. math::
\begin{align}
y_{i} &= \ln(\frac{z_{i}}{1 - z_{i}}) \\
z_{i} &= \begin{cases}
x_{i} & \text{if eps is None} \\
\text{eps} & \text{if } x_{i} < \text{eps} \\
x_{i} & \text{if } \text{eps} \leq x_{i} \leq 1 - \text{eps} \\
1 - \text{eps} & \text{if } x_{i} > 1 - \text{eps}
\end{cases}
\end{align}
""" + r"""
Args:
{input}
eps (float, optional): the epsilon for input clamp bound. Default: ``None``
Keyword args:
{out}
Example::
>>> a = torch.rand(5)
>>> a
tensor([0.2796, 0.9331, 0.6486, 0.1523, 0.6516])
>>> torch.special.logit(a, eps=1e-6)
tensor([-0.9466, 2.6352, 0.6131, -1.7169, 0.6261])
""".format(**common_args))
logsumexp = _add_docstr(_special.special_logsumexp,
r"""
logsumexp(input, dim, keepdim=False, *, out=None)
Alias for :func:`torch.logsumexp`.
""".format(**multi_dim_common))
expit = _add_docstr(_special.special_expit,
r"""
expit(input, *, out=None) -> Tensor
Computes the expit (also known as the logistic sigmoid function) of the elements of :attr:`input`.
.. math::
\text{out}_{i} = \frac{1}{1 + e^{-\text{input}_{i}}}
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> t = torch.randn(4)
>>> t
tensor([ 0.9213, 1.0887, -0.8858, -1.7683])
>>> torch.special.expit(t)
tensor([ 0.7153, 0.7481, 0.2920, 0.1458])
""".format(**common_args))
exp2 = _add_docstr(_special.special_exp2,
r"""
exp2(input, *, out=None) -> Tensor
Computes the base two exponential function of :attr:`input`.
.. math::
y_{i} = 2^{x_{i}}
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.exp2(torch.tensor([0, math.log2(2.), 3, 4]))
tensor([ 1., 2., 8., 16.])
""".format(**common_args))
expm1 = _add_docstr(_special.special_expm1,
r"""
expm1(input, *, out=None) -> Tensor
Computes the exponential of the elements minus 1
of :attr:`input`.
.. math::
y_{i} = e^{x_{i}} - 1
.. note:: This function provides greater precision than exp(x) - 1 for small values of x.
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.expm1(torch.tensor([0, math.log(2.)]))
tensor([ 0., 1.])
""".format(**common_args))
xlog1py = _add_docstr(_special.special_xlog1py,
r"""
xlog1py(input, other, *, out=None) -> Tensor
Computes ``input * log1p(other)`` with the following cases.
.. math::
\text{out}_{i} = \begin{cases}
\text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\
0 & \text{if } \text{input}_{i} = 0.0 \text{ and } \text{other}_{i} != \text{NaN} \\
\text{input}_{i} * \text{log1p}(\text{other}_{i})& \text{otherwise}
\end{cases}
Similar to SciPy's `scipy.special.xlog1py`.
""" + r"""
Args:
input (Number or Tensor) : Multiplier
other (Number or Tensor) : Argument
.. note:: At least one of :attr:`input` or :attr:`other` must be a tensor.
Keyword args:
{out}
Example::
>>> x = torch.zeros(5,)
>>> y = torch.tensor([-1, 0, 1, float('inf'), float('nan')])
>>> torch.special.xlog1py(x, y)
tensor([0., 0., 0., 0., nan])
>>> x = torch.tensor([1, 2, 3])
>>> y = torch.tensor([3, 2, 1])
>>> torch.special.xlog1py(x, y)
tensor([1.3863, 2.1972, 2.0794])
>>> torch.special.xlog1py(x, 4)
tensor([1.6094, 3.2189, 4.8283])
>>> torch.special.xlog1py(2, y)
tensor([2.7726, 2.1972, 1.3863])
""".format(**common_args))
xlogy = _add_docstr(_special.special_xlogy,
r"""
xlogy(input, other, *, out=None) -> Tensor
Computes ``input * log(other)`` with the following cases.
.. math::
\text{out}_{i} = \begin{cases}
\text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\
0 & \text{if } \text{input}_{i} = 0.0 \\
\text{input}_{i} * \log{(\text{other}_{i})} & \text{otherwise}
\end{cases}
Similar to SciPy's `scipy.special.xlogy`.
""" + r"""
Args:
input (Number or Tensor) : Multiplier
other (Number or Tensor) : Argument
.. note:: At least one of :attr:`input` or :attr:`other` must be a tensor.
Keyword args:
{out}
Example::
>>> x = torch.zeros(5,)
>>> y = torch.tensor([-1, 0, 1, float('inf'), float('nan')])
>>> torch.special.xlogy(x, y)
tensor([0., 0., 0., 0., nan])
>>> x = torch.tensor([1, 2, 3])
>>> y = torch.tensor([3, 2, 1])
>>> torch.special.xlogy(x, y)
tensor([1.0986, 1.3863, 0.0000])
>>> torch.special.xlogy(x, 4)
tensor([1.3863, 2.7726, 4.1589])
>>> torch.special.xlogy(2, y)
tensor([2.1972, 1.3863, 0.0000])
""".format(**common_args))
i0 = _add_docstr(_special.special_i0,
r"""
i0(input, *, out=None) -> Tensor
Computes the zeroth order modified Bessel function of the first kind for each element of :attr:`input`.
.. math::
\text{out}_{i} = I_0(\text{input}_{i}) = \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}
""" + r"""
Args:
input (Tensor): the input tensor
Keyword args:
{out}
Example::
>>> torch.i0(torch.arange(5, dtype=torch.float32))
tensor([ 1.0000, 1.2661, 2.2796, 4.8808, 11.3019])
""".format(**common_args))
i0e = _add_docstr(_special.special_i0e,
r"""
i0e(input, *, out=None) -> Tensor
Computes the exponentially scaled zeroth order modified Bessel function of the first kind (as defined below)
for each element of :attr:`input`.
.. math::
\text{out}_{i} = \exp(-|x|) * i0(x) = \exp(-|x|) * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.i0e(torch.arange(5, dtype=torch.float32))
tensor([1.0000, 0.4658, 0.3085, 0.2430, 0.2070])
""".format(**common_args))
i1 = _add_docstr(_special.special_i1,
r"""
i1(input, *, out=None) -> Tensor
Computes the first order modified Bessel function of the first kind (as defined below)
for each element of :attr:`input`.
.. math::
\text{out}_{i} = \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.i1(torch.arange(5, dtype=torch.float32))
tensor([0.0000, 0.5652, 1.5906, 3.9534, 9.7595])
""".format(**common_args))
i1e = _add_docstr(_special.special_i1e,
r"""
i1e(input, *, out=None) -> Tensor
Computes the exponentially scaled first order modified Bessel function of the first kind (as defined below)
for each element of :attr:`input`.
.. math::
\text{out}_{i} = \exp(-|x|) * i1(x) =
\exp(-|x|) * \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.i1e(torch.arange(5, dtype=torch.float32))
tensor([0.0000, 0.2079, 0.2153, 0.1968, 0.1788])
""".format(**common_args))
ndtr = _add_docstr(_special.special_ndtr,
r"""
ndtr(input, *, out=None) -> Tensor
Computes the area under the standard Gaussian probability density function,
integrated from minus infinity to :attr:`input`, elementwise.
.. math::
\text{ndtr}(x) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x} e^{-\frac{1}{2}t^2} dt
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.ndtr(torch.tensor([-3., -2, -1, 0, 1, 2, 3]))
tensor([0.0013, 0.0228, 0.1587, 0.5000, 0.8413, 0.9772, 0.9987])
""".format(**common_args))
ndtri = _add_docstr(_special.special_ndtri,
r"""
ndtri(input, *, out=None) -> Tensor
Computes the argument, x, for which the area under the Gaussian probability density function
(integrated from minus infinity to x) is equal to :attr:`input`, elementwise.
.. math::
\text{ndtri}(p) = \sqrt{2}\text{erf}^{-1}(2p - 1)
.. note::
Also known as quantile function for Normal Distribution.
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.ndtri(torch.tensor([0, 0.25, 0.5, 0.75, 1]))
tensor([ -inf, -0.6745, 0.0000, 0.6745, inf])
""".format(**common_args))
log_ndtr = _add_docstr(_special.special_log_ndtr,
r"""
log_ndtr(input, *, out=None) -> Tensor
Computes the log of the area under the standard Gaussian probability density function,
integrated from minus infinity to :attr:`input`, elementwise.
.. math::
\text{log\_ndtr}(x) = \log\left(\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x} e^{-\frac{1}{2}t^2} dt \right)
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> torch.special.log_ndtr(torch.tensor([-3., -2, -1, 0, 1, 2, 3]))
tensor([-6.6077 -3.7832 -1.841 -0.6931 -0.1728 -0.023 -0.0014])
""".format(**common_args))
log1p = _add_docstr(_special.special_log1p,
r"""
log1p(input, *, out=None) -> Tensor
Alias for :func:`torch.log1p`.
""")
sinc = _add_docstr(_special.special_sinc,
r"""
sinc(input, *, out=None) -> Tensor
Computes the normalized sinc of :attr:`input.`
.. math::
\text{out}_{i} =
\begin{cases}
1, & \text{if}\ \text{input}_{i}=0 \\
\sin(\pi \text{input}_{i}) / (\pi \text{input}_{i}), & \text{otherwise}
\end{cases}
""" + r"""
Args:
{input}
Keyword args:
{out}
Example::
>>> t = torch.randn(4)
>>> t
tensor([ 0.2252, -0.2948, 1.0267, -1.1566])
>>> torch.special.sinc(t)
tensor([ 0.9186, 0.8631, -0.0259, -0.1300])
""".format(**common_args))
round = _add_docstr(_special.special_round,
r"""
round(input, *, out=None) -> Tensor
Alias for :func:`torch.round`.
""")
softmax = _add_docstr(_special.special_softmax,
r"""
softmax(input, dim, *, dtype=None) -> Tensor
Computes the softmax function.
Softmax is defined as:
:math:`\text{Softmax}(x_{i}) = \frac{\exp(x_i)}{\sum_j \exp(x_j)}`
It is applied to all slices along dim, and will re-scale them so that the elements
lie in the range `[0, 1]` and sum to 1.
Args:
input (Tensor): input
dim (int): A dimension along which softmax will be computed.
dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
If specified, the input tensor is cast to :attr:`dtype` before the operation
is performed. This is useful for preventing data type overflows. Default: None.
Examples::
>>> t = torch.ones(2, 2)
>>> torch.special.softmax(t, 0)
tensor([[0.5000, 0.5000],
[0.5000, 0.5000]])
""")
log_softmax = _add_docstr(_special.special_log_softmax,
r"""
log_softmax(input, dim, *, dtype=None) -> Tensor
Computes softmax followed by a logarithm.
While mathematically equivalent to log(softmax(x)), doing these two
operations separately is slower and numerically unstable. This function
is computed as:
.. math::
\text{log\_softmax}(x_{i}) = \log\left(\frac{\exp(x_i) }{ \sum_j \exp(x_j)} \right)
""" + r"""
Args:
input (Tensor): input
dim (int): A dimension along which log_softmax will be computed.
dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
If specified, the input tensor is cast to :attr:`dtype` before the operation
is performed. This is useful for preventing data type overflows. Default: None.
Example::
>>> t = torch.ones(2, 2)
>>> torch.special.log_softmax(t, 0)
tensor([[-0.6931, -0.6931],
[-0.6931, -0.6931]])
""")
zeta = _add_docstr(_special.special_zeta,
r"""
zeta(input, other, *, out=None) -> Tensor
Computes the Hurwitz zeta function, elementwise.
.. math::
\zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x}
""" + r"""
Args:
input (Tensor): the input tensor corresponding to `x`.
other (Tensor): the input tensor corresponding to `q`.
.. note::
The Riemann zeta function corresponds to the case when `q = 1`
Keyword args:
{out}
Example::
>>> x = torch.tensor([2., 4.])
>>> torch.special.zeta(x, 1)
tensor([1.6449, 1.0823])
>>> torch.special.zeta(x, torch.tensor([1., 2.]))
tensor([1.6449, 0.0823])
>>> torch.special.zeta(2, torch.tensor([1., 2.]))
tensor([1.6449, 0.6449])
""".format(**common_args))
multigammaln = _add_docstr(_special.special_multigammaln,
r"""
multigammaln(input, p, *, out=None) -> Tensor
Computes the `multivariate log-gamma function
<https://en.wikipedia.org/wiki/Multivariate_gamma_function>`_ with dimension
:math:`p` element-wise, given by
.. math::
\log(\Gamma_{p}(a)) = C + \displaystyle \sum_{i=1}^{p} \log\left(\Gamma\left(a - \frac{i - 1}{2}\right)\right)
where :math:`C = \log(\pi) \cdot \frac{p (p - 1)}{4}` and :math:`\Gamma(-)` is the Gamma function.
All elements must be greater than :math:`\frac{p - 1}{2}`, otherwise the behavior is undefiend.
""" + """
Args:
input (Tensor): the tensor to compute the multivariate log-gamma function
p (int): the number of dimensions
Keyword args:
{out}
Example::
>>> a = torch.empty(2, 3).uniform_(1, 2)
>>> a
tensor([[1.6835, 1.8474, 1.1929],
[1.0475, 1.7162, 1.4180]])
>>> torch.special.multigammaln(a, 2)
tensor([[0.3928, 0.4007, 0.7586],
[1.0311, 0.3901, 0.5049]])
""".format(**common_args))
gammainc = _add_docstr(_special.special_gammainc,
r"""
gammainc(input, other, *, out=None) -> Tensor
Computes the regularized lower incomplete gamma function:
.. math::
\text{out}_{i} = \frac{1}{\Gamma(\text{input}_i)} \int_0^{\text{other}_i} t^{\text{input}_i-1} e^{-t} dt
where both :math:`\text{input}_i` and :math:`\text{other}_i` are weakly positive
and at least one is strictly positive.
If both are zero or either is negative then :math:`\text{out}_i=\text{nan}`.
:math:`\Gamma(\cdot)` in the equation above is the gamma function,
.. math::
\Gamma(\text{input}_i) = \int_0^\infty t^{(\text{input}_i-1)} e^{-t} dt.
See :func:`torch.special.gammaincc` and :func:`torch.special.gammaln` for related functions.
Supports :ref:`broadcasting to a common shape <broadcasting-semantics>`
and float inputs.
.. note::
The backward pass with respect to :attr:`input` is not yet supported.
Please open an issue on PyTorch's Github to request it.
""" + r"""
Args:
input (Tensor): the first non-negative input tensor
other (Tensor): the second non-negative input tensor
Keyword args:
{out}
Example::
>>> a1 = torch.tensor([4.0])
>>> a2 = torch.tensor([3.0, 4.0, 5.0])
>>> a = torch.special.gammaincc(a1, a2)
tensor([0.3528, 0.5665, 0.7350])
tensor([0.3528, 0.5665, 0.7350])
>>> b = torch.special.gammainc(a1, a2) + torch.special.gammaincc(a1, a2)
tensor([1., 1., 1.])
""".format(**common_args))
gammaincc = _add_docstr(_special.special_gammaincc,
r"""
gammaincc(input, other, *, out=None) -> Tensor
Computes the regularized upper incomplete gamma function:
.. math::
\text{out}_{i} = \frac{1}{\Gamma(\text{input}_i)} \int_{\text{other}_i}^{\infty} t^{\text{input}_i-1} e^{-t} dt
where both :math:`\text{input}_i` and :math:`\text{other}_i` are weakly positive
and at least one is strictly positive.
If both are zero or either is negative then :math:`\text{out}_i=\text{nan}`.
:math:`\Gamma(\cdot)` in the equation above is the gamma function,
.. math::
\Gamma(\text{input}_i) = \int_0^\infty t^{(\text{input}_i-1)} e^{-t} dt.
See :func:`torch.special.gammainc` and :func:`torch.special.gammaln` for related functions.
Supports :ref:`broadcasting to a common shape <broadcasting-semantics>`
and float inputs.
.. note::
The backward pass with respect to :attr:`input` is not yet supported.
Please open an issue on PyTorch's Github to request it.
""" + r"""
Args:
input (Tensor): the first non-negative input tensor
other (Tensor): the second non-negative input tensor
Keyword args:
{out}
Example::
>>> a1 = torch.tensor([4.0])
>>> a2 = torch.tensor([3.0, 4.0, 5.0])
>>> a = torch.special.gammaincc(a1, a2)
tensor([0.6472, 0.4335, 0.2650])
>>> b = torch.special.gammainc(a1, a2) + torch.special.gammaincc(a1, a2)
tensor([1., 1., 1.])
""".format(**common_args))
airy_ai = _add_docstr(_special.special_airy_ai,
r"""
airy_ai(input, *, out=None) -> Tensor
Airy function :math:`\text{Ai}\left(\text{input}\right)`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
bessel_j0 = _add_docstr(_special.special_bessel_j0,
r"""
bessel_j0(input, *, out=None) -> Tensor
Bessel function of the first kind of order :math:`0`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
bessel_j1 = _add_docstr(_special.special_bessel_j1,
r"""
bessel_j1(input, *, out=None) -> Tensor
Bessel function of the first kind of order :math:`1`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
bessel_y0 = _add_docstr(_special.special_bessel_y0,
r"""
bessel_y0(input, *, out=None) -> Tensor
Bessel function of the second kind of order :math:`0`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
bessel_y1 = _add_docstr(_special.special_bessel_y1,
r"""
bessel_y1(input, *, out=None) -> Tensor
Bessel function of the second kind of order :math:`1`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
chebyshev_polynomial_t = _add_docstr(_special.special_chebyshev_polynomial_t,
r"""
chebyshev_polynomial_t(input, n, *, out=None) -> Tensor
Chebyshev polynomial of the first kind :math:`T_{n}(\text{input})`.
If :math:`n = 0`, :math:`1` is returned. If :math:`n = 1`, :math:`\text{input}`
is returned. If :math:`n < 6` or :math:`|\text{input}| > 1` the recursion:
.. math::
T_{n + 1}(\text{input}) = 2 \times \text{input} \times T_{n}(\text{input}) - T_{n - 1}(\text{input})
is evaluated. Otherwise, the explicit trigonometric formula:
.. math::
T_{n}(\text{input}) = \text{cos}(n \times \text{arccos}(x))
is evaluated.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
chebyshev_polynomial_u = _add_docstr(_special.special_chebyshev_polynomial_u,
r"""
chebyshev_polynomial_t(input, n, *, out=None) -> Tensor
Chebyshev polynomial of the second kind :math:`U_{n}(\text{input})`.
If :math:`n = 0`, :math:`1` is returned. If :math:`n = 1`,
:math:`2 \times \text{input}` is returned. If :math:`n < 6` or
:math:`|\text{input}| > 1`, the recursion:
.. math::
T_{n + 1}(\text{input}) = 2 \times \text{input} \times T_{n}(\text{input}) - T_{n - 1}(\text{input})
is evaluated. Otherwise, the explicit trigonometric formula:
.. math::
\frac{\text{sin}((n + 1) \times \text{arccos}(\text{input}))}{\text{sin}(\text{arccos}(\text{input}))}
is evaluated.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
chebyshev_polynomial_v = _add_docstr(_special.special_chebyshev_polynomial_v,
r"""
chebyshev_polynomial_v(input, n, *, out=None) -> Tensor
Chebyshev polynomial of the third kind :math:`V_{n}^{\ast}(\text{input})`.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
chebyshev_polynomial_w = _add_docstr(_special.special_chebyshev_polynomial_w,
r"""
chebyshev_polynomial_w(input, n, *, out=None) -> Tensor
Chebyshev polynomial of the fourth kind :math:`W_{n}^{\ast}(\text{input})`.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
hermite_polynomial_h = _add_docstr(_special.special_hermite_polynomial_h,
r"""
hermite_polynomial_h(input, n, *, out=None) -> Tensor
Physicist’s Hermite polynomial :math:`H_{n}(\text{input})`.
If :math:`n = 0`, :math:`1` is returned. If :math:`n = 1`, :math:`\text{input}`
is returned. Otherwise, the recursion:
.. math::
H_{n + 1}(\text{input}) = 2 \times \text{input} \times H_{n}(\text{input}) - H_{n - 1}(\text{input})
is evaluated.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
hermite_polynomial_he = _add_docstr(_special.special_hermite_polynomial_he,
r"""
hermite_polynomial_he(input, n, *, out=None) -> Tensor
Probabilist’s Hermite polynomial :math:`He_{n}(\text{input})`.
If :math:`n = 0`, :math:`1` is returned. If :math:`n = 1`, :math:`\text{input}`
is returned. Otherwise, the recursion:
.. math::
He_{n + 1}(\text{input}) = 2 \times \text{input} \times He_{n}(\text{input}) - He_{n - 1}(\text{input})
is evaluated.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
laguerre_polynomial_l = _add_docstr(_special.special_laguerre_polynomial_l,
r"""
laguerre_polynomial_l(input, n, *, out=None) -> Tensor
Laguerre polynomial :math:`L_{n}(\text{input})`.
If :math:`n = 0`, :math:`1` is returned. If :math:`n = 1`, :math:`\text{input}`
is returned. Otherwise, the recursion:
.. math::
L_{n + 1}(\text{input}) = 2 \times \text{input} \times L_{n}(\text{input}) - L_{n - 1}(\text{input})
is evaluated.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
legendre_polynomial_p = _add_docstr(_special.special_legendre_polynomial_p,
r"""
legendre_polynomial_p(input, n, *, out=None) -> Tensor
Legendre polynomial :math:`P_{n}(\text{input})`.
If :math:`n = 0`, :math:`1` is returned. If :math:`n = 1`, :math:`\text{input}`
is returned. Otherwise, the recursion:
.. math::
P_{n + 1}(\text{input}) = 2 \times \text{input} \times P_{n}(\text{input}) - P_{n - 1}(\text{input})
is evaluated.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
modified_bessel_i0 = _add_docstr(_special.special_modified_bessel_i0,
r"""
modified_bessel_i0(input, *, out=None) -> Tensor
Modified Bessel function of the first kind of order :math:`0`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
modified_bessel_i1 = _add_docstr(_special.special_modified_bessel_i1,
r"""
modified_bessel_i1(input, *, out=None) -> Tensor
Modified Bessel function of the first kind of order :math:`1`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
modified_bessel_k0 = _add_docstr(_special.special_modified_bessel_k0,
r"""
modified_bessel_k0(input, *, out=None) -> Tensor
Modified Bessel function of the second kind of order :math:`0`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
modified_bessel_k1 = _add_docstr(_special.special_modified_bessel_k1,
r"""
modified_bessel_k1(input, *, out=None) -> Tensor
Modified Bessel function of the second kind of order :math:`1`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
scaled_modified_bessel_k0 = _add_docstr(_special.special_scaled_modified_bessel_k0,
r"""
scaled_modified_bessel_k0(input, *, out=None) -> Tensor
Scaled modified Bessel function of the second kind of order :math:`0`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
scaled_modified_bessel_k1 = _add_docstr(_special.special_scaled_modified_bessel_k1,
r"""
scaled_modified_bessel_k1(input, *, out=None) -> Tensor
Scaled modified Bessel function of the second kind of order :math:`1`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))
shifted_chebyshev_polynomial_t = _add_docstr(_special.special_shifted_chebyshev_polynomial_t,
r"""
shifted_chebyshev_polynomial_t(input, n, *, out=None) -> Tensor
Chebyshev polynomial of the first kind :math:`T_{n}^{\ast}(\text{input})`.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
shifted_chebyshev_polynomial_u = _add_docstr(_special.special_shifted_chebyshev_polynomial_u,
r"""
shifted_chebyshev_polynomial_u(input, n, *, out=None) -> Tensor
Chebyshev polynomial of the second kind :math:`U_{n}^{\ast}(\text{input})`.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
shifted_chebyshev_polynomial_v = _add_docstr(_special.special_shifted_chebyshev_polynomial_v,
r"""
shifted_chebyshev_polynomial_v(input, n, *, out=None) -> Tensor
Chebyshev polynomial of the third kind :math:`V_{n}^{\ast}(\text{input})`.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
shifted_chebyshev_polynomial_w = _add_docstr(_special.special_shifted_chebyshev_polynomial_w,
r"""
shifted_chebyshev_polynomial_w(input, n, *, out=None) -> Tensor
Chebyshev polynomial of the fourth kind :math:`W_{n}^{\ast}(\text{input})`.
""" + r"""
Args:
{input}
n (Tensor): Degree of the polynomial.
Keyword args:
{out}
""".format(**common_args))
spherical_bessel_j0 = _add_docstr(_special.special_spherical_bessel_j0,
r"""
spherical_bessel_j0(input, *, out=None) -> Tensor
Spherical Bessel function of the first kind of order :math:`0`.
""" + r"""
Args:
{input}
Keyword args:
{out}
""".format(**common_args))