Functions#

torch.special.airy_ai(input,*,out=None)→Tensor#

Airy function$\text{Ai}\left(\text{input}\right)$Ai(input).

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.bessel_j0(input,*,out=None)→Tensor#

Bessel function of the first kind of order$0$0.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.bessel_j1(input,*,out=None)→Tensor#

Bessel function of the first kind of order$1$1.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.bessel_y0(input,*,out=None)→Tensor#

Bessel function of the second kind of order$0$0.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.bessel_y1(input,*,out=None)→Tensor#

Bessel function of the second kind of order$1$1.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.chebyshev_polynomial_t(input,n,*,out=None)→Tensor#

Chebyshev polynomial of the first kind$T_n(\text{input})$Tn​(input).

If$n=0$n=0,$1$1 is returned. If$n=1$n=1,$\text{input}$inputis returned. Ifn<6 or$\mathrm{\mid }\text{input}\mathrm{\mid }>1$∣input∣>1 the recursion:

$$T_{n+1}(\text{input})=2\times \text{input}\times T_n(\text{input})-T_{n-1}(\text{input})$$Tn+1​(input)=2×input×Tn​(input)−Tn−1​(input)

is evaluated. Otherwise, the explicit trigonometric formula:

$$T_n(\text{input})=\text{cos}(n\times \text{arccos}(x))$$Tn​(input)=cos(n×arccos(x))

is evaluated.

Parameters

• input (Tensor) – the input tensor.

• n (Tensor) – Degree of the polynomial.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.chebyshev_polynomial_u(input,n,*,out=None)→Tensor#

Chebyshev polynomial of the second kind$U_n(\text{input})$Un​(input).

If$n=0$n=0,$1$1 is returned. If$n=1$n=1,$2\times \text{input}$2×input is returned. Ifn<6 or$\mathrm{\mid }\text{input}\mathrm{\mid }>1$∣input∣>1, the recursion:

$$U_{n+1}(\text{input})=2\times \text{input}\times U_n(\text{input})-U_{n-1}(\text{input})$$Un+1​(input)=2×input×Un​(input)−Un−1​(input)

is evaluated. Otherwise, the explicit trigonometric formula:

$$\frac{\text{sin}((n+1)\times \text{arccos}(\text{input}))}{\text{sin}(\text{arccos}(\text{input}))}$$sin(arccos(input))sin((n+1)×arccos(input))​

is evaluated.

Parameters

• input (Tensor) – the input tensor.

• n (Tensor) – Degree of the polynomial.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.chebyshev_polynomial_v(input,n,*,out=None)→Tensor#

Chebyshev polynomial of the third kind$V_n^{\ast }(\text{input})$Vn∗​(input).

Parameters

• input (Tensor) – the input tensor.

• n (Tensor) – Degree of the polynomial.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.chebyshev_polynomial_w(input,n,*,out=None)→Tensor#

Chebyshev polynomial of the fourth kind$W_n^{\ast }(\text{input})$Wn∗​(input).

Parameters

• input (Tensor) – the input tensor.

• n (Tensor) – Degree of the polynomial.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.digamma(input,*,out=None)→Tensor#

Computes the logarithmic derivative of the gamma function oninput.

$$\mathrm{ϝ}(x)=\frac{d}{dx}\ln \left(\mathrm{\Gamma }\left(x\right)\right)=\frac{{\mathrm{\Gamma }}^{\mathrm{\prime }}(x)}{\mathrm{\Gamma }(x)}$$ϝ(x)=dxd​ln(Γ(x))=Γ(x)Γ′(x)​

Parameters

input (Tensor) – the tensor to compute the digamma function on

Keyword Arguments

out (Tensor,optional) – the output tensor.

Note

This function is similar to SciPy’sscipy.special.digamma.

Note

From PyTorch 1.8 onwards, the digamma function returns-Inf for0.Previously it returnedNaN for0.

Example:

>>>a=torch.tensor([1,0.5])>>>torch.special.digamma(a)tensor([-0.5772, -1.9635])

torch.special.entr(input,*,out=None)→Tensor#

Computes the entropy oninput (as defined below), elementwise.

entr(x)=⎩⎨⎧​−x∗ln(x)0−∞​x>0x=0.0x<0​​​

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>a=torch.arange(-0.5,1,0.5)>>>atensor([-0.5000,  0.0000,  0.5000])>>>torch.special.entr(a)tensor([  -inf, 0.0000, 0.3466])

torch.special.erf(input,*,out=None)→Tensor#

Computes the error function ofinput. The error function is defined as follows:

$$\mathrm{e}\mathrm{r}\mathrm{f}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt$$erf(x)=π​2​∫0x​e−t2dt

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.special.erf(torch.tensor([0,-1.,10.]))tensor([ 0.0000, -0.8427,  1.0000])

torch.special.erfc(input,*,out=None)→Tensor#

Computes the complementary error function ofinput.The complementary error function is defined as follows:

$$\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(x)=1-\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt$$erfc(x)=1−π​2​∫0x​e−t2dt

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.special.erfc(torch.tensor([0,-1.,10.]))tensor([ 1.0000, 1.8427,  0.0000])

torch.special.erfcx(input,*,out=None)→Tensor#

Computes the scaled complementary error function for each element ofinput.The scaled complementary error function is defined as follows:

$$\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\mathrm{x}(x)=e^{x^2}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(x)$$erfcx(x)=ex2erfc(x)

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.special.erfcx(torch.tensor([0,-1.,10.]))tensor([ 1.0000, 5.0090, 0.0561])

torch.special.erfinv(input,*,out=None)→Tensor#

Computes the inverse error function ofinput.The inverse error function is defined in the range$(-1,1)$(−1,1) as:

$$\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{v}(\mathrm{e}\mathrm{r}\mathrm{f}(x))=x$$erfinv(erf(x))=x

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.special.erfinv(torch.tensor([0,0.5,-1.]))tensor([ 0.0000,  0.4769,    -inf])

torch.special.exp2(input,*,out=None)→Tensor#

Computes the base two exponential function ofinput.

$$y_i=2^{x_i}$$yi​=2xi​

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.special.exp2(torch.tensor([0,math.log2(2.),3,4]))tensor([ 1.,  2.,  8., 16.])

torch.special.expit(input,*,out=None)→Tensor#

Computes the expit (also known as the logistic sigmoid function) of the elements ofinput.

$${\text{out}}_i=\frac{1}{1+e^{-{\text{input}}_i}}$$outi​=1+e−inputi​1​

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>t=torch.randn(4)>>>ttensor([ 0.9213,  1.0887, -0.8858, -1.7683])>>>torch.special.expit(t)tensor([ 0.7153,  0.7481,  0.2920,  0.1458])

torch.special.expm1(input,*,out=None)→Tensor#

Computes the exponential of the elements minus 1ofinput.

$$y_i=e^{x_i}-1$$yi​=exi​−1

Note

This function provides greater precision than exp(x) - 1 for small values of x.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.special.expm1(torch.tensor([0,math.log(2.)]))tensor([ 0.,  1.])

torch.special.gammainc(input,other,*,out=None)→Tensor#

Computes the regularized lower incomplete gamma function:

$${\text{out}}_i=\frac{1}{\mathrm{\Gamma }({\text{input}}_i)}{\int }_0^{{\text{other}}_i}{t}^{{\text{input}}_i-1}{e}^{-t}dt$$outi​=Γ(inputi​)1​∫0otheri​​tinputi​−1e−tdt

where both${\text{input}}_i$inputi​ and${\text{other}}_i$otheri​ are weakly positiveand at least one is strictly positive.If both are zero or either is negative then${\text{out}}_i=\text{nan}$outi​=nan.$\mathrm{\Gamma }(\cdot )$Γ(⋅) in the equation above is the gamma function,

$$\mathrm{\Gamma }({\text{input}}_i)={\int }_0^{\mathrm{\infty }}{t}^{({\text{input}}_i-1)}{e}^{-t}dt\mathrm{.}$$Γ(inputi​)=∫0∞​t(inputi​−1)e−tdt.

Seetorch.special.gammaincc() andtorch.special.gammaln() for related functions.

Supportsbroadcasting to a common shapeand float inputs.

Note

The backward pass with respect toinput is not yet supported.Please open an issue on PyTorch’s Github to request it.

Parameters

• input (Tensor) – the first non-negative input tensor

• other (Tensor) – the second non-negative input tensor

Keyword Arguments

out (Tensor,optional) – the output tensor.

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.])

torch.special.gammaincc(input,other,*,out=None)→Tensor#

Computes the regularized upper incomplete gamma function:

$${\text{out}}_i=\frac{1}{\mathrm{\Gamma }({\text{input}}_i)}{\int }_{{\text{other}}_i}^{\mathrm{\infty }}{t}^{{\text{input}}_i-1}{e}^{-t}dt$$outi​=Γ(inputi​)1​∫otheri​∞​tinputi​−1e−tdt

where both${\text{input}}_i$inputi​ and${\text{other}}_i$otheri​ are weakly positiveand at least one is strictly positive.If both are zero or either is negative then${\text{out}}_i=\text{nan}$outi​=nan.$\mathrm{\Gamma }(\cdot )$Γ(⋅) in the equation above is the gamma function,

$$\mathrm{\Gamma }({\text{input}}_i)={\int }_0^{\mathrm{\infty }}{t}^{({\text{input}}_i-1)}{e}^{-t}dt\mathrm{.}$$Γ(inputi​)=∫0∞​t(inputi​−1)e−tdt.

Seetorch.special.gammainc() andtorch.special.gammaln() for related functions.

Supportsbroadcasting to a common shapeand float inputs.

Note

The backward pass with respect toinput is not yet supported.Please open an issue on PyTorch’s Github to request it.

Parameters

• input (Tensor) – the first non-negative input tensor

• other (Tensor) – the second non-negative input tensor

Keyword Arguments

out (Tensor,optional) – the output tensor.

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.])

torch.special.gammaln(input,*,out=None)→Tensor#

Computes the natural logarithm of the absolute value of the gamma function oninput.

$${\text{out}}_i=\ln \mathrm{\Gamma }(\mathrm{\mid }{\text{input}}_i\mathrm{\mid })$$outi​=lnΓ(∣inputi​∣)

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>a=torch.arange(0.5,2,0.5)>>>torch.special.gammaln(a)tensor([ 0.5724,  0.0000, -0.1208])

torch.special.hermite_polynomial_h(input,n,*,out=None)→Tensor#

Physicist’s Hermite polynomial$H_n(\text{input})$Hn​(input).

If$n=0$n=0,$1$1 is returned. If$n=1$n=1,$\text{input}$inputis returned. Otherwise, the recursion:

$$H_{n+1}(\text{input})=2\times \text{input}\times H_n(\text{input})-H_{n-1}(\text{input})$$Hn+1​(input)=2×input×Hn​(input)−Hn−1​(input)

is evaluated.

Parameters

• input (Tensor) – the input tensor.

• n (Tensor) – Degree of the polynomial.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.hermite_polynomial_he(input,n,*,out=None)→Tensor#

Probabilist’s Hermite polynomial$H{e}_n(\text{input})$Hen​(input).

If$n=0$n=0,$1$1 is returned. If$n=1$n=1,$\text{input}$inputis returned. Otherwise, the recursion:

$$H{e}_{n+1}(\text{input})=2\times \text{input}\times H{e}_n(\text{input})-H{e}_{n-1}(\text{input})$$Hen+1​(input)=2×input×Hen​(input)−Hen−1​(input)

is evaluated.

Parameters

• input (Tensor) – the input tensor.

• n (Tensor) – Degree of the polynomial.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.i0(input,*,out=None)→Tensor#

Computes the zeroth order modified Bessel function of the first kind for each element ofinput.

$${\text{out}}_i=I_0({\text{input}}_i)=\sum _{k=0}^{\mathrm{\infty }}\frac{({\text{input}}_i^2\mathrm{/}4{)}^k}{(k!{)}^2}$$outi​=I0​(inputi​)=k=0∑∞​(k!)2(inputi2​/4)k​

Parameters

input (Tensor) – the input tensor

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.i0(torch.arange(5,dtype=torch.float32))tensor([ 1.0000,  1.2661,  2.2796,  4.8808, 11.3019])

torch.special.i0e(input,*,out=None)→Tensor#

Computes the exponentially scaled zeroth order modified Bessel function of the first kind (as defined below)for each element ofinput.

$${\text{out}}_i=\exp (-\mathrm{\mid }x\mathrm{\mid })\ast i0(x)=\exp (-\mathrm{\mid }x\mathrm{\mid })\ast \sum _{k=0}^{\mathrm{\infty }}\frac{({\text{input}}_i^2\mathrm{/}4{)}^k}{(k!{)}^2}$$outi​=exp(−∣x∣)∗i0(x)=exp(−∣x∣)∗k=0∑∞​(k!)2(inputi2​/4)k​

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.special.i0e(torch.arange(5,dtype=torch.float32))tensor([1.0000, 0.4658, 0.3085, 0.2430, 0.2070])

torch.special.i1(input,*,out=None)→Tensor#

Computes the first order modified Bessel function of the first kind (as defined below)for each element ofinput.

$${\text{out}}_i=\frac{({\text{input}}_i)}{2}\ast \sum _{k=0}^{\mathrm{\infty }}\frac{({\text{input}}_i^2\mathrm{/}4{)}^k}{(k!)\ast (k+1)!}$$outi​=2(inputi​)​∗k=0∑∞​(k!)∗(k+1)!(inputi2​/4)k​

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.special.i1(torch.arange(5,dtype=torch.float32))tensor([0.0000, 0.5652, 1.5906, 3.9534, 9.7595])

torch.special.i1e(input,*,out=None)→Tensor#

Computes the exponentially scaled first order modified Bessel function of the first kind (as defined below)for each element ofinput.

$${\text{out}}_i=\exp (-\mathrm{\mid }x\mathrm{\mid })\ast i1(x)=\exp (-\mathrm{\mid }x\mathrm{\mid })\ast \frac{({\text{input}}_i)}{2}\ast \sum _{k=0}^{\mathrm{\infty }}\frac{({\text{input}}_i^2\mathrm{/}4{)}^k}{(k!)\ast (k+1)!}$$outi​=exp(−∣x∣)∗i1(x)=exp(−∣x∣)∗2(inputi​)​∗k=0∑∞​(k!)∗(k+1)!(inputi2​/4)k​

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>torch.special.i1e(torch.arange(5,dtype=torch.float32))tensor([0.0000, 0.2079, 0.2153, 0.1968, 0.1788])

torch.special.laguerre_polynomial_l(input,n,*,out=None)→Tensor#

Laguerre polynomial$L_n(\text{input})$Ln​(input).

If$n=0$n=0,$1$1 is returned. If$n=1$n=1,$\text{input}$inputis returned. Otherwise, the recursion:

$$L_{n+1}(\text{input})=2\times \text{input}\times L_n(\text{input})-L_{n-1}(\text{input})$$Ln+1​(input)=2×input×Ln​(input)−Ln−1​(input)

is evaluated.

Parameters

• input (Tensor) – the input tensor.

• n (Tensor) – Degree of the polynomial.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.legendre_polynomial_p(input,n,*,out=None)→Tensor#

Legendre polynomial$P_n(\text{input})$Pn​(input).

If$n=0$n=0,$1$1 is returned. If$n=1$n=1,$\text{input}$inputis returned. Otherwise, the recursion:

$$P_{n+1}(\text{input})=2\times \text{input}\times P_n(\text{input})-P_{n-1}(\text{input})$$Pn+1​(input)=2×input×Pn​(input)−Pn−1​(input)

is evaluated.

Parameters

• input (Tensor) – the input tensor.

• n (Tensor) – Degree of the polynomial.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.log1p(input,*,out=None)→Tensor#

Alias fortorch.log1p().

torch.special.log_ndtr(input,*,out=None)→Tensor#

Computes the log of the area under the standard Gaussian probability density function,integrated from minus infinity toinput, elementwise.

$$\text{log\_ndtr}(x)=\log \left(\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^x{e}^{-\frac{1}{2}{t}^2}dt\right)$$log_ndtr(x)=log(2π​1​∫−∞x​e−21​t2dt)

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

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])

torch.special.log_softmax(input,dim,*,dtype=None)→Tensor#

Computes softmax followed by a logarithm.

While mathematically equivalent to log(softmax(x)), doing these twooperations separately is slower and numerically unstable. This functionis computed as:

$$\text{log\_softmax}(x_i)=\log \left(\frac{\exp (x_i)}{\sum _j\exp (x_j)}\right)$$log_softmax(xi​)=log(∑j​exp(xj​)exp(xi​)​)

Parameters

• input (Tensor) – input

• dim (int) – A dimension along which log_softmax will be computed.

• dtype (torch.dtype, optional) – the desired data type of returned tensor.If specified, the input tensor is cast todtype before the operationis 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]])

torch.special.logit(input,eps=None,*,out=None)→Tensor#

Returns a new tensor with the logit of the elements ofinput.input is clamped to [eps, 1 - eps] when eps is not None.When eps is None andinput < 0 orinput > 1, the function will yields NaN.

yi​zi​​=ln(1−zi​zi​​)=⎩⎨⎧​xi​epsxi​1−eps​if eps is Noneif xi​<epsif eps≤xi​≤1−epsif xi​>1−eps​​​

Parameters

• input (Tensor) – the input tensor.

• eps (float,optional) – the epsilon for input clamp bound. Default:None

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>a=torch.rand(5)>>>atensor([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])

torch.special.logsumexp(input,dim,keepdim=False,*,out=None)#

Alias fortorch.logsumexp().

torch.special.modified_bessel_i0(input,*,out=None)→Tensor#

Modified Bessel function of the first kind of order$0$0.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.modified_bessel_i1(input,*,out=None)→Tensor#

Modified Bessel function of the first kind of order$1$1.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.modified_bessel_k0(input,*,out=None)→Tensor#

Modified Bessel function of the second kind of order$0$0.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.modified_bessel_k1(input,*,out=None)→Tensor#

Modified Bessel function of the second kind of order$1$1.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor,optional) – the output tensor.

torch.special.multigammaln(input,p,*,out=None)→Tensor#

Computes themultivariate log-gamma function with dimension$p$p element-wise, given by

$$\log ({\mathrm{\Gamma }}_p(a))=C+{\displaystyle \sum _{i=1}^p\log \left(\mathrm{\Gamma }(a-\frac{i-1}{2})\right)}$$log(Γp​(a))=C+i=1∑p​log(Γ(a−2i−1​))

where$C=\log (\pi )\cdot \frac{p(p-1)}{4}$C=log(π)⋅4p(p−1)​ and$\mathrm{\Gamma }(-)$Γ(−) is the Gamma function.

All elements must be greater than$\frac{p-1}{2}$2p−1​, otherwise the behavior is undefined.

Parameters

• input (Tensor) – the tensor to compute the multivariate log-gamma function

• p (int) – the number of dimensions

Keyword Arguments

out (Tensor,optional) – the output tensor.

Example:

>>>a=torch.empty(2,3).uniform_(1,2)>>>atensor([[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]])