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<li class="toctree-l1"><a class="reference internal" href="notes/amp_examples.html">CUDA Automatic Mixed Precision examples</a></li>
<li class="toctree-l1"><a class="reference internal" href="notes/autograd.html">Autograd mechanics</a></li>
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<section id="torch-special">
<h1>torch.special<a class="headerlink" href="#torch-special" title="Permalink to this heading">¶</a></h1>
<p>The torch.special module, modeled after SciPy’s <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/special.html">special</a> module.</p>
<span class="target" id="module-torch.special"></span><section id="functions">
<h2>Functions<a class="headerlink" href="#functions" title="Permalink to this heading">¶</a></h2>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.airy_ai">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">airy_ai</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.airy_ai" title="Permalink to this definition">¶</a></dt>
<dd><p>Airy function <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ai</mtext><mrow><mo fence="true">(</mo><mtext>input</mtext><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\text{Ai}\left(\text{input}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Ai</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord text"><span class="mord">input</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></span>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.bessel_j0">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">bessel_j0</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.bessel_j0" title="Permalink to this definition">¶</a></dt>
<dd><p>Bessel function of the first kind of order <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.bessel_j1">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">bessel_j1</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.bessel_j1" title="Permalink to this definition">¶</a></dt>
<dd><p>Bessel function of the first kind of order <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.digamma">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">digamma</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.digamma" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the logarithmic derivative of the gamma function on <cite>input</cite>.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">ϝ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>ln</mi><mo></mo><mrow><mo fence="true">(</mo><mi mathvariant="normal">Γ</mi><mrow><mo fence="true">(</mo><mi>x</mi><mo fence="true">)</mo></mrow><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">Γ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\digamma(x) = \frac{d}{dx} \ln\left(\Gamma\left(x\right)\right) = \frac{\Gamma'(x)}{\Gamma(x)}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord amsrm">ϝ</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord">Γ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">x</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3649em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4289em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Γ</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">Γ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the tensor to compute the digamma function on</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function is similar to SciPy’s <cite>scipy.special.digamma</cite>.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>From PyTorch 1.8 onwards, the digamma function returns <cite>-Inf</cite> for <cite>0</cite>.
Previously it returned <cite>NaN</cite> for <cite>0</cite>.</p>
</div>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">digamma</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([-0.5772, -1.9635])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.entr">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">entr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.entr" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the entropy on <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> (as defined below), elementwise.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>entr(x)</mtext><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>x</mi><mo>∗</mo><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>=</mo><mn>0.0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo><</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
\text{entr(x)} = \begin{cases}
-x * \ln(x) & x > 0 \\
0 & x = 0.0 \\
-\infty & x < 0
\end{cases}
\end{align}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.62em;vertical-align:-2.06em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.56em;"><span style="top:-4.56em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"><span class="mord text"><span class="mord">entr(x)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-2.2em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.192em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width='0.8889em' height='0.316em' style='width:0.8889em' viewBox='0 0 888.89 316' preserveAspectRatio='xMinYMin'><path d='M384 0 H504 V316 H384z M384 0 H504 V316 H384z'/></svg></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width='0.8889em' height='0.316em' style='width:0.8889em' viewBox='0 0 888.89 316' preserveAspectRatio='xMinYMin'><path d='M384 0 H504 V316 H384z M384 0 H504 V316 H384z'/></svg></span></span><span style="top:-4.6em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">−</span><span class="mord">∞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0.0</span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.06em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.56em;"><span style="top:-4.56em;"><span class="pstrut" style="height:4.41em;"></span><span class="eqn-num"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.06em;"><span></span></span></span></span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<dl>
<dt>Example::</dt><dd><div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([-0.5000, 0.0000, 0.5000])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">entr</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([ -inf, 0.0000, 0.3466])</span>
</pre></div>
</div>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.erf">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">erf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.erf" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the error function of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>. The error function is defined as follows:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">f</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>2</mn><msqrt><mi>π</mi></msqrt></mfrac><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></msup><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm" style="margin-right:0.07778em;">erf</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3443em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
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s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
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H400000v40H845.2724
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">erf</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">10.</span><span class="p">]))</span>
<span class="go">tensor([ 0.0000, -0.8427, 1.0000])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.erfc">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">erfc</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.erfc" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the complementary error function of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.
The complementary error function is defined as follows:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">c</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mn>2</mn><msqrt><mi>π</mi></msqrt></mfrac><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></msup><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\mathrm{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">erfc</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.3443em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">erfc</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">10.</span><span class="p">]))</span>
<span class="go">tensor([ 1.0000, 1.8427, 0.0000])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.erfcx">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">erfcx</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.erfcx" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the scaled complementary error function for each element of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.
The scaled complementary error function is defined as follows:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">x</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi><msup><mi>x</mi><mn>2</mn></msup></msup><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">c</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{erfcx}(x) = e^{x^2} \mathrm{erfc}(x)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">erfcx</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2869em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">erfc</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">erfcx</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">10.</span><span class="p">]))</span>
<span class="go">tensor([ 1.0000, 5.0090, 0.0561])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.erfinv">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">erfinv</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.erfinv" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the inverse error function of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.
The inverse error function is defined in the range <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span> as:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">v</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">f</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\mathrm{erfinv}(\mathrm{erf}(x)) = x
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm" style="margin-right:0.01389em;">erfinv</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm" style="margin-right:0.07778em;">erf</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">erfinv</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]))</span>
<span class="go">tensor([ 0.0000, 0.4769, -inf])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.exp2">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">exp2</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.exp2" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the base two exponential function of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><msup><mn>2</mn><msub><mi>x</mi><mi>i</mi></msub></msup></mrow><annotation encoding="application/x-tex">y_{i} = 2^{x_{i}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7144em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">exp2</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="n">math</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="mf">2.</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]))</span>
<span class="go">tensor([ 1., 2., 8., 16.])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.expit">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">expit</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.expit" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the expit (also known as the logistic sigmoid function) of the elements of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mtext>input</mtext><mi>i</mi></msub></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text{out}_{i} = \frac{1}{1 + e^{-\text{input}_{i}}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7614em;"><span style="top:-2.9939em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2052em;"><span style="top:-2.2341em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2659em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">t</span>
<span class="go">tensor([ 0.9213, 1.0887, -0.8858, -1.7683])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">expit</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="go">tensor([ 0.7153, 0.7481, 0.2920, 0.1458])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.expm1">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">expm1</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.expm1" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the exponential of the elements minus 1
of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><msup><mi>e</mi><msub><mi>x</mi><mi>i</mi></msub></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y_{i} = e^{x_{i}} - 1
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7977em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></div><div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function provides greater precision than exp(x) - 1 for small values of x.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">expm1</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mf">2.</span><span class="p">)]))</span>
<span class="go">tensor([ 0., 1.])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.gammainc">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">gammainc</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">other</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.gammainc" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the regularized lower incomplete gamma function:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><msub><mtext>input</mtext><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><msubsup><mo>∫</mo><mn>0</mn><msub><mtext>other</mtext><mi>i</mi></msub></msubsup><msup><mi>t</mi><mrow><msub><mtext>input</mtext><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\text{out}_{i} = \frac{1}{\Gamma(\text{input}_i)} \int_0^{\text{other}_i} t^{\text{input}_i-1} e^{-t} dt
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.535em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Γ</span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.599em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">other</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8805em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2052em;"><span style="top:-2.2341em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2659em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></div><p>where both <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>input</mtext><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\text{input}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.912em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span></span> and <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>other</mtext><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\text{other}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">other</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> are weakly positive
and at least one is strictly positive.
If both are zero or either is negative then <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><mtext>nan</mtext></mrow><annotation encoding="application/x-tex">\text{out}_i=\text{nan}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">nan</span></span></span></span></span></span>.
<span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Γ</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span></span> in the equation above is the gamma function,</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><msub><mtext>input</mtext><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><msup><mi>t</mi><mrow><mo stretchy="false">(</mo><msub><mtext>input</mtext><mi>i</mi></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\Gamma(\text{input}_i) = \int_0^\infty t^{(\text{input}_i-1)} e^{-t} dt.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Γ</span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2052em;"><span style="top:-2.2341em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2659em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mord">.</span></span></span></span></span></div><p>See <a class="reference internal" href="#torch.special.gammaincc" title="torch.special.gammaincc"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.special.gammaincc()</span></code></a> and <a class="reference internal" href="#torch.special.gammaln" title="torch.special.gammaln"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.special.gammaln()</span></code></a> for related functions.</p>
<p>Supports <a class="reference internal" href="notes/broadcasting.html#broadcasting-semantics"><span class="std std-ref">broadcasting to a common shape</span></a>
and float inputs.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The backward pass with respect to <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is not yet supported.
Please open an issue on PyTorch’s Github to request it.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the first non-negative input tensor</p></li>
<li><p><strong>other</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the second non-negative input tensor</p></li>
</ul>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">4.0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">a2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">gammaincc</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">)</span>
<span class="go">tensor([0.3528, 0.5665, 0.7350])</span>
<span class="go">tensor([0.3528, 0.5665, 0.7350])</span>
<span class="gp">>>> </span><span class="n">b</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">gammainc</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">)</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">gammaincc</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">)</span>
<span class="go">tensor([1., 1., 1.])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.gammaincc">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">gammaincc</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">other</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.gammaincc" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the regularized upper incomplete gamma function:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><msub><mtext>input</mtext><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><msubsup><mo>∫</mo><msub><mtext>other</mtext><mi>i</mi></msub><mi mathvariant="normal">∞</mi></msubsup><msup><mi>t</mi><mrow><msub><mtext>input</mtext><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\text{out}_{i} = \frac{1}{\Gamma(\text{input}_i)} \int_{\text{other}_i}^{\infty} t^{\text{input}_i-1} e^{-t} dt
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4263em;vertical-align:-1.012em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Γ</span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">other</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.012em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8805em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2052em;"><span style="top:-2.2341em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2659em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></div><p>where both <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>input</mtext><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\text{input}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.912em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span></span> and <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>other</mtext><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\text{other}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">other</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> are weakly positive
and at least one is strictly positive.
If both are zero or either is negative then <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><mtext>nan</mtext></mrow><annotation encoding="application/x-tex">\text{out}_i=\text{nan}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">nan</span></span></span></span></span></span>.
<span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Γ</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span></span> in the equation above is the gamma function,</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><msub><mtext>input</mtext><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><msup><mi>t</mi><mrow><mo stretchy="false">(</mo><msub><mtext>input</mtext><mi>i</mi></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\Gamma(\text{input}_i) = \int_0^\infty t^{(\text{input}_i-1)} e^{-t} dt.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Γ</span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2052em;"><span style="top:-2.2341em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2659em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8436em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mord">.</span></span></span></span></span></div><p>See <a class="reference internal" href="#torch.special.gammainc" title="torch.special.gammainc"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.special.gammainc()</span></code></a> and <a class="reference internal" href="#torch.special.gammaln" title="torch.special.gammaln"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.special.gammaln()</span></code></a> for related functions.</p>
<p>Supports <a class="reference internal" href="notes/broadcasting.html#broadcasting-semantics"><span class="std std-ref">broadcasting to a common shape</span></a>
and float inputs.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The backward pass with respect to <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is not yet supported.
Please open an issue on PyTorch’s Github to request it.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the first non-negative input tensor</p></li>
<li><p><strong>other</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the second non-negative input tensor</p></li>
</ul>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">4.0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">a2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">gammaincc</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">)</span>
<span class="go">tensor([0.6472, 0.4335, 0.2650])</span>
<span class="gp">>>> </span><span class="n">b</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">gammainc</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">)</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">gammaincc</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">)</span>
<span class="go">tensor([1., 1., 1.])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.gammaln">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">gammaln</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.gammaln" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the natural logarithm of the absolute value of the gamma function on <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><mi>ln</mi><mo></mo><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><msub><mtext>input</mtext><mi>i</mi></msub><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{out}_{i} = \ln \Gamma(|\text{input}_{i}|)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">Γ</span><span class="mopen">(</span><span class="mord">∣</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mclose">)</span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">gammaln</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([ 0.5724, 0.0000, -0.1208])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.i0">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">i0</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.i0" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the zeroth order modified Bessel function of the first kind for each element of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><msub><mi>I</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msub><mtext>input</mtext><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><msubsup><mtext>input</mtext><mi>i</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>4</mn><msup><mo stretchy="false">)</mo><mi>k</mi></msup></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">!</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text{out}_{i} = I_0(\text{input}_{i}) = \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5489em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">!</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/4</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">i0</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float32</span><span class="p">))</span>
<span class="go">tensor([ 1.0000, 1.2661, 2.2796, 4.8808, 11.3019])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.i0e">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">i0e</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.i0e" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the exponentially scaled zeroth order modified Bessel function of the first kind (as defined below)
for each element of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo><mo>∗</mo><mi>i</mi><mn>0</mn><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo><mo>∗</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><msubsup><mtext>input</mtext><mi>i</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>4</mn><msup><mo stretchy="false">)</mo><mi>k</mi></msup></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">!</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text{out}_{i} = \exp(-|x|) * i0(x) = \exp(-|x|) * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">i</span><span class="mord">0</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5489em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">!</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/4</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<dl>
<dt>Example::</dt><dd><div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">i0e</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float32</span><span class="p">))</span>
<span class="go">tensor([1.0000, 0.4658, 0.3085, 0.2430, 0.2070])</span>
</pre></div>
</div>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.i1">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">i1</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.i1" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the first order modified Bessel function of the first kind (as defined below)
for each element of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mtext>input</mtext><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo>∗</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><msubsup><mtext>input</mtext><mi>i</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>4</mn><msup><mo stretchy="false">)</mo><mi>k</mi></msup></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">!</mo><mo stretchy="false">)</mo><mo>∗</mo><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">!</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text{out}_{i} = \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5489em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">!)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/4</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<dl>
<dt>Example::</dt><dd><div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">i1</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float32</span><span class="p">))</span>
<span class="go">tensor([0.0000, 0.5652, 1.5906, 3.9534, 9.7595])</span>
</pre></div>
</div>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.i1e">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">i1e</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.i1e" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the exponentially scaled first order modified Bessel function of the first kind (as defined below)
for each element of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>out</mtext><mi>i</mi></msub><mo>=</mo><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo><mo>∗</mo><mi>i</mi><mn>1</mn><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo><mo>∗</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mtext>input</mtext><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo>∗</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><msubsup><mtext>input</mtext><mi>i</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>4</mn><msup><mo stretchy="false">)</mo><mi>k</mi></msup></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">!</mo><mo stretchy="false">)</mo><mo>∗</mo><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">!</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\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)!}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">out</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">i</span><span class="mord">1</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5489em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">!)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">input</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-2.453em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/4</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<dl>
<dt>Example::</dt><dd><div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">i1e</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float32</span><span class="p">))</span>
<span class="go">tensor([0.0000, 0.2079, 0.2153, 0.1968, 0.1788])</span>
</pre></div>
</div>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.log1p">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">log1p</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.log1p" title="Permalink to this definition">¶</a></dt>
<dd><p>Alias for <a class="reference internal" href="generated/torch.log1p.html#torch.log1p" title="torch.log1p"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.log1p()</span></code></a>.</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="torch.special.log_ndtr">
<span class="sig-prename descclassname"><span class="pre">torch.special.</span></span><span class="sig-name descname"><span class="pre">log_ndtr</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a></span></span><a class="headerlink" href="#torch.special.log_ndtr" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the log of the area under the standard Gaussian probability density function,
integrated from minus infinity to <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>, elementwise.</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>log_ndtr</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>log</mi><mo></mo><mrow><mo fence="true">(</mo><mfrac><mn>1</mn><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup></mrow></msup><mi>d</mi><mi>t</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\text{log\_ndtr}(x) = \log\left(\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x} e^{-\frac{1}{2}t^2} dt \right)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.06em;vertical-align:-0.31em;"></span><span class="mord text"><span class="mord">log_ndtr</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4203em;vertical-align:-0.9703em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.2028em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9072em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span><span style="top:-2.8672em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1328em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></div><dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor.</p>
</dd>
<dt class="field-even">Keyword Arguments<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output tensor.</p>
</dd>
</dl>
<dl>