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<div class="section" id="torch-linalg-solve">
<h1>torch.linalg.solve<a class="headerlink" href="#torch-linalg-solve" title="Permalink to this headline">¶</a></h1>
<dl class="py function">
<dt id="torch.linalg.solve">
<code class="sig-prename descclassname"><span class="pre">torch.linalg.</span></code><code class="sig-name descname"><span class="pre">solve</span></code><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">B</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="pre">Tensor</span><a class="headerlink" href="#torch.linalg.solve" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the solution of a square system of linear equations with a unique solution.</p>
<p>Letting <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">K</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68889em;vertical-align:0em;"></span><span class="mord mathbb">K</span></span></span></span></span> be <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68889em;vertical-align:0em;"></span><span class="mord mathbb">R</span></span></span></span></span> or <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68889em;vertical-align:0em;"></span><span class="mord mathbb">C</span></span></span></span></span>,
this function computes the solution <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∈</mo><msup><mi mathvariant="double-struck">K</mi><mrow><mi>n</mi><mo>×</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">X \in \mathbb{K}^{n \times k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbb">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><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 mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span></span> of the <strong>linear system</strong> associated to
<span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi mathvariant="double-struck">K</mi><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup><mo separator="true">,</mo><mi>B</mi><mo>∈</mo><msup><mi mathvariant="double-struck">K</mi><mrow><mi>m</mi><mo>×</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A \in \mathbb{K}^{n \times n}, B \in \mathbb{K}^{m \times k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9657709999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathbb">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.771331em;"><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 mathnormal mtight">n</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbb">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><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 mathnormal mtight">m</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span></span>, which is defined 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><mi>A</mi><mi>X</mi><mo>=</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">AX = B
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></span></div><p>This system of linear equations has one solution if and only if <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span></span></span></span></span> is <a class="reference external" href="https://en.wikipedia.org/wiki/Invertible_matrix#The_invertible_matrix_theorem">invertible</a>.
This function assumes that <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span></span></span></span></span> is invertible.</p>
<p>Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batches of matrices, and if the inputs are batches of matrices then
the output has the same batch dimensions.</p>
<p>Letting <cite>*</cite> be zero or more batch dimensions,</p>
<ul class="simple">
<li><p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code> has shape <cite>(*, n, n)</cite> and <code class="xref py py-attr docutils literal notranslate"><span class="pre">B</span></code> has shape <cite>(*, n)</cite> (a batch of vectors) or shape
<cite>(*, n, k)</cite> (a batch of matrices or “multiple right-hand sides”), this function returns <cite>X</cite> of shape
<cite>(*, n)</cite> or <cite>(*, n, k)</cite> respectively.</p></li>
<li><p>Otherwise, if <code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code> has shape <cite>(*, n, n)</cite> and <code class="xref py py-attr docutils literal notranslate"><span class="pre">B</span></code> has shape <cite>(n,)</cite> or <cite>(n, k)</cite>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">B</span></code>
is broadcasted to have shape <cite>(*, n)</cite> or <cite>(*, n, k)</cite> respectively.
This function then returns the solution of the resulting batch of systems of linear equations.</p></li>
</ul>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function computes <cite>X = </cite><code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code><cite>.inverse() @ </cite><code class="xref py py-attr docutils literal notranslate"><span class="pre">B</span></code> in a faster and
more numerically stable way than performing the computations separately.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When inputs are on a CUDA device, this function synchronizes that device with the CPU.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>A</strong> (<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – tensor of shape <cite>(*, n, n)</cite> where <cite>*</cite> is zero or more batch dimensions.</p></li>
<li><p><strong>B</strong> (<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – right-hand side tensor of shape <cite>(*, n)</cite> or <cite>(*, n, k)</cite> or <cite>(n,)</cite> or <cite>(n, k)</cite>
according to the rules described above</p></li>
</ul>
</dd>
<dt class="field-even">Keyword Arguments</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>) – output tensor. Ignored if <cite>None</cite>. Default: <cite>None</cite>.</p>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><p><a class="reference external" href="https://docs.python.org/3/library/exceptions.html#RuntimeError" title="(in Python v3.10)"><strong>RuntimeError</strong></a> – if the <code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code> matrix is not invertible or any matrix in a batched <code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code>
is not invertible.</p>
</dd>
</dl>
<p>Examples:</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">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</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">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">x</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="go">True</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">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</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">randn</span><span class="p">(</span><span class="mi">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="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X</span><span class="o">.</span><span class="n">shape</span>
<span class="go">torch.Size([2, 3, 4])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">X</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
<span class="go">True</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">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</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">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="c1"># b is broadcasted to size (2, 3, 1)</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">shape</span>
<span class="go">torch.Size([2, 3, 1])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">A</span> <span class="o">@</span> <span class="n">x</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="go">True</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">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="c1"># b is broadcasted to size (2, 3)</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">shape</span>
<span class="go">torch.Size([2, 3])</span>
<span class="gp">>>> </span><span class="n">Ax</span> <span class="o">=</span> <span class="n">A</span> <span class="o">@</span> <span class="n">x</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">Ax</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">expand_as</span><span class="p">(</span><span class="n">Ax</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
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