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finite-difference.html
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finite-difference.html
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<li class="toctree-l3"><a class="reference internal" href="#devito.finite_differences.finite_difference.cross_derivative"><code class="docutils literal notranslate"><span class="pre">cross_derivative()</span></code></a></li>
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<li class="toctree-l3"><a class="reference internal" href="#devito.finite_differences.finite_difference.generic_derivative"><code class="docutils literal notranslate"><span class="pre">generic_derivative()</span></code></a></li>
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<section id="module-devito.finite_differences.finite_difference">
<span id="finite-difference-approximations"></span><h1>Finite-difference approximations<a class="headerlink" href="#module-devito.finite_differences.finite_difference" title="Link to this heading"></a></h1>
<dl class="py function">
<dt class="sig sig-object py" id="devito.finite_differences.finite_difference.cross_derivative">
<span class="sig-prename descclassname"><span class="pre">devito.finite_differences.finite_difference.</span></span><span class="sig-name descname"><span class="pre">cross_derivative</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">expr</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dims</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fd_order</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">deriv_order</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">kwargs</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/devito/finite_differences/finite_difference.html#cross_derivative"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#devito.finite_differences.finite_difference.cross_derivative" title="Link to this definition"></a></dt>
<dd><p>Arbitrary-order cross derivative of a given expression.</p>
<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>expr</strong> (<em>expr-like</em>) – Expression for which the cross derivative is produced.</p></li>
<li><p><strong>dims</strong> (<em>tuple</em><em> of </em><a class="reference internal" href="dimension.html#devito.types.dimension.Dimension" title="devito.types.dimension.Dimension"><em>Dimension</em></a>) – Dimensions w.r.t. which to differentiate.</p></li>
<li><p><strong>fd_order</strong> (<em>tuple</em><em> of </em><em>ints</em>) – Coefficient discretization order. Note: this impacts the width of
the resulting stencil.</p></li>
<li><p><strong>deriv_order</strong> (<em>tuple</em><em> of </em><em>ints</em>) – Derivative order, e.g. 2 for a second-order derivative.</p></li>
<li><p><strong>matvec</strong> (<em>Transpose</em><em>, </em><em>optional</em>) – Forward (matvec=direct) or transpose (matvec=transpose) mode of the
finite difference. Defaults to <cite>direct</cite>.</p></li>
<li><p><strong>x0</strong> (<em>dict</em><em>, </em><em>optional</em>) – Origin of the finite-difference scheme as a map dim: origin_dim.</p></li>
<li><p><strong>symbolic</strong> (<em>bool</em><em>, </em><em>optional</em>) – Use default or custom coefficients (weights). Defaults to False.</p></li>
<li><p><strong>expand</strong> (<em>bool</em><em>, </em><em>optional</em>) – If True, the derivative is fully expanded as a sum of products,
otherwise an IndexSum is returned. Defaults to True.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>Cross-derivative of <code class="docutils literal notranslate"><span class="pre">expr</span></code>.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>expr-like</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">devito</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Grid</span>
<span class="gp">>>> </span><span class="n">grid</span> <span class="o">=</span> <span class="n">Grid</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">4</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="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">grid</span><span class="o">.</span><span class="n">dimensions</span>
<span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">name</span><span class="o">=</span><span class="s1">'f'</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">,</span> <span class="n">space_order</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">name</span><span class="o">=</span><span class="s1">'g'</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">,</span> <span class="n">space_order</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">cross_derivative</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">fd_order</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">deriv_order</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="go">(-1/h_y)*(-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x) + (-f(x, y + h_y)*g(x, y + h_y)/h_x + f(x + h_x, y + h_y)*g(x + h_x, y + h_y)/h_x)/h_y</span>
</pre></div>
</div>
<p>Semantically, this is equivalent to</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">dxdy</span>
<span class="go">Derivative(f(x, y)*g(x, y), x, y)</span>
</pre></div>
</div>
<p>The only difference is that in the latter case derivatives remain unevaluated.
The expanded form is obtained via <code class="docutils literal notranslate"><span class="pre">evaluate</span></code></p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">dxdy</span><span class="o">.</span><span class="n">evaluate</span>
<span class="go">(-1/h_y)*(-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x) + (-f(x, y + h_y)*g(x, y + h_y)/h_x + f(x + h_x, y + h_y)*g(x + h_x, y + h_y)/h_x)/h_y</span>
</pre></div>
</div>
<p>Finally the x0 argument allows to choose the origin of the finite-difference</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">cross_derivative</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">,</span> <span class="n">dims</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">fd_order</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">deriv_order</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x0</span><span class="o">=</span><span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">2</span><span class="p">})</span>
<span class="go">(-1/h_y)*(-f(1, 2)*g(1, 2)/h_x + f(h_x + 1, 2)*g(h_x + 1, 2)/h_x) + (-f(1, h_y + 2)*g(1, h_y + 2)/h_x + f(h_x + 1, h_y + 2)*g(h_x + 1, h_y + 2)/h_x)/h_y</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="devito.finite_differences.finite_difference.first_derivative">
<span class="sig-prename descclassname"><span class="pre">devito.finite_differences.finite_difference.</span></span><span class="sig-name descname"><span class="pre">first_derivative</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">expr</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fd_order</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">side</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">centered[0]</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">matvec</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">direct[1]</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">symbolic</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">expand</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/devito/finite_differences/finite_difference.html#first_derivative"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#devito.finite_differences.finite_difference.first_derivative" title="Link to this definition"></a></dt>
<dd><p>First-order derivative of a given expression.</p>
<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>expr</strong> (<em>expr-like</em>) – Expression for which the first-order derivative is produced.</p></li>
<li><p><strong>dim</strong> (<a class="reference internal" href="dimension.html#devito.types.dimension.Dimension" title="devito.types.dimension.Dimension"><em>Dimension</em></a>) – The Dimension w.r.t. which to differentiate.</p></li>
<li><p><strong>fd_order</strong> (<em>int</em><em>, </em><em>optional</em>) – Coefficient discretization order. Note: this impacts the width of
the resulting stencil. Defaults to <cite>expr.space_order</cite>.</p></li>
<li><p><strong>side</strong> (<em>Side</em><em>, </em><em>optional</em>) – Side of the finite difference location, centered (at x), left (at x - 1)
or right (at x +1). Defaults to <cite>centered</cite>.</p></li>
<li><p><strong>matvec</strong> (<em>Transpose</em><em>, </em><em>optional</em>) – Forward (matvec=direct) or transpose (matvec=transpose) mode of the
finite difference. Defaults to <cite>direct</cite>.</p></li>
<li><p><strong>x0</strong> (<em>dict</em><em>, </em><em>optional</em>) – Origin of the finite-difference scheme as a map dim: origin_dim.</p></li>
<li><p><strong>symbolic</strong> (<em>bool</em><em>, </em><em>optional</em>) – Use default or custom coefficients (weights). Defaults to False.</p></li>
<li><p><strong>expand</strong> (<em>bool</em><em>, </em><em>optional</em>) – If True, the derivative is fully expanded as a sum of products,
otherwise an IndexSum is returned. Defaults to True.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>First-order derivative of <code class="docutils literal notranslate"><span class="pre">expr</span></code>.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>expr-like</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">devito</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Grid</span><span class="p">,</span> <span class="n">first_derivative</span><span class="p">,</span> <span class="n">transpose</span>
<span class="gp">>>> </span><span class="n">grid</span> <span class="o">=</span> <span class="n">Grid</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">4</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="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">grid</span><span class="o">.</span><span class="n">dimensions</span>
<span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">name</span><span class="o">=</span><span class="s1">'f'</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">name</span><span class="o">=</span><span class="s1">'g'</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">first_derivative</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="n">x</span><span class="p">)</span>
<span class="go">-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x</span>
</pre></div>
</div>
<p>Semantically, this is equivalent to</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">dx</span>
<span class="go">Derivative(f(x, y)*g(x, y), x)</span>
</pre></div>
</div>
<p>The only difference is that in the latter case derivatives remain unevaluated.
The expanded form is obtained via <code class="docutils literal notranslate"><span class="pre">evaluate</span></code></p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">dx</span><span class="o">.</span><span class="n">evaluate</span>
<span class="go">-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x</span>
</pre></div>
</div>
<p>For the adjoint mode of the first derivative, pass <code class="docutils literal notranslate"><span class="pre">matvec=transpose</span></code></p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">name</span><span class="o">=</span><span class="s1">'g'</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">first_derivative</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">matvec</span><span class="o">=</span><span class="n">transpose</span><span class="p">)</span>
<span class="go">-f(x, y)*g(x, y)/h_x + f(x - h_x, y)*g(x - h_x, y)/h_x</span>
</pre></div>
</div>
<p>This is also accessible via the .T shortcut</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">dx</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">evaluate</span>
<span class="go">-f(x, y)*g(x, y)/h_x + f(x - h_x, y)*g(x - h_x, y)/h_x</span>
</pre></div>
</div>
<p>Finally the x0 argument allows to choose the origin of the finite-difference</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">first_derivative</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="p">})</span>
<span class="go">-f(1, y)/h_x + f(h_x + 1, y)/h_x</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="devito.finite_differences.finite_difference.generate_indices">
<span class="sig-prename descclassname"><span class="pre">devito.finite_differences.finite_difference.</span></span><span class="sig-name descname"><span class="pre">generate_indices</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">expr</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">order</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">side</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">matvec</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</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><a class="reference internal" href="_modules/devito/finite_differences/tools.html#generate_indices"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#devito.finite_differences.finite_difference.generate_indices" title="Link to this definition"></a></dt>
<dd><p>Indices for the finite-difference scheme.</p>
<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>expr</strong> (<em>expr-like</em>) – Expression that is differentiated.</p></li>
<li><p><strong>dim</strong> (<a class="reference internal" href="dimension.html#devito.types.dimension.Dimension" title="devito.types.dimension.Dimension"><em>Dimension</em></a>) – Dimensions w.r.t which the derivative is taken.</p></li>
<li><p><strong>order</strong> (<em>int</em>) – Order of the finite-difference scheme.</p></li>
<li><p><strong>side</strong> (<em>Side</em><em>, </em><em>optional</em>) – Side of the scheme (centered, left, right).</p></li>
<li><p><strong>matvec</strong> (<em>Transpose</em><em>, </em><em>optional</em>) – Forward (matvec=direct) or transpose (matvec=transpose) mode of the
finite difference. Defaults to <cite>direct</cite>.</p></li>
<li><p><strong>x0</strong> (<em>dict</em><em> of </em><em>{Dimension: Dimension</em><em> or </em><em>Expr</em><em> or </em><em>Number}</em><em>, </em><em>optional</em>) – Origin of the scheme, ie. <cite>x</cite>, <cite>x + .5 * x.spacing</cite>, …</p></li>
</ul>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p>An IndexSet, representing an ordered list of indices.</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="devito.finite_differences.finite_difference.generic_derivative">
<span class="sig-prename descclassname"><span class="pre">devito.finite_differences.finite_difference.</span></span><span class="sig-name descname"><span class="pre">generic_derivative</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">expr</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fd_order</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">deriv_order</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">matvec</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">direct[1]</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">symbolic</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">expand</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/devito/finite_differences/finite_difference.html#generic_derivative"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#devito.finite_differences.finite_difference.generic_derivative" title="Link to this definition"></a></dt>
<dd><p>Arbitrary-order derivative of a given expression.</p>
<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>expr</strong> (<em>expr-like</em>) – Expression for which the derivative is produced.</p></li>
<li><p><strong>dim</strong> (<a class="reference internal" href="dimension.html#devito.types.dimension.Dimension" title="devito.types.dimension.Dimension"><em>Dimension</em></a>) – The Dimension w.r.t. which to differentiate.</p></li>
<li><p><strong>fd_order</strong> (<em>int</em>) – Coefficient discretization order. Note: this impacts the width of
the resulting stencil.</p></li>
<li><p><strong>deriv_order</strong> (<em>int</em>) – Derivative order, e.g. 2 for a second-order derivative.</p></li>
<li><p><strong>matvec</strong> (<em>Transpose</em><em>, </em><em>optional</em>) – Forward (matvec=direct) or transpose (matvec=transpose) mode of the
finite difference. Defaults to <cite>direct</cite>.</p></li>
<li><p><strong>x0</strong> (<em>dict</em><em>, </em><em>optional</em>) – Origin of the finite-difference scheme as a map dim: origin_dim.</p></li>
<li><p><strong>symbolic</strong> (<em>bool</em><em>, </em><em>optional</em>) – Use default or custom coefficients (weights). Defaults to False.</p></li>
<li><p><strong>expand</strong> (<em>bool</em><em>, </em><em>optional</em>) – If True, the derivative is fully expanded as a sum of products,
otherwise an IndexSum is returned. Defaults to True.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><code class="docutils literal notranslate"><span class="pre">deriv-order</span></code> derivative of <code class="docutils literal notranslate"><span class="pre">expr</span></code>.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>expr-like</p>
</dd>
</dl>
</dd></dl>
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