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1_quadrature.html

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@@ -324,7 +324,7 @@ <h2><span class="section-number">1.8. </span>Implementing quadrature rules in Py
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<p>You can implement
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<code class="xref py py-meth docutils literal notranslate"><span class="pre">integrate()</span></code> in one line
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using a <a class="reference external" href="https://docs.python.org/3/tutorial/datastructures.html#list-comprehensions">list
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comprehension</a> and <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot" title="(in NumPy v1.24)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.dot()</span></code></a>.</p>
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comprehension</a> and <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot" title="(in NumPy v1.26)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.dot()</span></code></a>.</p>
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</div>
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<div class="admonition hint">
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<p class="admonition-title">Hint</p>

2_finite_elements.html

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@@ -361,7 +361,7 @@ <h2><span class="section-number">2.5. </span>Implementing finite elements in Pyt
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<code class="docutils literal notranslate"><span class="pre">test/test_04_init_finite_element.py</span></code>.</p>
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</div></div><div class="admonition hint">
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<p class="admonition-title">Hint</p>
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<p>The <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.linalg.inv.html#numpy.linalg.inv" title="(in NumPy v1.24)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.linalg.inv()</span></code></a> function may be
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<p>The <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.linalg.inv.html#numpy.linalg.inv" title="(in NumPy v1.26)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.linalg.inv()</span></code></a> function may be
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used to invert the matrix.</p>
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</div>
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</section>
@@ -507,15 +507,15 @@ <h2><span class="section-number">2.8. </span>Gradients of basis functions<a clas
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<code class="docutils literal notranslate"><span class="pre">test/test_03_vandermonde_matrix.py</span></code>.</p>
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</div></div><div class="admonition hint">
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<p class="admonition-title">Hint</p>
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<p>The <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.transpose.html#numpy.ndarray.transpose" title="(in NumPy v1.24)"><code class="xref py py-meth docutils literal notranslate"><span class="pre">transpose()</span></code></a> method of numpy arrays enables
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<p>The <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.transpose.html#numpy.ndarray.transpose" title="(in NumPy v1.26)"><code class="xref py py-meth docutils literal notranslate"><span class="pre">transpose()</span></code></a> method of numpy arrays enables
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generalised transposes swapping any dimensions.</p>
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</div>
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<div class="admonition hint">
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<p class="admonition-title">Hint</p>
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<p>At least one of the natural ways of implementing this function
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results in a whole load of <code class="xref py py-data docutils literal notranslate"><span class="pre">nan</span></code> values in the generalised
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Vandermonde matrix. In this case, you might find
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<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.nan_to_num.html#numpy.nan_to_num" title="(in NumPy v1.24)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.nan_to_num()</span></code></a> useful.</p>
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<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.nan_to_num.html#numpy.nan_to_num" title="(in NumPy v1.26)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.nan_to_num()</span></code></a> useful.</p>
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</div>
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<div class="proof proof-type-exercise" id="id13">
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@@ -552,7 +552,7 @@ <h2><span class="section-number">2.8. </span>Gradients of basis functions<a clas
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</div>
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</details><div class="admonition hint">
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<p class="admonition-title">Hint</p>
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<p>The <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.einsum.html#numpy.einsum" title="(in NumPy v1.24)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.einsum()</span></code></a> function implements generalised tensor
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<p>The <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.einsum.html#numpy.einsum" title="(in NumPy v1.26)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.einsum()</span></code></a> function implements generalised tensor
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contractions using <a class="reference external" href="http://mathworld.wolfram.com/EinsteinSummation.html">Einstein summation notation</a>. For
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example:</p>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">A</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s2">&quot;ijk,jl-&gt;ilk&quot;</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span>

4_function_spaces.html

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@@ -334,11 +334,11 @@ <h2><span class="section-number">4.5. </span>Implementing function spaces in Pyt
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<div class="admonition hint">
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<p class="admonition-title">Hint</p>
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<p><code class="xref py py-attr docutils literal notranslate"><span class="pre">cell_nodes</span></code> needs to
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be integer-valued. If you choose to use <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="(in NumPy v1.24)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.zeros()</span></code></a>
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be integer-valued. If you choose to use <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="(in NumPy v1.26)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.zeros()</span></code></a>
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to create a matrix which you then populate with values, you
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need to explicitly specify that you want a matrix of
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integers. This can be achieved by passing the <code class="docutils literal notranslate"><span class="pre">dtype</span></code> argument
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to <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="(in NumPy v1.24)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.zeros()</span></code></a>. For example <code class="docutils literal notranslate"><span class="pre">numpy.zeros((nrows,</span> <span class="pre">ncols),</span> <span class="pre">dtype=int)</span></code>.</p>
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to <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="(in NumPy v1.26)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.zeros()</span></code></a>. For example <code class="docutils literal notranslate"><span class="pre">numpy.zeros((nrows,</span> <span class="pre">ncols),</span> <span class="pre">dtype=int)</span></code>.</p>
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</div>
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<p class="rubric">Footnotes</p>
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<dl class="footnote brackets">

5_functions.html

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@@ -385,10 +385,10 @@ <h3><span class="section-number">5.3.3. </span>Implementing integration<a class=
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basis functions at each quadrature point.</p></li>
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<li><p>Visit each cell in turn.</p></li>
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<li><p>Construct the <code class="xref py py-meth docutils literal notranslate"><span class="pre">jacobian()</span></code> for that cell
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and take the absolute value of its determinant (<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.absolute.html#numpy.absolute" title="(in NumPy v1.24)"><code class="xref py py-data docutils literal notranslate"><span class="pre">numpy.absolute</span></code></a>
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and <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.linalg.det.html#numpy.linalg.det" title="(in NumPy v1.24)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.linalg.det()</span></code></a> will be useful here).</p></li>
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and take the absolute value of its determinant (<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.absolute.html#numpy.absolute" title="(in NumPy v1.26)"><code class="xref py py-data docutils literal notranslate"><span class="pre">numpy.absolute</span></code></a>
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and <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.linalg.det.html#numpy.linalg.det" title="(in NumPy v1.26)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.linalg.det()</span></code></a> will be useful here).</p></li>
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<li><p>Sum all of the arrays you have constructed over the correct
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indices to a contribution to the integral (<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.einsum.html#numpy.einsum" title="(in NumPy v1.24)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.einsum()</span></code></a>
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indices to a contribution to the integral (<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.einsum.html#numpy.einsum" title="(in NumPy v1.26)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.einsum()</span></code></a>
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may be useful for this).</p></li>
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</ol>
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</div>

6_finite_element_problems.html

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@@ -256,11 +256,11 @@ <h3><span class="section-number">6.2.3. </span>A note on matrix insertion<a clas
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">A</span><span class="p">[</span><span class="n">nodes</span><span class="p">,</span> <span class="n">nodes</span><span class="p">]</span> <span class="o">+=</span> <span class="n">m</span> <span class="c1"># DON&#39;T DO THIS!</span>
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</pre></div>
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</div>
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<p>Unfortunately, <a class="reference external" href="https://numpy.org/doc/stable/reference/index.html#module-numpy" title="(in NumPy v1.24)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">numpy</span></code></a> interprets this as an instruction to
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<p>Unfortunately, <a class="reference external" href="https://numpy.org/doc/stable/reference/index.html#module-numpy" title="(in NumPy v1.26)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">numpy</span></code></a> interprets this as an instruction to
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insert a vector into the diagonal of <code class="docutils literal notranslate"><span class="pre">A</span></code>, and will complain that
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the two-dimensional right hand side does not match the
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one-dimensional left hand side. Instead, one has to employ the
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<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ix_.html#numpy.ix_" title="(in NumPy v1.24)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.ix_()</span></code></a> function:</p>
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<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ix_.html#numpy.ix_" title="(in NumPy v1.26)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.ix_()</span></code></a> function:</p>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">A</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ix_</span><span class="p">(</span><span class="n">nodes</span><span class="p">,</span> <span class="n">nodes</span><span class="p">)]</span> <span class="o">+=</span> <span class="n">m</span> <span class="c1"># DO THIS!</span>
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</pre></div>
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are avoided.</p>
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<div class="admonition hint">
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<p class="admonition-title">Hint</p>
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<p>The <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.11.0.dev0+2027.1651835)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a> package provides convenient interfaces
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<p>The <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.13.0.dev)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a> package provides convenient interfaces
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which enable Python code to employ a variety of sparse matrix
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formats using essentially identical operations to the dense matrix
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case. The skeleton code already contains commands to construct
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empty sparse matrices and to solve the resulting linear system. You
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may, if you wish, experiment with choosing other sparse formats
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from <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.11.0.dev0+2027.1651835)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a>, but it is very strongly suggested that
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from <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.13.0.dev)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a>, but it is very strongly suggested that
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you do <strong>not</strong> switch to a dense numpy array; unless, that is, you
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particularly enjoy running out of memory on your computer!</p>
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</div>

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