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Computationallinearalgebracourse.pdf

@@ -687,7 +687,7 @@ <h2><span class="section-number">2.6. </span>Householder triangulation<a class="
 done so, you will need to modified
 <a class="reference internal" href="cla_utils.html#cla_utils.exercises3.householder" title="cla_utils.exercises3.householder"><code class="xref py py-func docutils literal notranslate"><span class="pre">cla_utils.exercises3.householder()</span></code></a> to use the <code class="docutils literal notranslate"><span class="pre">kmax</span></code>
 argument. You may make use of the built-in triangular solve
-algorithm <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.linalg.solve_triangular.html#scipy.linalg.solve_triangular" title="(in SciPy v1.8.0.dev0+1987.d7fa27f)"><code class="xref py py-func docutils literal notranslate"><span class="pre">scipy.linalg.solve_triangular()</span></code></a> (we shall consider
+algorithm <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.linalg.solve_triangular.html#scipy.linalg.solve_triangular" title="(in SciPy v1.8.0.dev0+2093.40fc5b3)"><code class="xref py py-func docutils literal notranslate"><span class="pre">scipy.linalg.solve_triangular()</span></code></a> (we shall consider
 triangular matrix algorithms briefly later). The test script
 <code class="docutils literal notranslate"><span class="pre">test_exercises3.py</span></code> in the <code class="docutils literal notranslate"><span class="pre">test</span></code> directory will also test this
 function.</p>
@@ -799,7 +799,7 @@ <h2><span class="section-number">2.7. </span>Application: Least squares problems
 appropriate augmented matrix <span class="math notranslate nohighlight">\(\hat{A}\)</span>, calling
 <a class="reference internal" href="cla_utils.html#cla_utils.exercises3.householder" title="cla_utils.exercises3.householder"><code class="xref py py-func docutils literal notranslate"><span class="pre">cla_utils.exercises3.householder()</span></code></a> and extracting appropriate
 subarrays using slice notation, before using
-<a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.linalg.solve_triangular.html#scipy.linalg.solve_triangular" title="(in SciPy v1.8.0.dev0+1987.d7fa27f)"><code class="xref py py-func docutils literal notranslate"><span class="pre">scipy.linalg.solve_triangular()</span></code></a> to solve the resulting upper triangular
+<a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.linalg.solve_triangular.html#scipy.linalg.solve_triangular" title="(in SciPy v1.8.0.dev0+2093.40fc5b3)"><code class="xref py py-func docutils literal notranslate"><span class="pre">scipy.linalg.solve_triangular()</span></code></a> to solve the resulting upper triangular
 system, before returning the solution <span class="math notranslate nohighlight">\(x\)</span>. The test script
 <code class="docutils literal notranslate"><span class="pre">test_exercises3.py</span></code> in the <code class="docutils literal notranslate"><span class="pre">test</span></code> directory will also test this
 function.</p>

L2_QR_factorisation.html

@@ -66,8 +66,8 @@ <h2><span class="section-number">4.1. </span>An algorithm for LU decomposition<a
 <iframe allowfullscreen="true" src="https://player.vimeo.com/video/454095315" style="border: 0; height: 345px; width: 560px">
 </iframe></div></details><p>The computational way to view Gaussian elimination is through the LU
 decomposition of an invertible matrix, <span class="math notranslate nohighlight">\(A=LU\)</span>, where <span class="math notranslate nohighlight">\(L\)</span> is lower
-triangular (<span class="math notranslate nohighlight">\(l_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(j&lt;i\)</span>) and <span class="math notranslate nohighlight">\(U\)</span> is upper triangular
-(<span class="math notranslate nohighlight">\(u_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(j&gt;i\)</span>). Here we use the symbol <span class="math notranslate nohighlight">\(U\)</span> instead of <span class="math notranslate nohighlight">\(R\)</span> to
+triangular (<span class="math notranslate nohighlight">\(l_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(j&gt;i\)</span>) and <span class="math notranslate nohighlight">\(U\)</span> is upper triangular
+(<span class="math notranslate nohighlight">\(u_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(j&lt;i\)</span>). Here we use the symbol <span class="math notranslate nohighlight">\(U\)</span> instead of <span class="math notranslate nohighlight">\(R\)</span> to
 emphasise that we are looking as square matrices.  The process of
 obtaining the <span class="math notranslate nohighlight">\(LU\)</span> decomposition is very similar to the Householder
 algorithm, in that we repeatedly left multiply <span class="math notranslate nohighlight">\(A\)</span> by matrices to
@@ -121,7 +121,7 @@ <h2><span class="section-number">4.1. </span>An algorithm for LU decomposition<a
 <li><p><span class="math notranslate nohighlight">\(x_1 \gets b_1/L_{11}\)</span></p></li>
 <li><p>FOR <span class="math notranslate nohighlight">\(i= 2\)</span> TO <span class="math notranslate nohighlight">\(m\)</span></p>
 <ul>
-<li><p><span class="math notranslate nohighlight">\(x_i \gets (b_i - \sum_{k=1}^iL_{ik}x_k)/L_{ii}\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(x_i \gets (b_i - \sum_{k=1}^{i-1}L_{ik}x_k)/L_{ii}\)</span></p></li>
 </ul>
 </li>
 </ul>
@@ -519,12 +519,13 @@ <h2><span class="section-number">4.2. </span>Pivoting<a class="headerlink" href=
     </div><div class="proof-content">
 <p>The function <a class="reference internal" href="cla_utils.html#cla_utils.exercises7.perm" title="cla_utils.exercises7.perm"><code class="xref py py-func docutils literal notranslate"><span class="pre">cla_utils.exercises7.perm()</span></code></a> has been left
 unimplemented. It should take an <span class="math notranslate nohighlight">\(m\times m\)</span> permutation matrix
-<span class="math notranslate nohighlight">\(P\)</span>, stored as a vector of indices <span class="math notranslate nohighlight">\(p\in\mathbb{N}^m\)</span> so that
-<span class="math notranslate nohighlight">\((Px)_i = x_{p_i}\)</span>, <span class="math notranslate nohighlight">\(i=1,2,\ldots, m\)</span>, and replace it with the
-matrix <span class="math notranslate nohighlight">\(P_{i,j}P\)</span> (also stored as a vector of indices) where
-<span class="math notranslate nohighlight">\(P_{i,j}\)</span> is the permutation matrix that exchanges the entries <span class="math notranslate nohighlight">\(i\)</span>
-and <span class="math notranslate nohighlight">\(j\)</span>. The test script <code class="docutils literal notranslate"><span class="pre">test_exercises7.py</span></code> in the <code class="docutils literal notranslate"><span class="pre">test</span></code>
-directory will test this function.</p>
+<span class="math notranslate nohighlight">\(P\)</span>, stored as an (integer-valued) array of indices
+<span class="math notranslate nohighlight">\(p\in\mathbb{N}^m\)</span> so that <span class="math notranslate nohighlight">\((Px)_i = x_{p_i}\)</span>, <span class="math notranslate nohighlight">\(i=1,2,\ldots, m\)</span>,
+and replace it with the matrix <span class="math notranslate nohighlight">\(P_{i,j}P\)</span> (also stored as a array
+of indices) where <span class="math notranslate nohighlight">\(P_{i,j}\)</span> is the permutation matrix that
+exchanges the entries <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(j\)</span>. The test script
+<code class="docutils literal notranslate"><span class="pre">test_exercises7.py</span></code> in the <code class="docutils literal notranslate"><span class="pre">test</span></code> directory will test this
+function.</p>
 </div></div><div class="proof proof-type-exercise" id="id8">
 
     <div class="proof-title">

L4_LU_decomposition.html

@@ -499,7 +499,7 @@ <h2><span class="section-number">5.4. </span>Rayleigh quotient<a class="headerli
 <li><p>Finding an eigenvector <span class="math notranslate nohighlight">\(v\)</span> of <span class="math notranslate nohighlight">\(A\)</span> with eigenvalue <span class="math notranslate nohighlight">\(\lambda\)</span> (you can use <a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.linalg.eig.html#numpy.linalg.eig" title="(in NumPy v1.21)"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.linalg.eig()</span></code></a> for this),</p></li>
 <li><p>Choosing a perturbation vector <span class="math notranslate nohighlight">\(r\)</span>, and perturbation parameter <span class="math notranslate nohighlight">\(\epsilon&gt;0\)</span>,</p></li>
 <li><p>Comparing the Rayleigh quotient of <span class="math notranslate nohighlight">\(v + \epsilon r\)</span> with <span class="math notranslate nohighlight">\(\lambda\)</span>,</p></li>
-<li><p>Plotting (on a log-log graph, use <a class="reference external" href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.loglog.html#matplotlib.pyplot.loglog" title="(in Matplotlib v3.4.3)"><code class="xref py py-func docutils literal notranslate"><span class="pre">matplotlib.pyplot.loglog()</span></code></a>) the error in estimating the eigenvalue as a function of <span class="math notranslate nohighlight">\(\epsilon\)</span>.</p></li>
+<li><p>Plotting (on a log-log graph, use <a class="reference external" href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.loglog.html#matplotlib.pyplot.loglog" title="(in Matplotlib v3.5.0)"><code class="xref py py-func docutils literal notranslate"><span class="pre">matplotlib.pyplot.loglog()</span></code></a>) the error in estimating the eigenvalue as a function of <span class="math notranslate nohighlight">\(\epsilon\)</span>.</p></li>
 </ol>
 <p>The best way to do this is to plot the computed data values as points,
 and then superpose a line plot of <span class="math notranslate nohighlight">\(a\epsilon^k\)</span> for appropriate
@@ -773,7 +773,7 @@ <h2><span class="section-number">5.8. </span>The pure QR algorithm<a class="head
 <a class="reference internal" href="cla_utils.html#cla_utils.exercises9.get_B100" title="cla_utils.exercises9.get_B100"><code class="xref py py-func docutils literal notranslate"><span class="pre">cla_utils.exercises9.get_B100()</span></code></a>,
 <a class="reference internal" href="cla_utils.html#cla_utils.exercises9.get_C100" title="cla_utils.exercises9.get_C100"><code class="xref py py-func docutils literal notranslate"><span class="pre">cla_utils.exercises9.get_C100()</span></code></a>, and
 <a class="reference internal" href="cla_utils.html#cla_utils.exercises9.get_D100" title="cla_utils.exercises9.get_D100"><code class="xref py py-func docutils literal notranslate"><span class="pre">cla_utils.exercises9.get_D100()</span></code></a>. You can use
-<a class="reference external" href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.pcolor.html#matplotlib.pyplot.pcolor" title="(in Matplotlib v3.4.3)"><code class="xref py py-func docutils literal notranslate"><span class="pre">matplotlib.pyplot.pcolor()</span></code></a> to visualise the entries,
+<a class="reference external" href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.pcolor.html#matplotlib.pyplot.pcolor" title="(in Matplotlib v3.5.0)"><code class="xref py py-func docutils literal notranslate"><span class="pre">matplotlib.pyplot.pcolor()</span></code></a> to visualise the entries,
 or compute norms of the components of the matrices below the diagonal,
 for example. What do you observe? How does this relate to the structure
 of the four matrices?</p>

L5_eigenvalues.html

@@ -54,7 +54,7 @@ <h1>Source code for cla_utils.exercises6</h1><div class="highlight"><pre>
 <span class="sd">    in column k given by lvec (k is inferred from the size of lvec).</span>
 
 <span class="sd">    :param m: integer giving the dimensions of L.</span>
-<span class="sd">    :param lvec: a m-k-1 dimensional numpy array.</span>
+<span class="sd">    :param lvec: a m-k dimensional numpy array.</span>
 <span class="sd">    :return Lk: an mxm dimensional numpy array.</span>
 
 <span class="sd">    &quot;&quot;&quot;</span>

_modules/cla_utils/exercises6.html

@@ -21,8 +21,8 @@ An algorithm for LU decomposition
 
 The computational way to view Gaussian elimination is through the LU
 decomposition of an invertible matrix, `A=LU`, where `L` is lower
-triangular (`l_{ij}=0` for `j<i`) and `U` is upper triangular
-(`u_{ij}=0` for `j>i`). Here we use the symbol `U` instead of `R` to
+triangular (`l_{ij}=0` for `j>i`) and `U` is upper triangular
+(`u_{ij}=0` for `j<i`). Here we use the symbol `U` instead of `R` to
 emphasise that we are looking as square matrices.  The process of
 obtaining the `LU` decomposition is very similar to the Householder
 algorithm, in that we repeatedly left multiply `A` by matrices to
@@ -91,7 +91,7 @@ way, written in pseudo-code as follows.
 * `x_1 \gets b_1/L_{11}`
 * FOR `i= 2` TO `m`
 
-  * `x_i \gets (b_i - \sum_{k=1}^iL_{ik}x_k)/L_{ii}`
+  * `x_i \gets (b_i - \sum_{k=1}^{i-1}L_{ik}x_k)/L_{ii}`
 
 Forward substitution has an operation count that is identical to back
 substitution, by symmetry, i.e. `\mathcal{O}(m^2)`. In contrast, we
@@ -498,12 +498,13 @@ complete pivoting, `PAQ=LU`.
 
    The function :func:`cla_utils.exercises7.perm` has been left
    unimplemented. It should take an `m\times m` permutation matrix
-   `P`, stored as a vector of indices `p\in\mathbb{N}^m` so that
-   `(Px)_i = x_{p_i}`, `i=1,2,\ldots, m`, and replace it with the
-   matrix `P_{i,j}P` (also stored as a vector of indices) where
-   `P_{i,j}` is the permutation matrix that exchanges the entries `i`
-   and `j`. The test script ``test_exercises7.py`` in the ``test``
-   directory will test this function.
+   `P`, stored as an (integer-valued) array of indices
+   `p\in\mathbb{N}^m` so that `(Px)_i = x_{p_i}`, `i=1,2,\ldots, m`,
+   and replace it with the matrix `P_{i,j}P` (also stored as a array
+   of indices) where `P_{i,j}` is the permutation matrix that
+   exchanges the entries `i` and `j`. The test script
+   ``test_exercises7.py`` in the ``test`` directory will test this
+   function.
 
 
 .. proof:exercise::

_sources/L4_LU_decomposition.rst.txt

@@ -16,7 +16,19 @@ repositories for Autumn 2021. An up to date version is in the
 2. The skeleton code for exercises2.GS_classical was amended as the
    function should just return R, not Q and R.
 
-3. Updated the sign of the inequality in the formula for $r$ in Section 2.2.
+3. Updated the sign of the inequality in the formula for `r` in Section 2.2.
 
-4. Replaced $r_{11}/r_{11}$ by $r_{1n}/r_{11}$ in the final column of the
-   right-multiplied $R$ matrix in Section 2.5.
+4. Replaced `r_{11}/r_{11}` by `r_{1n}/r_{11}` in the final column of the
+   right-multiplied `R` matrix in Section 2.5.
+
+5. Updated docstring in :code:`cla_utils:get_Lk` and the corresponding
+   test so that `k` corresponds to the column being operated on.
+
+6. Corrected the upper limit of the sum in the algorithm for forward
+   substitution.
+
+7. Corrected the sign of inequalities in :code:`test_solve_U` and
+   :code:`test_solve_L`. And then uncorrected them again, they were right
+   in the first place.
+
+8. Fixed the sign of the inequalities in the discussion of $L$ and $U$ matrices.

_sources/errata.rst.txt

@@ -621,7 +621,7 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 <dt class="field-odd">Parameters</dt>
 <dd class="field-odd"><ul class="simple">
 <li><p><strong>m</strong> – integer giving the dimensions of L.</p></li>
-<li><p><strong>lvec</strong> – a m-k-1 dimensional numpy array.</p></li>
+<li><p><strong>lvec</strong> – a m-k dimensional numpy array.</p></li>
 </ul>
 </dd>
 <dt class="field-even">Return Lk</dt>

cla_utils.html

@@ -15,6 +15,7 @@
     <script src="_static/underscore.js"></script>
     <script src="_static/doctools.js"></script>
     <script src="_static/proof.js"></script>
+    <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
     <link rel="index" title="Index" href="genindex.html" />
     <link rel="search" title="Search" href="search.html" />
     <link rel="next" title="1. Preliminaries" href="L1_preliminaries.html" />
@@ -59,9 +60,17 @@ <h1>Errata for 2021/22<a class="headerlink" href="#errata-for-2021-22" title="Pe
 version.</p></li>
 <li><p>The skeleton code for exercises2.GS_classical was amended as the
 function should just return R, not Q and R.</p></li>
-<li><p>Updated the sign of the inequality in the formula for $r$ in Section 2.2.</p></li>
-<li><p>Replaced $r_{11}/r_{11}$ by $r_{1n}/r_{11}$ in the final column of the
-right-multiplied $R$ matrix in Section 2.5.</p></li>
+<li><p>Updated the sign of the inequality in the formula for <span class="math notranslate nohighlight">\(r\)</span> in Section 2.2.</p></li>
+<li><p>Replaced <span class="math notranslate nohighlight">\(r_{11}/r_{11}\)</span> by <span class="math notranslate nohighlight">\(r_{1n}/r_{11}\)</span> in the final column of the
+right-multiplied <span class="math notranslate nohighlight">\(R\)</span> matrix in Section 2.5.</p></li>
+<li><p>Updated docstring in <code class="code docutils literal notranslate"><span class="pre">cla_utils:get_Lk</span></code> and the corresponding
+test so that <span class="math notranslate nohighlight">\(k\)</span> corresponds to the column being operated on.</p></li>
+<li><p>Corrected the upper limit of the sum in the algorithm for forward
+substitution.</p></li>
+<li><p>Corrected the sign of inequalities in <code class="code docutils literal notranslate"><span class="pre">test_solve_U</span></code> and
+<code class="code docutils literal notranslate"><span class="pre">test_solve_L</span></code>. And then uncorrected them again, they were right
+in the first place.</p></li>
+<li><p>Fixed the sign of the inequalities in the discussion of $L$ and $U$ matrices.</p></li>
 </ol>
 </section>