Website build

Author: colinjcotter

Computationallinearalgebracourse.pdf

@@ -812,7 +812,7 @@ <h2><span class="section-number">5.9. </span>Simultaneous iteration<a class="hea
 first one of the remaining projected vectors as an approximation of
 <span class="math notranslate nohighlight">\(q_2\)</span> and project it again from the rest.</p>
 <p>We can translate this idea to matrices by defining <span class="math notranslate nohighlight">\(V^{(0)}\)</span> to be the
-matrix with columns given by the set of initial <span class="math notranslate nohighlight">\(v`s. Then after `k\)</span>
+matrix with columns given by the set of initial <span class="math notranslate nohighlight">\(v\)</span> s. Then after <span class="math notranslate nohighlight">\(k\)</span>
 applications of <span class="math notranslate nohighlight">\(A\)</span>, we have <span class="math notranslate nohighlight">\(V^{(k)}=A^{k} V^{(0)}\)</span>. By the column space
 interpretation of matrix-matrix multiplication, each column of <span class="math notranslate nohighlight">\(V^{(k)}\)</span>
 is <span class="math notranslate nohighlight">\(A^{k}\)</span> multiplied by the corresponding column of <span class="math notranslate nohighlight">\(V^{(0)}\)</span>. To make the
@@ -962,12 +962,12 @@ <h2><span class="section-number">5.11. </span>Connections between power iteratio
 <div class="math notranslate nohighlight">
 \[A^k = {Q'}^{(k)}{R'}^{(k)},\]</div>
 <p>from the above theorem. (Remember that <span class="math notranslate nohighlight">\({Q'}^{(k)}\)</span> and <span class="math notranslate nohighlight">\({R'}^{(k)}\)</span>,
-are different from <span class="math notranslate nohighlight">\({Q'}^{(k)}\)</span> and <span class="math notranslate nohighlight">\({R'}^{(k)}\)</span>.)  In particular, the
-first column of <span class="math notranslate nohighlight">\(R^{(k)}\)</span> is <span class="math notranslate nohighlight">\(e_1r_{11}^{(k)}\)</span> (because <span class="math notranslate nohighlight">\(R^{(k)}\)</span> is
+are different from <span class="math notranslate nohighlight">\({Q}^{(k)}\)</span> and <span class="math notranslate nohighlight">\({R}^{(k)}\)</span>.)  In particular, the
+first column of <span class="math notranslate nohighlight">\({R'}^{(k)}\)</span> is <span class="math notranslate nohighlight">\(e_1r_{11}^{(k)}\)</span> (because <span class="math notranslate nohighlight">\({R'}^{(k)}\)</span> is
 an upper triangular matrix), so the first column of <span class="math notranslate nohighlight">\(A^k\)</span> is</p>
 <div class="math notranslate nohighlight">
-\[A^ke_1 = r_{11}^{(k)}Q^{(k)}e_1.\]</div>
-<p>In other words, the first column of <span class="math notranslate nohighlight">\(Q^{(k)}\)</span> is the result of <span class="math notranslate nohighlight">\(k\)</span>
+\[A^ke_1 = r_{11}^{(k)}{Q'}^{(k)}e_1.\]</div>
+<p>In other words, the first column of <span class="math notranslate nohighlight">\({Q'}^{(k)}\)</span> is the result of <span class="math notranslate nohighlight">\(k\)</span>
 iterations of power iteration starting at <span class="math notranslate nohighlight">\(e_1\)</span>. (We already knew this
 from the previous theorem, but here we are introducing ways to look at
 different components of <span class="math notranslate nohighlight">\({Q'}^{(k)}\)</span> and <span class="math notranslate nohighlight">\({R'}^{(k)}\)</span>). This means that

L5_eigenvalues.html

@@ -16,7 +16,7 @@
     <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="cla_utils package" href="cla_utils.html" />
+    <link rel="next" title="7. Preconditioning Krylov methods" href="L7_preconditioning.html" />
     <link rel="prev" title="5. Finding eigenvalues of matrices" href="L5_eigenvalues.html" />
 <!--[if lte IE 6]>
 <link rel="stylesheet" href="_static/ie6.css" type="text/css" media="screen" charset="utf-8" />
@@ -126,7 +126,7 @@ <h2><span class="section-number">6.2. </span>Arnoldi iteration<a class="headerli
 we see that the <span class="math notranslate nohighlight">\(n\)</span>-th column is</p>
 <blockquote>
 <div><div class="math notranslate nohighlight">
-\[Aq_n = h_{1n}q_1 + h_{2n}q_n + \ldots + h_{n,n}q_n + h_{n+1,n}q_{n+1}.\]</div>
+\[Aq_n = h_{1n}q_1 + h_{2n}q_2 + \ldots + h_{n,n}q_n + h_{n+1,n}q_{n+1}.\]</div>
 </div></blockquote>
 <p>This formula shows us how to construct the non-zero entries of the
 nth column of <span class="math notranslate nohighlight">\(H\)</span>; this defines the Arnoldi algorithm which we