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2018-06-08 13:05:59 -07:00
parent f5fe7d9726
commit b98e9f2107
18 changed files with 463 additions and 2825 deletions
@@ -157,6 +157,9 @@
 <span class="s1">dudipd == dup dipd</span>
 <span class="s1">primrec == [i] genrec</span>
 <span class="s1">step_zero == 0 roll&gt; step</span>
+<span class="s1">codireco == cons dip rest cons</span>
+<span class="s1">make_generator == [codireco] ccons</span>
+<span class="s1">ccons == cons cons</span>
 <span class="s1">&#39;&#39;&#39;</span>

 <span class="c1">##Zipper</span>
@@ -17,7 +17,7 @@
    <link rel="index" title="Index" href="../genindex.html" />
    <link rel="search" title="Search" href="../search.html" />
    <link rel="next" title="Quadratic formula" href="Quadratic.html" />
-    <link rel="prev" title="Treating Trees" href="Trees.html" />
+    <link rel="prev" title="Using x to Generate Values" href="Generator_Programs.html" />
   
  <link rel="stylesheet" href="../_static/custom.css" type="text/css" />
  
@@ -34,16 +34,18 @@
            
  <div class="section" id="newton-s-method">
 <h1><a class="reference external" href="https://en.wikipedia.org/wiki/Newton%27s_method">Newton’s method</a><a class="headerlink" href="#newton-s-method" title="Permalink to this headline">¶</a></h1>
-<p>Newton-Raphson for finding the root of an equation.</p>
+<p>Let’s use the Newton-Raphson method for finding the root of an equation
+to write a function that can compute the square root of a number.</p>
+<p>Cf. <a class="reference external" href="https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf">“Why Functional Programming Matters” by John
+Hughes</a></p>
 <div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">notebook_preamble</span> <span class="k">import</span> <span class="n">J</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">define</span>
 </pre></div>
 </div>
-<p>Cf. <a class="reference external" href="https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf">“Why Functional Programming Matters” by John
-Hughes</a></p>
 <div class="section" id="a-generator-for-approximations">
 <h2>A Generator for Approximations<a class="headerlink" href="#a-generator-for-approximations" title="Permalink to this headline">¶</a></h2>
-<p>In <a class="reference internal" href="Generator Programs.html"><span class="doc">Using x to Generate Values</span></a> we derive a function (called <code class="docutils literal notranslate"><span class="pre">make_generator</span></code> in the dictionary) that accepts an initial value and a quoted program and returns a new quoted program that, when driven by the <code class="docutils literal notranslate"><span class="pre">x</span></code> combinator (<a class="reference internal" href="../library.html#joy.library.x" title="joy.library.x"><code class="xref py py-func docutils literal notranslate"><span class="pre">joy.library.x()</span></code></a>), acts like a lazy stream.</p>
-<p>To make a generator that generates successive approximations let’s start by assuming an initial approximation and then derive the function that computes the next approximation:</p>
+<p>To make a generator that generates successive approximations let’s start
+by assuming an initial approximation and then derive the function that
+computes the next approximation:</p>
 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span>   <span class="n">a</span> <span class="n">F</span>
 <span class="o">---------</span>
    <span class="n">a</span><span class="s1">&#39;</span>
@@ -51,7 +53,8 @@ Hughes</a></p>
 </div>
 <div class="section" id="a-function-to-compute-the-next-approximation">
 <h3>A Function to Compute the Next Approximation<a class="headerlink" href="#a-function-to-compute-the-next-approximation" title="Permalink to this headline">¶</a></h3>
-<p>Looking at the equation again:</p>
+<p>This is the equation for computing the next approximate value of the
+square root:</p>
 <p><span class="math notranslate nohighlight">\(a_{i+1} = \frac{(a_i+\frac{n}{a_i})}{2}\)</span></p>
 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">a</span> <span class="n">n</span> <span class="n">over</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span>
 <span class="n">a</span> <span class="n">n</span> <span class="n">a</span>    <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span>
@@ -71,7 +74,8 @@ Hughes</a></p>
 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">a</span> <span class="p">[</span><span class="n">dup</span> <span class="n">F</span><span class="p">]</span> <span class="n">make_generator</span>
 </pre></div>
 </div>
-<p>With <code class="docutils literal notranslate"><span class="pre">n</span></code> as part of the function <code class="docutils literal notranslate"><span class="pre">F</span></code>, but <code class="docutils literal notranslate"><span class="pre">n</span></code> is the input to the <code class="docutils literal notranslate"><span class="pre">sqrt</span></code> function we’re writing.  If we let 1 be the initial approximation:</p>
+<p>With n as part of the function F, but n is the input to the sqrt
+function we’re writing. If we let 1 be the initial approximation:</p>
 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="n">n</span> <span class="mi">1</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span>
 <span class="mi">1</span> <span class="n">n</span><span class="o">/</span><span class="mi">1</span>   <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span>
 <span class="mi">1</span> <span class="n">n</span>     <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span>
@@ -80,48 +84,51 @@ Hughes</a></p>
 </pre></div>
 </div>
 <p>The generator can be written as:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="n">swap</span> <span class="p">[</span><span class="n">over</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="p">[</span><span class="n">dup</span><span class="p">]</span> <span class="n">swoncat</span> <span class="n">make_generator</span>
-</pre></div>
-</div>
-<p>Example:</p>
 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">23</span> <span class="mi">1</span> <span class="n">swap</span>  <span class="p">[</span><span class="n">over</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="p">[</span><span class="n">dup</span><span class="p">]</span> <span class="n">swoncat</span> <span class="n">make_generator</span>
 <span class="mi">1</span> <span class="mi">23</span>       <span class="p">[</span><span class="n">over</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="p">[</span><span class="n">dup</span><span class="p">]</span> <span class="n">swoncat</span> <span class="n">make_generator</span>
 <span class="mi">1</span>       <span class="p">[</span><span class="mi">23</span> <span class="n">over</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span><span class="p">]</span>      <span class="p">[</span><span class="n">dup</span><span class="p">]</span> <span class="n">swoncat</span> <span class="n">make_generator</span>
 <span class="mi">1</span>   <span class="p">[</span><span class="n">dup</span> <span class="mi">23</span> <span class="n">over</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span><span class="p">]</span>                    <span class="n">make_generator</span>
-<span class="o">.</span>
-<span class="o">.</span>
-<span class="o">.</span>
-<span class="p">[</span><span class="mi">1</span> <span class="n">swap</span> <span class="p">[</span><span class="n">dup</span> <span class="mi">23</span> <span class="n">over</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span>
 </pre></div>
 </div>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;gsra == 1 swap [over / + 2 /] cons [dup] swoncat make_generator&#39;</span><span class="p">)</span>
+</pre></div>
 </div>
-<div class="section" id="a-generator-of-square-root-approximations">
-<h3>A Generator of Square Root Approximations<a class="headerlink" href="#a-generator-of-square-root-approximations" title="Permalink to this headline">¶</a></h3>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">gsra</span> <span class="o">==</span> <span class="mi">1</span> <span class="n">swap</span> <span class="p">[</span><span class="n">over</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">cons</span> <span class="p">[</span><span class="n">dup</span><span class="p">]</span> <span class="n">swoncat</span> <span class="n">make_generator</span>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;23 gsra&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">1</span> <span class="p">[</span><span class="n">dup</span> <span class="mi">23</span> <span class="n">over</span> <span class="o">/</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span><span class="p">]</span> <span class="n">codireco</span><span class="p">]</span>
+</pre></div>
+</div>
+<p>Let’s drive the generator a few time (with the <code class="docutils literal notranslate"><span class="pre">x</span></code> combinator) and
+square the approximation to see how well it works…</p>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;23 gsra 6 [x popd] times first sqr&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">23.0000000001585</span>
 </pre></div>
 </div>
 </div>
 </div>
 <div class="section" id="finding-consecutive-approximations-within-a-tolerance">
-<h2>Finding Consecutive Approximations <code class="docutils literal notranslate"><span class="pre">within</span></code> a Tolerance<a class="headerlink" href="#finding-consecutive-approximations-within-a-tolerance" title="Permalink to this headline">¶</a></h2>
-<blockquote>
-<div>The remainder of a square root finder is a function <em>within</em>, which takes a tolerance and a list of approximations and looks down the list for two successive approximations that differ by no more than the given tolerance.</div></blockquote>
+<h2>Finding Consecutive Approximations within a Tolerance<a class="headerlink" href="#finding-consecutive-approximations-within-a-tolerance" title="Permalink to this headline">¶</a></h2>
 <p>From <a class="reference external" href="https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf">“Why Functional Programming Matters” by John
-Hughes</a></p>
+Hughes</a>:</p>
+<blockquote>
+<div>The remainder of a square root finder is a function <em>within</em>, which
+takes a tolerance and a list of approximations and looks down the
+list for two successive approximations that differ by no more than
+the given tolerance.</div></blockquote>
 <p>(And note that by “list” he means a lazily-evaluated list.)</p>
-<p>Using the <em>output</em> <code class="docutils literal notranslate"><span class="pre">[a</span> <span class="pre">G]</span></code> of the above <a class="reference internal" href="Generator Programs.html"><span class="doc">generator</span></a> for square root approximations, and further assuming that the first term <code class="docutils literal notranslate"><span class="pre">a</span></code> has been generated already and epsilon <code class="docutils literal notranslate"><span class="pre">ε</span></code> is handy on the stack…</p>
+<p>Using the <em>output</em> <code class="docutils literal notranslate"><span class="pre">[a</span> <span class="pre">G]</span></code> of the above generator for square root
+approximations, and further assuming that the first term a has been
+generated already and epsilon ε is handy on the stack…</p>
 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span>   <span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span> <span class="n">ε</span> <span class="n">within</span>
 <span class="o">----------------------</span> <span class="n">a</span> <span class="n">b</span> <span class="o">-</span> <span class="nb">abs</span> <span class="n">ε</span> <span class="o">&lt;=</span>
      <span class="n">b</span>
-</pre></div>
-</div>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>   <span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span> <span class="n">ε</span> <span class="n">within</span>
+
+
+   <span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span> <span class="n">ε</span> <span class="n">within</span>
 <span class="o">----------------------</span> <span class="n">a</span> <span class="n">b</span> <span class="o">-</span> <span class="nb">abs</span> <span class="n">ε</span> <span class="o">&gt;</span>
-       <span class="o">.</span>
-   <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span> <span class="n">x</span> <span class="n">ε</span> <span class="o">...</span>
-   <span class="n">b</span> <span class="p">[</span><span class="n">c</span> <span class="n">G</span><span class="p">]</span> <span class="n">ε</span> <span class="o">...</span>
-       <span class="o">.</span>
-<span class="o">----------------------</span>
   <span class="n">b</span> <span class="p">[</span><span class="n">c</span> <span class="n">G</span><span class="p">]</span> <span class="n">ε</span> <span class="n">within</span>
 </pre></div>
 </div>
@@ -135,7 +142,7 @@ Hughes</a></p>
 <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="p">)</span><span class="o">&lt;=</span><span class="n">ε</span><span class="p">)</span>
 </pre></div>
 </div>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">P</span> <span class="o">==</span> <span class="p">[</span><span class="n">first</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">]</span> <span class="n">dip</span> <span class="o">&lt;=</span>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;_within_P == [first - abs] dip &lt;=&#39;</span><span class="p">)</span>
 </pre></div>
 </div>
 </div>
@@ -147,7 +154,7 @@ Hughes</a></p>
   <span class="n">b</span>
 </pre></div>
 </div>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">B</span> <span class="o">==</span> <span class="n">roll</span><span class="o">&lt;</span> <span class="n">popop</span> <span class="n">first</span>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;_within_B == roll&lt; popop first&#39;</span><span class="p">)</span>
 </pre></div>
 </div>
 </div>
@@ -157,10 +164,11 @@ Hughes</a></p>
 </pre></div>
 </div>
 <ol class="arabic simple">
-<li>Discard <code class="docutils literal notranslate"><span class="pre">a</span></code>.</li>
+<li>Discard a.</li>
 <li>Use <code class="docutils literal notranslate"><span class="pre">x</span></code> combinator to generate next term from <code class="docutils literal notranslate"><span class="pre">G</span></code>.</li>
 <li>Run <code class="docutils literal notranslate"><span class="pre">within</span></code> with <code class="docutils literal notranslate"><span class="pre">i</span></code> (it is a <code class="docutils literal notranslate"><span class="pre">primrec</span></code> function.)</li>
 </ol>
+<p>Pretty straightforward:</p>
 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span>        <span class="n">ε</span> <span class="n">R0</span>           <span class="p">[</span><span class="n">within</span><span class="p">]</span> <span class="n">R1</span>
 <span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span>        <span class="n">ε</span> <span class="p">[</span><span class="n">popd</span> <span class="n">x</span><span class="p">]</span> <span class="n">dip</span> <span class="p">[</span><span class="n">within</span><span class="p">]</span> <span class="n">i</span>
 <span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span> <span class="n">popd</span> <span class="n">x</span> <span class="n">ε</span>              <span class="p">[</span><span class="n">within</span><span class="p">]</span> <span class="n">i</span>
@@ -171,34 +179,51 @@ Hughes</a></p>
 <span class="n">b</span> <span class="p">[</span><span class="n">c</span> <span class="n">G</span><span class="p">]</span> <span class="n">ε</span> <span class="n">within</span>
 </pre></div>
 </div>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">R0</span> <span class="o">==</span> <span class="p">[</span><span class="n">popd</span> <span class="n">x</span><span class="p">]</span> <span class="n">dip</span>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;_within_R == [popd x] dip&#39;</span><span class="p">)</span>
 </pre></div>
 </div>
 </div>
 <div class="section" id="setting-up">
 <h3>Setting up<a class="headerlink" href="#setting-up" title="Permalink to this headline">¶</a></h3>
-<p>The recursive function we have defined so far needs a slight preamble: <code class="docutils literal notranslate"><span class="pre">x</span></code> to prime the generator and the epsilon value to use:</p>
+<p>The recursive function we have defined so far needs a slight preamble:
+<code class="docutils literal notranslate"><span class="pre">x</span></code> to prime the generator and the epsilon value to use:</p>
 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">a</span> <span class="n">G</span><span class="p">]</span> <span class="n">x</span> <span class="n">ε</span> <span class="o">...</span>
 <span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span> <span class="n">ε</span> <span class="o">...</span>
 </pre></div>
 </div>
-</div>
-<div class="section" id="within">
-<h3><code class="docutils literal notranslate"><span class="pre">within</span></code><a class="headerlink" href="#within" title="Permalink to this headline">¶</a></h3>
-<p>Giving us the following definitions:</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">_within_P</span> <span class="o">==</span> <span class="p">[</span><span class="n">first</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">]</span> <span class="n">dip</span> <span class="o">&lt;=</span>
-<span class="n">_within_B</span> <span class="o">==</span> <span class="n">roll</span><span class="o">&lt;</span> <span class="n">popop</span> <span class="n">first</span>
-<span class="n">_within_R</span> <span class="o">==</span> <span class="p">[</span><span class="n">popd</span> <span class="n">x</span><span class="p">]</span> <span class="n">dip</span>
-<span class="n">within</span> <span class="o">==</span> <span class="n">x</span> <span class="n">ε</span> <span class="p">[</span><span class="n">_within_P</span><span class="p">]</span> <span class="p">[</span><span class="n">_within_B</span><span class="p">]</span> <span class="p">[</span><span class="n">_within_R</span><span class="p">]</span> <span class="n">primrec</span>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;within == x 0.000000001 [_within_P] [_within_B] [_within_R] primrec&#39;</span><span class="p">)</span>
+<span class="n">define</span><span class="p">(</span><span class="s1">&#39;sqrt == gsra within&#39;</span><span class="p">)</span>
 </pre></div>
 </div>
-</div>
-</div>
-<div class="section" id="finding-square-roots">
-<h2>Finding Square Roots<a class="headerlink" href="#finding-square-roots" title="Permalink to this headline">¶</a></h2>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">sqrt</span> <span class="o">==</span> <span class="n">gsra</span> <span class="n">within</span>
+<p>Try it out…</p>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;36 sqrt&#39;</span><span class="p">)</span>
 </pre></div>
 </div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">6.0</span>
+</pre></div>
+</div>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;23 sqrt&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">4.795831523312719</span>
+</pre></div>
+</div>
+<p>Check it.</p>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">4.795831523312719</span><span class="o">**</span><span class="mi">2</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">22.999999999999996</span>
+</pre></div>
+</div>
+<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">sqrt</span>
+
+<span class="n">sqrt</span><span class="p">(</span><span class="mi">23</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">4.795831523312719</span>
+</pre></div>
+</div>
+</div>
 </div>
 </div>

@@ -214,18 +239,15 @@ Hughes</a></p>
 <li><a class="reference internal" href="#a-generator-for-approximations">A Generator for Approximations</a><ul>
 <li><a class="reference internal" href="#a-function-to-compute-the-next-approximation">A Function to Compute the Next Approximation</a></li>
 <li><a class="reference internal" href="#make-it-into-a-generator">Make it into a Generator</a></li>
-<li><a class="reference internal" href="#a-generator-of-square-root-approximations">A Generator of Square Root Approximations</a></li>
 </ul>
 </li>
-<li><a class="reference internal" href="#finding-consecutive-approximations-within-a-tolerance">Finding Consecutive Approximations <code class="docutils literal notranslate"><span class="pre">within</span></code> a Tolerance</a><ul>
+<li><a class="reference internal" href="#finding-consecutive-approximations-within-a-tolerance">Finding Consecutive Approximations within a Tolerance</a><ul>
 <li><a class="reference internal" href="#predicate">Predicate</a></li>
 <li><a class="reference internal" href="#base-case">Base-Case</a></li>
 <li><a class="reference internal" href="#recur">Recur</a></li>
 <li><a class="reference internal" href="#setting-up">Setting up</a></li>
-<li><a class="reference internal" href="#within"><code class="docutils literal notranslate"><span class="pre">within</span></code></a></li>
 </ul>
 </li>
-<li><a class="reference internal" href="#finding-square-roots">Finding Square Roots</a></li>
 </ul>
 </li>
 </ul>
@@ -234,7 +256,7 @@ Hughes</a></p>
 <ul>
  <li><a href="../index.html">Documentation overview</a><ul>
  <li><a href="index.html">Essays about Programming in Joy</a><ul>
-      <li>Previous: <a href="Trees.html" title="previous chapter">Treating Trees</a></li>
+      <li>Previous: <a href="Generator_Programs.html" title="previous chapter">Using <code class="docutils literal notranslate"><span class="pre">x</span></code> to Generate Values</a></li>
      <li>Next: <a href="Quadratic.html" title="next chapter">Quadratic formula</a></li>
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@@ -16,7 +16,7 @@
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    <link rel="index" title="Index" href="../genindex.html" />
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-    <link rel="next" title="No Updates" href="NoUpdates.html" />
+    <link rel="next" title="Traversing Datastructures with Zippers" href="Zipper.html" />
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@@ -213,7 +213,7 @@ already.)</p>
  <li><a href="../index.html">Documentation overview</a><ul>
  <li><a href="index.html">Essays about Programming in Joy</a><ul>
      <li>Previous: <a href="Newton-Raphson.html" title="previous chapter">Newton’s method</a></li>
-      <li>Next: <a href="NoUpdates.html" title="next chapter">No Updates</a></li>
+      <li>Next: <a href="Zipper.html" title="next chapter">Traversing Datastructures with Zippers</a></li>
  </ul></li>
  </ul></li>
 </ul>
@@ -16,6 +16,8 @@
    <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/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="No Updates" href="NoUpdates.html" />
+    <link rel="prev" title="Quadratic formula" href="Quadratic.html" />
   
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@@ -323,6 +325,10 @@ i d i d i d d Bingo!
 <h3>Related Topics</h3>
 <ul>
  <li><a href="../index.html">Documentation overview</a><ul>
+  <li><a href="index.html">Essays about Programming in Joy</a><ul>
+      <li>Previous: <a href="Quadratic.html" title="previous chapter">Quadratic formula</a></li>
+      <li>Next: <a href="NoUpdates.html" title="next chapter">No Updates</a></li>
+  </ul></li>
  </ul></li>
 </ul>
 </div>
@@ -86,8 +86,7 @@
 </li>
 <li class="toctree-l1"><a class="reference internal" href="Newton-Raphson.html">Newton’s method</a><ul>
 <li class="toctree-l2"><a class="reference internal" href="Newton-Raphson.html#a-generator-for-approximations">A Generator for Approximations</a></li>
-<li class="toctree-l2"><a class="reference internal" href="Newton-Raphson.html#finding-consecutive-approximations-within-a-tolerance">Finding Consecutive Approximations <code class="docutils literal notranslate"><span class="pre">within</span></code> a Tolerance</a></li>
-<li class="toctree-l2"><a class="reference internal" href="Newton-Raphson.html#finding-square-roots">Finding Square Roots</a></li>
+<li class="toctree-l2"><a class="reference internal" href="Newton-Raphson.html#finding-consecutive-approximations-within-a-tolerance">Finding Consecutive Approximations within a Tolerance</a></li>
 </ul>
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