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Update some of the docs.

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Simon Forman
2020-05-20 19:15:47 -07:00
parent 6a6b63bf62
commit ffabda0407
60 changed files with 5182 additions and 5853 deletions
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docs/sphinx_docs/_build/html/notebooks/Newton-Raphson.html
+16 -16
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@@ -32,13 +32,13 @@
<div class="bodywrapper">
<div class="body" role="main">
<div class="section" id="newtons-method">
<h1><a class="reference external" href="https://en.wikipedia.org/wiki/Newton%27s_method">Newton’s method</a><a class="headerlink" href="#newtons-method" title="Permalink to this headline"></a></h1>
<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>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>
<div class="code ipython3 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>
<div class="section" id="a-generator-for-approximations">
@@ -90,10 +90,10 @@ function we’re writing. If we let 1 be the initial approximation:</p>
<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>
</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>
<div class="code ipython3 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="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>
<div class="code ipython3 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>
@@ -101,7 +101,7 @@ function we’re writing. If we let 1 be the initial approximation:</p>
</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>
<div class="code ipython3 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>
@@ -142,7 +142,7 @@ generated already and epsilon ε is handy on the stack…</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="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>
<div class="code ipython3 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>
@@ -154,7 +154,7 @@ generated already and epsilon ε is handy on the stack…</p>
<span class="n">b</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;_within_B == roll&lt; popop first&#39;</span><span class="p">)</span>
<div class="code ipython3 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>
@@ -166,7 +166,7 @@ generated already and epsilon ε is handy on the stack…</p>
<ol class="arabic simple">
<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>
<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 “tail-recursive” 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>
@@ -179,7 +179,7 @@ generated already and epsilon ε is handy on the stack…</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="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>
<div class="code ipython3 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>
@@ -191,31 +191,31 @@ generated already and epsilon ε is handy on the stack…</p>
<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 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>
<div class="code ipython3 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] tailrec&#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>
<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>
<div class="code ipython3 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>
<div class="code ipython3 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>
<div class="code ipython3 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>
<div class="code ipython3 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>