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<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>
<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>
<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>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>
</pre></div>
</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>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>
<span class="n">a</span> <span class="n">n</span><span class="o">/</span><span class="n">a</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span>
<span class="n">a</span><span class="o">+</span><span class="n">n</span><span class="o">/</span><span class="n">a</span> <span class="mi">2</span> <span class="o">/</span>
<span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">n</span><span class="o">/</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
</pre></div>
</div>
<p>The function we want has the argument <code class="docutils literal notranslate"><span class="pre">n</span></code> in it:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">F</span> <span class="o">==</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>
</pre></div>
</div>
</div>
<div class="section" id="make-it-into-a-generator">
<h3>Make it into a Generator<a class="headerlink" href="#make-it-into-a-generator" title="Permalink to this headline"></a></h3>
<p>Our generator would be created by:</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 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>
<span class="n">n</span><span class="o">+</span><span class="mi">1</span> <span class="mi">2</span> <span class="o">/</span>
<span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
</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">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>
</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="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 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>
<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 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>
<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="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="section" id="predicate">
<h3>Predicate<a class="headerlink" href="#predicate" title="Permalink to this headline"></a></h3>
<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="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">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span> <span class="n">first</span> <span class="o">-</span> <span class="nb">abs</span> <span class="n">ε</span> <span class="o">&lt;=</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">a</span><span class="o">-</span><span class="n">b</span> <span class="nb">abs</span> <span class="n">ε</span> <span class="o">&lt;=</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="n">ε</span> <span class="o">&lt;=</span>
<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>
</pre></div>
</div>
</div>
<div class="section" id="base-case">
<h3>Base-Case<a class="headerlink" href="#base-case" title="Permalink to this headline"></a></h3>
<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">roll</span><span class="o">&lt;</span> <span class="n">popop</span> <span class="n">first</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">a</span> <span class="n">popop</span> <span class="n">first</span>
<span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span> <span class="n">first</span>
<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>
</pre></div>
</div>
</div>
<div class="section" id="recur">
<h3>Recur<a class="headerlink" href="#recur" title="Permalink to this headline"></a></h3>
<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>
</pre></div>
</div>
<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>
</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>
<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="p">[</span><span class="n">within</span><span class="p">]</span> <span class="n">i</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="p">[</span><span class="n">within</span><span class="p">]</span> <span class="n">i</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>
<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>
</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>
<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 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>
<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>
</div>
</div>
</div>
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<h3><a href="../index.html">Table Of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">Newton’s method</a><ul>
<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>
</ul>
</li>
<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>
</ul>
</li>
</ul>
</li>
</ul>
<div class="relations">
<h3>Related Topics</h3>
<ul>
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