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<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>
<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="finding-the-square-root-of-a-number">
<h2>Finding the Square-Root of a Number<a class="headerlink" href="#finding-the-square-root-of-a-number" title="Permalink to this headline"></a></h2>
<p>Let’s define a function that computes this equation:</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">n</span> <span class="n">a</span> <span class="n">Q</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>
<span class="n">n</span> <span class="n">a</span> <span class="n">tuck</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>We want it to leave n but replace a, so we execute it with <code class="docutils literal notranslate"><span class="pre">unary</span></code>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Q</span> <span class="o">==</span> <span class="p">[</span><span class="n">tuck</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">unary</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;Q == [tuck / + 2 /] unary&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="compute-the-error">
<h2>Compute the Error<a class="headerlink" href="#compute-the-error" title="Permalink to this headline"></a></h2>
<p>And a function to compute the error:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="n">a</span> <span class="n">sqr</span> <span class="o">-</span> <span class="nb">abs</span>
<span class="o">|</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>
</pre></div>
</div>
<p>This should be <code class="docutils literal notranslate"><span class="pre">nullary</span></code> so as to leave both n and a on the stack
below the error.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">err</span> <span class="o">==</span> <span class="p">[</span><span class="n">sqr</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">]</span> <span class="n">nullary</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;err == [sqr - abs] nullary&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="square-root">
<h2><code class="docutils literal notranslate"><span class="pre">square-root</span></code><a class="headerlink" href="#square-root" title="Permalink to this headline"></a></h2>
<p>Now we can define a recursive program that expects a number <code class="docutils literal notranslate"><span class="pre">n</span></code>, an
initial estimate <code class="docutils literal notranslate"><span class="pre">a</span></code>, and an epsilon value <code class="docutils literal notranslate"><span class="pre">ε</span></code>, and that leaves on
the stack the square root of <code class="docutils literal notranslate"><span class="pre">n</span></code> to within the precision of the
epsilon value. (Later on we’ll refine it to generate the initial
estimate and hard-code an epsilon value.)</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>n a ε square-root
-----------------
√n
</pre></div>
</div>
<p>If we apply the two functions <code class="docutils literal notranslate"><span class="pre">Q</span></code> and <code class="docutils literal notranslate"><span class="pre">err</span></code> defined above we get the
next approximation and the error on the stack below the epsilon.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="n">a</span> <span class="n">ε</span> <span class="p">[</span><span class="n">Q</span> <span class="n">err</span><span class="p">]</span> <span class="n">dip</span>
<span class="n">n</span> <span class="n">a</span> <span class="n">Q</span> <span class="n">err</span> <span class="n">ε</span>
<span class="n">n</span> <span class="n">a</span><span class="s1">&#39; err ε</span>
<span class="n">n</span> <span class="n">a</span><span class="s1">&#39; e ε</span>
</pre></div>
</div>
<p>Let’s define a recursive function <code class="docutils literal notranslate"><span class="pre">K</span></code> from here.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="n">a</span><span class="s1">&#39; e ε K</span>
<span class="n">K</span> <span class="o">==</span> <span class="p">[</span><span class="n">P</span><span class="p">]</span> <span class="p">[</span><span class="n">E</span><span class="p">]</span> <span class="p">[</span><span class="n">R0</span><span class="p">]</span> <span class="p">[</span><span class="n">R1</span><span class="p">]</span> <span class="n">genrec</span>
</pre></div>
</div>
<div class="section" id="base-case">
<h3>Base-case<a class="headerlink" href="#base-case" title="Permalink to this headline"></a></h3>
<p>The predicate and the base case are obvious:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">K</span> <span class="o">==</span> <span class="p">[</span><span class="o">&lt;</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="n">popd</span><span class="p">]</span> <span class="p">[</span><span class="n">R0</span><span class="p">]</span> <span class="p">[</span><span class="n">R1</span><span class="p">]</span> <span class="n">genrec</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="n">a</span><span class="s1">&#39; e ε popop popd</span>
<span class="n">n</span> <span class="n">a</span><span class="s1">&#39; popd</span>
<span class="n">a</span><span class="s1">&#39;</span>
</pre></div>
</div>
</div>
<div class="section" id="recur">
<h3>Recur<a class="headerlink" href="#recur" title="Permalink to this headline"></a></h3>
<p>The recursive branch is pretty easy. Discard the error and recur.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">K</span> <span class="o">==</span> <span class="p">[</span><span class="o">&lt;</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="n">popd</span><span class="p">]</span> <span class="p">[</span><span class="n">R0</span><span class="p">]</span> <span class="p">[</span><span class="n">R1</span><span class="p">]</span> <span class="n">genrec</span>
<span class="n">K</span> <span class="o">==</span> <span class="p">[</span><span class="o">&lt;</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="n">popd</span><span class="p">]</span> <span class="p">[</span><span class="n">R0</span> <span class="p">[</span><span class="n">K</span><span class="p">]</span> <span class="n">R1</span><span class="p">]</span> <span class="n">ifte</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="n">a</span><span class="s1">&#39; e ε R0 [K] R1</span>
<span class="n">n</span> <span class="n">a</span><span class="s1">&#39; e ε popd [Q err] dip [K] i</span>
<span class="n">n</span> <span class="n">a</span><span class="s1">&#39; ε [Q err] dip [K] i</span>
<span class="n">n</span> <span class="n">a</span><span class="s1">&#39; Q err ε [K] i</span>
<span class="n">n</span> <span class="n">a</span><span class="s1">&#39;&#39;</span> <span class="n">e</span> <span class="n">ε</span> <span class="n">K</span>
</pre></div>
</div>
<p>This fragment alone is pretty useful. (<code class="docutils literal notranslate"><span class="pre">R1</span></code> is <code class="docutils literal notranslate"><span class="pre">i</span></code> so this is a <code class="docutils literal notranslate"><span class="pre">primrec</span></code> “primitive recursive” function.)</p>
<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;K == [&lt;] [popop popd] [popd [Q err] dip] primrec&#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;25 10 0.001 dup K&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">5.000000232305737</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;25 10 0.000001 dup K&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">5.000000000000005</span>
</pre></div>
</div>
</div>
<div class="section" id="initial-approximation-and-epsilon">
<h3>Initial Approximation and Epsilon<a class="headerlink" href="#initial-approximation-and-epsilon" title="Permalink to this headline"></a></h3>
<p>So now all we need is a way to generate an initial approximation and an
epsilon value:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">square</span><span class="o">-</span><span class="n">root</span> <span class="o">==</span> <span class="n">dup</span> <span class="mi">3</span> <span class="o">/</span> <span class="mf">0.000001</span> <span class="n">dup</span> <span class="n">K</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;square-root == dup 3 / 0.000001 dup K&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="examples">
<h3>Examples<a class="headerlink" href="#examples" title="Permalink to this headline"></a></h3>
<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 square-root&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">6.000000000000007</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;4895048365636 square-root&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">2212475.6192184356</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">2212475.6192184356</span> <span class="o">*</span> <span class="mf">2212475.6192184356</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">4895048365636.0</span>
</pre></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="#finding-the-square-root-of-a-number">Finding the Square-Root of a Number</a></li>
<li><a class="reference internal" href="#compute-the-error">Compute the Error</a></li>
<li><a class="reference internal" href="#square-root"><code class="docutils literal notranslate"><span class="pre">square-root</span></code></a><ul>
<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="#initial-approximation-and-epsilon">Initial Approximation and Epsilon</a></li>
<li><a class="reference internal" href="#examples">Examples</a></li>
</ul>
</li>
</ul>
</li>
</ul>
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