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  37   <section id="newton-s-method">
  38 <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>
  39 <p>Let’s use the Newton-Raphson method for finding the root of an equation
  40 to write a function that can compute the square root of a number.</p>
  41 <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
  42 Hughes</a></p>
  43 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">notebook_preamble</span> <span class="kn">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>
  44 </pre></div>
  45 </div>
  46 <section id="a-generator-for-approximations">
  47 <h2>A Generator for Approximations<a class="headerlink" href="#a-generator-for-approximations" title="Permalink to this headline">¶</a></h2>
  48 <p>To make a generator that generates successive approximations let’s start
  49 by assuming an initial approximation and then derive the function that
  50 computes the next approximation:</p>
  51 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span>   <span class="n">a</span> <span class="n">F</span>
  52 <span class="o">---------</span>
  53     <span class="n">a</span><span class="s1">&#39;</span>
  54 </pre></div>
  55 </div>
  56 <section id="a-function-to-compute-the-next-approximation">
  57 <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>
  58 <p>This is the equation for computing the next approximate value of the
  59 square root:</p>
  60 <p><span class="math notranslate nohighlight">\(a_{i+1} = \frac{(a_i+\frac{n}{a_i})}{2}\)</span></p>
  61 <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>
  62 <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>
  63 <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>
  64 <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>
  65 <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>
  66 </pre></div>
  67 </div>
  68 <p>The function we want has the argument <code class="docutils literal notranslate"><span class="pre">n</span></code> in it:</p>
  69 <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>
  70 </pre></div>
  71 </div>
  72 </section>
  73 <section id="make-it-into-a-generator">
  74 <h3>Make it into a Generator<a class="headerlink" href="#make-it-into-a-generator" title="Permalink to this headline">¶</a></h3>
  75 <p>Our generator would be created by:</p>
  76 <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>
  77 </pre></div>
  78 </div>
  79 <p>With n as part of the function F, but n is the input to the sqrt
  80 function we’re writing. If we let 1 be the initial approximation:</p>
  81 <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>
  82 <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>
  83 <span class="mi">1</span> <span class="n">n</span>     <span class="o">+</span> <span class="mi">2</span> <span class="o">/</span>
  84 <span class="n">n</span><span class="o">+</span><span class="mi">1</span>       <span class="mi">2</span> <span class="o">/</span>
  85 <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>
  86 </pre></div>
  87 </div>
  88 <p>The generator can be written as:</p>
  89 <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>
  90 <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>
  91 <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>
  92 <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>
  93 </pre></div>
  94 </div>
  95 <div class="highlight-ipython3 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>
  96 </pre></div>
  97 </div>
  98 <div class="highlight-ipython3 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>
  99 </pre></div>
 100 </div>
 101 <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>
 102 </pre></div>
 103 </div>
 104 <p>Let’s drive the generator a few time (with the <code class="docutils literal notranslate"><span class="pre">x</span></code> combinator) and
 105 square the approximation to see how well it works…</p>
 106 <div class="highlight-ipython3 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>
 107 </pre></div>
 108 </div>
 109 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">23.0000000001585</span>
 110 </pre></div>
 111 </div>
 112 </section>
 113 </section>
 114 <section id="finding-consecutive-approximations-within-a-tolerance">
 115 <h2>Finding Consecutive Approximations within a Tolerance<a class="headerlink" href="#finding-consecutive-approximations-within-a-tolerance" title="Permalink to this headline">¶</a></h2>
 116 <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
 117 Hughes</a>:</p>
 118 <blockquote>
 119 <div><p>The remainder of a square root finder is a function <em>within</em>, which
 120 takes a tolerance and a list of approximations and looks down the
 121 list for two successive approximations that differ by no more than
 122 the given tolerance.</p>
 123 </div></blockquote>
 124 <p>(And note that by “list” he means a lazily-evaluated list.)</p>
 125 <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
 126 approximations, and further assuming that the first term a has been
 127 generated already and epsilon ε is handy on the stack…</p>
 128 <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>
 129 <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>
 130       <span class="n">b</span>
 131
 132
 133    <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>
 134 <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>
 135    <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>
 136 </pre></div>
 137 </div>
 138 <section id="predicate">
 139 <h3>Predicate<a class="headerlink" href="#predicate" title="Permalink to this headline">¶</a></h3>
 140 <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>
 141 <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>
 142 <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>
 143 <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>
 144 <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>
 145 <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>
 146 </pre></div>
 147 </div>
 148 <div class="highlight-ipython3 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>
 149 </pre></div>
 150 </div>
 151 </section>
 152 <section id="base-case">
 153 <h3>Base-Case<a class="headerlink" href="#base-case" title="Permalink to this headline">¶</a></h3>
 154 <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>
 155   <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>
 156   <span class="p">[</span><span class="n">b</span> <span class="n">G</span><span class="p">]</span>               <span class="n">first</span>
 157    <span class="n">b</span>
 158 </pre></div>
 159 </div>
 160 <div class="highlight-ipython3 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>
 161 </pre></div>
 162 </div>
 163 </section>
 164 <section id="recur">
 165 <h3>Recur<a class="headerlink" href="#recur" title="Permalink to this headline">¶</a></h3>
 166 <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>
 167 </pre></div>
 168 </div>
 169 <ol class="arabic simple">
 170 <li><p>Discard a.</p></li>
 171 <li><p>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>.</p></li>
 172 <li><p>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.)</p></li>
 173 </ol>
 174 <p>Pretty straightforward:</p>
 175 <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>
 176 <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>
 177 <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>
 178   <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>
 179 <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>
 180 <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>
 181
 182 <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>
 183 </pre></div>
 184 </div>
 185 <div class="highlight-ipython3 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>
 186 </pre></div>
 187 </div>
 188 </section>
 189 <section id="setting-up">
 190 <h3>Setting up<a class="headerlink" href="#setting-up" title="Permalink to this headline">¶</a></h3>
 191 <p>The recursive function we have defined so far needs a slight preamble:
 192 <code class="docutils literal notranslate"><span class="pre">x</span></code> to prime the generator and the epsilon value to use:</p>
 193 <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>
 194 <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>
 195 </pre></div>
 196 </div>
 197 <div class="highlight-ipython3 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>
 198 <span class="n">define</span><span class="p">(</span><span class="s1">&#39;sqrt gsra within&#39;</span><span class="p">)</span>
 199 </pre></div>
 200 </div>
 201 <p>Try it out…</p>
 202 <div class="highlight-ipython3 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>
 203 </pre></div>
 204 </div>
 205 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">6.0</span>
 206 </pre></div>
 207 </div>
 208 <div class="highlight-ipython3 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>
 209 </pre></div>
 210 </div>
 211 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">4.795831523312719</span>
 212 </pre></div>
 213 </div>
 214 <p>Check it.</p>
 215 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="mf">4.795831523312719</span><span class="o">**</span><span class="mi">2</span>
 216 </pre></div>
 217 </div>
 218 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">22.999999999999996</span>
 219 </pre></div>
 220 </div>
 221 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">sqrt</span>
 222
 223 <span class="n">sqrt</span><span class="p">(</span><span class="mi">23</span><span class="p">)</span>
 224 </pre></div>
 225 </div>
 226 <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">4.795831523312719</span>
 227 </pre></div>
 228 </div>
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