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diff --git a/docs/doc/prefixes.html b/docs/doc/prefixes.html index 59e94b21..dd27b18c 100644 --- a/docs/doc/prefixes.html +++ b/docs/doc/prefixes.html @@ -4,7 +4,7 @@ <title>BQN: Prefixes and Suffixes</title> </head> <div class="nav">(<a href="https://github.com/mlochbaum/BQN">github</a>) / <a href="../index.html">BQN</a> / <a href="index.html">doc</a></div> -<h1 id="prefixes-and-suffixes">Prefixes and Suffixes</h1> +<h1 id="prefixes-and-suffixes"><a class="header" href="#prefixes-and-suffixes">Prefixes and Suffixes</a></h1> <p>The Prefixes (<code><span class='Function'>↑</span></code>) function gives a list of all prefixes of its argument array along the <a href="leading.html">first axis</a>, and Suffixes (<code><span class='Function'>↓</span></code>) gives a similar list for suffixes. Because the result can be much larger than the argument, these functions may not be used often in high-performance code, but they are a powerful conceptual tool and can make sense for algorithms that are inherently quadratic.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oaRICJhYmNkZSIK4oaTICJhYmNkZSI=">↗️</a><pre> <span class='Function'>↑</span> <span class='String'>"abcde"</span> ⟨ ⟨⟩ "a" "ab" "abc" "abcd" "abcde" ⟩ @@ -26,9 +26,9 @@ ⟨ "abcde" "abcde" "abcde" "abcde" "abcde" "abcde" ⟩ </pre> <p>Joining corresponding elements of <code><span class='Function'>↑</span><span class='Value'>𝕩</span></code> and <code><span class='Function'>↓</span><span class='Value'>𝕩</span></code> gives <code><span class='Value'>𝕩</span></code> again. This is because <code><span class='Value'>i</span><span class='Function'>⊑</span><span class='Paren'>(</span><span class='Function'>↑∾</span><span class='Modifier'>¨</span><span class='Function'>↓</span><span class='Paren'>)</span><span class='Value'>𝕩</span></code> is <code><span class='Paren'>(</span><span class='Value'>i</span><span class='Function'>⊑↑</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>∾</span><span class='Paren'>(</span><span class='Value'>i</span><span class='Function'>⊑↓</span><span class='Value'>𝕩</span><span class='Paren'>)</span></code>, or, using the Take and Drop correspondences, <code><span class='Paren'>(</span><span class='Value'>i</span><span class='Function'>↑</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>∾</span><span class='Paren'>(</span><span class='Value'>i</span><span class='Function'>↓</span><span class='Value'>𝕩</span><span class='Paren'>)</span></code>, which is <code><span class='Value'>𝕩</span></code>. Element-wise, we are combining the first <code><span class='Value'>i</span></code> cells of <code><span class='Value'>𝕩</span></code> with all but the first <code><span class='Value'>i</span></code>. Looking at the entire result, we now know that <code><span class='Paren'>(</span><span class='Function'>↑∾</span><span class='Modifier'>¨</span><span class='Function'>↓</span><span class='Paren'>)</span><span class='Value'>𝕩</span></code> is <code><span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>+≠</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>⥊<</span><span class='Value'>𝕩</span></code>. The total number of cells in this combined array is therefore <code><span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>+≠</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>×≠</span><span class='Value'>𝕩</span></code>, or <code><span class='Number'>1</span><span class='Modifier2'>⊸</span><span class='Function'>+</span><span class='Modifier2'>⊸</span><span class='Function'>×≠</span><span class='Value'>𝕩</span></code>. Each of Prefixes and Suffixes must have the same total number of cells (in fact, <code><span class='Function'>↑</span><span class='Value'>𝕩</span></code> is <code><span class='Function'>⌽</span><span class='Modifier'>¨</span><span class='Modifier2'>∘</span><span class='Function'>↓</span><span class='Modifier2'>⌾</span><span class='Function'>⌽</span><span class='Value'>𝕩</span></code>, and Reverse doesn't change an array's shape). So the total number in either one is <code><span class='Number'>2</span><span class='Function'>÷</span><span class='Modifier'>˜</span><span class='Number'>1</span><span class='Modifier2'>⊸</span><span class='Function'>+</span><span class='Modifier2'>⊸</span><span class='Function'>×≠</span><span class='Value'>𝕩</span></code>. With <code><span class='Value'>n</span><span class='Gets'>←</span><span class='Function'>≠</span><span class='Value'>𝕩</span></code>, it is <code><span class='Number'>2</span><span class='Function'>÷</span><span class='Modifier'>˜</span><span class='Value'>n</span><span class='Function'>×</span><span class='Number'>1</span><span class='Function'>+</span><span class='Value'>n</span></code>.</p> -<h2 id="definition">Definition</h2> +<h2 id="definition"><a class="header" href="#definition">Definition</a></h2> <p>Knowing the <a href="shape.html">length</a> and the elements, it's easy to define functions for Prefixes and Suffixes: <code><span class='Function'>↑</span></code> is equivalent to <code><span class='Paren'>(</span><span class='Function'>↕</span><span class='Number'>1</span><span class='Function'>+≠</span><span class='Paren'>)</span><span class='Function'>↑</span><span class='Modifier'>¨</span><span class='Function'><</span></code> while <code><span class='Function'>↓</span></code> is <code><span class='Paren'>(</span><span class='Function'>↕</span><span class='Number'>1</span><span class='Function'>+≠</span><span class='Paren'>)</span><span class='Function'>↓</span><span class='Modifier'>¨</span><span class='Function'><</span></code>. Each primitive is defined only on arrays with at least one axis.</p> -<h2 id="working-with-pairs">Working with pairs</h2> +<h2 id="working-with-pairs"><a class="header" href="#working-with-pairs">Working with pairs</a></h2> <p>Sometimes it's useful to apply an operation to every unordered pair of elements from a list. For example, consider all possible products of numbers between 1 and 6:</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=w5fijJzLnCAxK+KGlTY=">↗️</a><pre> <span class='Function'>×</span><span class='Modifier'>⌜˜</span> <span class='Number'>1</span><span class='Function'>+↕</span><span class='Number'>6</span> ┌─ @@ -70,7 +70,7 @@ <span class='Paren'>(</span><span class='Function'><</span><span class='Modifier'>˘</span> <span class='Function'>≍</span><span class='Modifier'>¨¨</span> <span class='Function'>≠</span> <span class='Function'>↑</span> <span class='Function'>↑</span><span class='Paren'>)</span> <span class='String'>"abc"</span> ⟨ ⟨⟩ ⟨ "ba" ⟩ ⟨ "ca" "cb" ⟩ ⟩ </pre> -<h2 id="slices">Slices</h2> +<h2 id="slices"><a class="header" href="#slices">Slices</a></h2> <p>Prefixes and Suffixes give certain restricted slices of the argument array, where a slice is a contiguous selection of cells. If we want all slices along the first axis, we can take the suffixes of each prefix, or vice-versa:</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oaTwqjihpEgImFiYyIK4oaRwqjihpMgImFiYyI=">↗️</a><pre> <span class='Function'>↓</span><span class='Modifier'>¨</span><span class='Function'>↑</span> <span class='String'>"abc"</span> ┌─ |