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diff --git a/docs/doc/transpose.html b/docs/doc/transpose.html index 42d3dd71..b2319106 100644 --- a/docs/doc/transpose.html +++ b/docs/doc/transpose.html @@ -4,9 +4,9 @@ <title>BQN: Transpose</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="transpose">Transpose</h1> +<h1 id="transpose"><a class="header" href="#transpose">Transpose</a></h1> <p>Transpose (<code><span class='Function'>⍉</span></code>) is a tool for rearranging the axes of an array. BQN's version is tweaked relative to APL to align better with the leading axis model and make common operations easier.</p> -<h2 id="transpose-basics">Transpose basics</h2> +<h2 id="transpose-basics"><a class="header" href="#transpose-basics">Transpose basics</a></h2> <p>The name for the primitive <code><span class='Function'>⍉</span></code> comes from the <a href="https://en.wikipedia.org/wiki/Transpose">Transpose</a> operation on matrices. Given a matrix as an array of rank 2, <code><span class='Function'>⍉</span></code> will transpose it:</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqiIG1hdCDihpAgMuKAvzMg4qWKIOKGlTYK4o2JIG1hdA==">↗️</a><pre> <span class='Function'>⊢</span> <span class='Value'>mat</span> <span class='Gets'>←</span> <span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span> <span class='Function'>⥊</span> <span class='Function'>↕</span><span class='Number'>6</span> ┌─ @@ -27,7 +27,7 @@ 3 </pre> <p>With two axes the only interesting operation of this sort is to swap them (and with one or zero axes there's nothing interesting to do, and <code><span class='Function'>⍉</span></code> just returns the argument array). But a BQN programmer may well want to work with higher-rank arrays—although such a programmer might call them "tensors"—and this means there are many more ways to rearrange the axes. Transpose extends to high-rank arrays to allow some useful special cases as well as completely general axis rearrangement, as described below.</p> -<h2 id="monadic-transpose">Monadic Transpose</h2> +<h2 id="monadic-transpose"><a class="header" href="#monadic-transpose">Monadic Transpose</a></h2> <p>APL extends matrix transposition to any rank by reversing all axes for its monadic <code><span class='Function'>⍉</span></code>, but this generalization isn't very natural and is almost never used. The main reason for it is to maintain the equivalence <code><span class='Value'>a</span> <span class='Function'>MP</span> <span class='Value'>b</span> <span class='Gets'>←→</span> <span class='Value'>b</span> <span class='Function'>MP</span><span class='Modifier2'>⌾</span><span class='Function'>⍉</span> <span class='Value'>a</span></code>, where <code><span class='Function'>MP</span> <span class='Gets'>←</span> <span class='Function'>+</span><span class='Modifier'>˝</span><span class='Modifier2'>∘</span><span class='Function'>×</span><span class='Modifier2'>⎉</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>∞</span></code> is the generalized matrix product. But even here APL's Transpose is suspect. It does much more work than it needs to, as we'll see.</p> <p>BQN's transpose takes the first axis of <code><span class='Value'>𝕩</span></code> and moves it to the end.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4omiIGEyMzQ1NiDihpAg4oaVMuKAvzPigL804oC/NeKAvzYKCuKJoiDijYkgYTIzNDU2">↗️</a><pre> <span class='Function'>≢</span> <span class='Value'>a23456</span> <span class='Gets'>←</span> <span class='Function'>↕</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>5</span><span class='Ligature'>‿</span><span class='Number'>6</span> @@ -88,7 +88,7 @@ </span>⟨ 3 4 2 5 6 ⟩ </pre> <p>In a case like this BQN's Dyadic transpose is much easier.</p> -<h2 id="dyadic-transpose">Dyadic Transpose</h2> +<h2 id="dyadic-transpose"><a class="header" href="#dyadic-transpose">Dyadic Transpose</a></h2> <p>Transpose also allows a left argument that specifies a permutation of <code><span class='Value'>𝕩</span></code>'s axes. For each index <code><span class='Value'>p</span><span class='Gets'>←</span><span class='Value'>i</span><span class='Function'>⊑</span><span class='Value'>𝕨</span></code> in the left argument, axis <code><span class='Value'>i</span></code> of <code><span class='Value'>𝕩</span></code> is used for axis <code><span class='Value'>p</span></code> of the result. Multiple argument axes can be sent to the same result axis, in which case that axis goes along a diagonal of <code><span class='Value'>𝕩</span></code>, and the result will have a lower rank than <code><span class='Value'>𝕩</span></code>.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4omiIDHigL8z4oC/MuKAvzDigL80IOKNiSBhMjM0NTYKCuKJoiAx4oC/MuKAvzLigL8w4oC/MCDijYkgYTIzNDU2ICAjIERvbid0IHdvcnJ5IHRvbyBtdWNoIGFib3V0IHRoaXMgY2FzZSB0aG91Z2g=">↗️</a><pre> <span class='Function'>≢</span> <span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>4</span> <span class='Function'>⍉</span> <span class='Value'>a23456</span> ⟨ 5 2 4 3 6 ⟩ @@ -109,7 +109,7 @@ </span>⟨ 3 4 2 5 6 ⟩ </pre> <p>Finally, it's worth noting that, as monadic Transpose moves the first axis to the end, it's equivalent to dyadic Transpose with a "default" left argument: <code><span class='Paren'>(</span><span class='Function'>=-</span><span class='Number'>1</span><span class='Modifier'>˙</span><span class='Paren'>)</span><span class='Modifier2'>⊸</span><span class='Function'>⍉</span></code>.</p> -<h2 id="definitions">Definitions</h2> +<h2 id="definitions"><a class="header" href="#definitions">Definitions</a></h2> <p>Here we define the two valences of Transpose more precisely.</p> <p>An atom right argument to either valence of Transpose is always enclosed to get an array before doing anything else.</p> <p>Monadic transpose is identical to <code><span class='Paren'>(</span><span class='Function'>=-</span><span class='Number'>1</span><span class='Modifier'>˙</span><span class='Paren'>)</span><span class='Modifier2'>⊸</span><span class='Function'>⍉</span></code>, except that if <code><span class='Value'>𝕩</span></code> is a unit it is returned unchanged (after enclosing, if it's an atom) rather than giving an error.</p> |