From 2afb23928e1984d475cc460e1672e8f6fa0e4dbe Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Wed, 11 Aug 2021 17:21:31 -0400 Subject: Allow clicking on header to get fragment link --- docs/doc/prefixes.html | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) (limited to 'docs/doc/prefixes.html') 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 @@ BQN: Prefixes and Suffixes
(github) / BQN / doc
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Prefixes and Suffixes

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Prefixes and Suffixes

The Prefixes (↑) function gives a list of all prefixes of its argument array along the first axis, and Suffixes (↓) 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.

↗️
    ↑ "abcde"
 ⟨ ⟨⟩ "a" "ab" "abc" "abcd" "abcde" ⟩
@@ -26,9 +26,9 @@
 ⟨ "abcde" "abcde" "abcde" "abcde" "abcde" "abcde" ⟩

Joining corresponding elements of ↑𝕩 and ↓𝕩 gives 𝕩 again. This is because i⊑(↑∾¨↓)𝕩 is (i⊑↑𝕩)∾(i⊑↓𝕩), or, using the Take and Drop correspondences, (i↑𝕩)∾(i↓𝕩), which is 𝕩. Element-wise, we are combining the first i cells of 𝕩 with all but the first i. Looking at the entire result, we now know that (↑∾¨↓)𝕩 is (1+≠𝕩)⥊<𝕩. The total number of cells in this combined array is therefore (1+≠𝕩)×≠𝕩, or 1⊸+⊸×≠𝕩. Each of Prefixes and Suffixes must have the same total number of cells (in fact, ↑𝕩 is ⌽¨∘↓⌾⌽𝕩, and Reverse doesn't change an array's shape). So the total number in either one is 2÷˜1⊸+⊸×≠𝕩. With n←≠𝕩, it is 2÷˜n×1+n.

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Definition

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Definition

Knowing the length and the elements, it's easy to define functions for Prefixes and Suffixes: ↑ is equivalent to (↕1+≠)↑¨< while ↓ is (↕1+≠)↓¨<. Each primitive is defined only on arrays with at least one axis.

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Working with pairs

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Working with pairs

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:

↗️
    ×⌜˜ 1+↕6
 ┌─                  
@@ -70,7 +70,7 @@
     (<˘ ≍¨¨ ≠ ↑ ↑) "abc"
 ⟨ ⟨⟩ ⟨ "ba" ⟩ ⟨ "ca" "cb" ⟩ ⟩
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Slices

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Slices

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:

↗️
    ↓¨↑ "abc"
 ┌─                                                         
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