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<title>Direct function</title>

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Direct function

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</style><div role="note" class="hatnote navigation-not-searchable">"Dfns" redirects here. For the political entity sometimes known as the Democratic Federation of Northern Syria, see <a href="Autonomous_Administration_of_North_and_East_Syria" class="mw-redirect" title="Autonomous Administration of North and East Syria">Autonomous Administration of North and East Syria</a>.</div>

A direct function (dfn, pronounced "dee fun") is an alternative way to define a function and operator (a <a href="Higher-order_function" title="Higher-order function">higher-order function</a>) in the programming language <a href="APL_(programming_language)" title="APL (programming language)">APL</a>. A direct operator can also be called a dop (pronounced "dee op"). They were invented by <a href="John_M._Scholes" title="John M. Scholes">John Scholes</a> in 1996.<a href="#cite_note-Scholes1996-1">[1]</a> They are a unique combination of <a href="Array_programming" title="Array programming">array programming</a>, higher-order function, and <a href="Functional_programming" title="Functional programming">functional programming</a>, and are a major distinguishing advance of early 21st century APL over prior versions.

A dfn is a sequence of possibly <a href="Guard_(computer_science)" title="Guard (computer science)">guarded expressions</a> (or just a guard) between <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">{</code> and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">}</code>, separated by <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⋄</code> or new-lines, wherein <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺</code> denotes the left argument and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code> the right, and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">∇</code> denotes <a href="Recursion_(computer_science)" title="Recursion (computer science)">recursion</a> (function self-reference). For example, the function <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">PT</code> tests whether each row of <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code> is a <a href="Pythagorean_triplet" class="mw-redirect" title="Pythagorean triplet">Pythagorean triplet</a> (by testing whether the sum of squares equals twice the square of the maximum).

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> PT← {(+/⍵*2)=2×(⌈/⍵)*2}

PT 3 4 5

1

x

4 5 3

3 11 6

5 13 12

17 16 8

11 12 4

17 15 8

PT x

</pre></div>

The <a href="Factorial" title="Factorial">factorial</a> function as a dfn:

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> fact← {0=⍵:1 ⋄ ⍵×∇ ⍵-1}

fact 5

120

fact¨ ⍳10 ⍝ fact applied to each element of 0 to 9

1 1 2 6 24 120 720 5040 40320 362880

</pre></div>

<div class="mw-heading mw-heading2"><h2 id="Description">Description</h2></div>

The rules for dfns are summarized by the following "reference card":<a href="#cite_note-refcard-2">[2]</a>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">{⍺ function ⍵}</code>

</td>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">{⍺⍺ operator ⍵⍵}</code>

</td>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">:</code>   guard

</td></tr>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺</code>  left argument

</td>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺⍺</code>  left operand

</td>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">::</code>  error-guard

</td></tr>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code>  right argument

</td>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵⍵</code>  right operand

</td>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺←</code>  default left argument

</td></tr>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">∇</code>  self-reference

</td>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">∇∇</code>  self-reference

</td>

<td><code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">s←</code>  shy result

</td></tr></tbody></table>

A dfn is a sequence of possibly <a href="Guard_(computer_science)" title="Guard (computer science)">guarded expressions</a> (or just a guard) between <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">{</code> and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">}</code>, separated by <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⋄</code> or new-lines.

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>expression

guard: expression

guard:

</pre></div>

The expressions and/or guards are evaluated in sequence. A guard must evaluate to a 0 or 1; its associated expression is evaluated if the value is 1. A dfn terminates after the first unguarded expression which does not end in <a href="Assignment_(computer_science)" title="Assignment (computer science)">assignment</a>, or after the first guarded expression whose guard evaluates to 1, or if there are no more expressions. The result of a dfn is that of the last evaluated expression. If that last evaluated expression ends in assignment, the result is "shy"—not automatically displayed in the session.

Names assigned in a dfn are <a href="Local_variable" title="Local variable">local</a> by default, with <a href="Scope_(computer_science)#Lexical_scoping" title="Scope (computer science)">lexical scope</a>.

<code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺</code> denotes the left function argument and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code> the right; <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺⍺</code> denotes the left operand and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵⍵</code> the right. If <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵⍵</code> occurs in the definition, then the dfn is a dyadic <a href="Higher-order_function" title="Higher-order function">operator</a>; if only <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺⍺</code> occurs but not <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵⍵</code>, then it is a monadic operator; if neither <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺⍺</code> or <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵⍵</code> occurs, then the dfn is a function.

The special syntax <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺←expression</code> is used to give a default value to the left argument if a dfn is called monadically, that is, called with no left argument. The <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺←expression</code> is not evaluated otherwise.

<code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">∇</code> denotes <a href="Recursion_(computer_science)" title="Recursion (computer science)">recursion</a> or self-reference by the function, and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">∇∇</code> denotes self-reference by the operator. Such denotation permits <a href="Anonymous_recursion" title="Anonymous recursion">anonymous recursion</a>.

<a href="Exception_handling" title="Exception handling">Error trapping</a> is provided through error-guards, <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">errnums::expression</code>. When an error is generated, the system searches dynamically through the calling functions for an error-guard that matches the error. If one is found, the execution environment is unwound to its state immediately prior to the error-guard's execution and the associated expression of the error-guard is evaluated as the result of the dfn.

Additional descriptions, explanations, and tutorials on dfns are available in the cited articles.<a href="#cite_note-Scholes2001a-3">[3]</a><a href="#cite_note-Scholes2009b-4">[4]</a><a href="#cite_note-Scholes2009c-5">[5]</a><a href="#cite_note-Scholes2018v-6">[6]</a><a href="#cite_note-Scholes2018t-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

The examples here illustrate different aspects of dfns. Additional examples are found in the cited articles.<a href="#cite_note-dfnsWS-8">[8]</a><a href="#cite_note-APL1978-9">[9]</a><a href="#cite_note-history50-10">[10]</a>

<div class="mw-heading mw-heading3"><h3 id="Default_left_argument">Default left argument</h3></div>

The function <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">{⍺+0j1×⍵}</code> adds <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺</code> to <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">0j1</code> (i or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}">

<msqrt>

</msqrt>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}}</annotation>

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</math><img src="./4ea1ea9ac61e6e1e84ac39130f78143c18865719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" loading="lazy">) times <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code>.

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> 3 {⍺+0j1×⍵} 4

3J4

∘.{⍺+0j1×⍵}⍨ ¯2+⍳5

¯2J¯2 ¯2J¯1 ¯2 ¯2J1 ¯2J2

¯1J¯2 ¯1J¯1 ¯1 ¯1J1 ¯1J2

0J¯2 0J¯1 0 0J1 0J2

1J¯2 1J¯1 1 1J1 1J2

2J¯2 2J¯1 2 2J1 2J2

</pre></div>

The significance of this function can be seen as follows:

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</style><blockquote class="templatequote">Complex numbers can be constructed as ordered pairs of real numbers, similar to how integers can be constructed as ordered pairs of natural numbers and rational numbers as ordered pairs of integers. For complex numbers, <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">{⍺+0j1×⍵}</code> plays the same role as <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">-</code> for integers and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">÷</code> for rational numbers.<a href="#cite_note-Hui2016-11">[11]</a>: §8 </blockquote>

Moreover, analogous to that monadic <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">-⍵</code> ⇔ <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">0-⍵</code> (negate) and monadic <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">÷⍵</code> ⇔ <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">1÷⍵</code> (reciprocal), a monadic definition of the function is useful, effected by specifying a default value of 0 for <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺</code>: if <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">j←{⍺←0 ⋄ ⍺+0j1×⍵}</code>, then <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">j ⍵</code> ⇔ <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">0 j ⍵</code> ⇔ <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">0+0j1×⍵</code>.

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> j←{⍺←0 ⋄ ⍺+0j1×⍵}

3 j 4 ¯5.6 7.89

3J4 3J¯5.6 3J7.89

j 4 ¯5.6 7.89

0J4 0J¯5.6 0J7.89

sin← 1∘○

cos← 2∘○

Euler← {(*j ⍵) = (cos ⍵) j (sin ⍵)}

Euler (¯0.5+?10⍴0) j (¯0.5+?10⍴0)

1 1 1 1 1 1 1 1 1 1

</pre></div>

The last expression illustrates <a href="Euler's_formula" title="Euler's formula">Euler's formula</a> on ten random numbers with real and imaginary parts in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-0.5,0.5\right)}">

<mrow>

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</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \left(-0.5,0.5\right)}</annotation>

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</math><img src="./603e6507dd57b57ed41e527036b55b69e208c671.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.595ex; height:2.843ex;" alt="{\displaystyle \left(-0.5,0.5\right)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Single_recursion">Single recursion</h3></div>

The ternary construction of the <a href="Cantor_set" title="Cantor set">Cantor set</a> starts with the interval [0,1] and at each stage removes the middle third from each remaining subinterval:

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\biggl [}0,1{\biggr ]}\to }</annotation>

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</math><img src="./e7eb9d41d3abd96a641a6d0a42ef9f658e00206b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.783ex; height:6.176ex;" alt="{\displaystyle {\biggl [}0,1{\biggr ]}\to }" loading="lazy">

<mrow>

<mrow>

<mfrac>

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</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \left[0,{\frac {1}{3}}\right]\cup \left[{\frac {2}{3}},1\right]\to }</annotation>

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</math><img src="./14eda089dec5f40b78407470761df675f29eb29b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.851ex; height:6.176ex;" alt="{\displaystyle \left[0,{\frac {1}{3}}\right]\cup \left[{\frac {2}{3}},1\right]\to }" loading="lazy">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

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<mfrac>

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<mrow>

<mrow>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \left[0,{\frac {1}{9}}\right]\cup \left[{\frac {2}{9}},{\frac {1}{3}}\right]\cup \left[{\frac {2}{3}},{\frac {7}{9}}\right]\cup \left[{\frac {8}{9}},1\right]\to \cdots }</annotation>

</semantics>

</math><img src="./ca7e65312fc194a15cc9d6af51d003f1bf01f054.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.357ex; height:6.176ex;" alt="{\displaystyle \left[0,{\frac {1}{9}}\right]\cup \left[{\frac {2}{9}},{\frac {1}{3}}\right]\cup \left[{\frac {2}{3}},{\frac {7}{9}}\right]\cup \left[{\frac {8}{9}},1\right]\to \cdots }" loading="lazy">

The Cantor set of order <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code> defined as a dfn:<a href="#cite_note-Hui2016-11">[11]</a>: §2.5

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> Cantor← {0=⍵:,1 ⋄ ,1 0 1 ∘.∧ ∇ ⍵-1}

Cantor 0

1

Cantor 1

1 0 1

Cantor 2

Cantor 3

1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1

</pre></div>

Cantor 0 to Cantor 6 depicted as black bars:

The function <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">sieve ⍵</code> computes a bit vector of length <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code> so that bit <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">i</code> (for <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">0≤i</code> and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">i<⍵</code>) is 1 if and only if <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">i</code> is a <a href="Prime_number" title="Prime number">prime</a>.<a href="#cite_note-history50-10">[10]</a>: §46

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>sieve←{

4≥⍵:⍵⍴0 0 1 1

r←⌊0.5*⍨n←⍵

p←2 3 5 7 11 13 17 19 23 29 31 37 41 43

p←(1+(n≤×⍀p)⍳1)↑p

b← 0@1 ⊃ {(m⍴⍵)>m⍴⍺↑1 ⊣ m←n⌊⍺×≢⍵}⌿ ⊖1,p

{r<q←b⍳1:b⊣b[⍵]←1 ⋄ b[q,q×⍸b↑⍨⌈n÷q]←0 ⋄ ∇ ⍵,q}p

}

10 10 ⍴ sieve 100

0 0 1 1 0 1 0 1 0 0

0 1 0 1 0 0 0 1 0 1

0 0 0 1 0 0 0 0 0 1

0 1 0 0 0 0 0 1 0 0

0 1 0 1 0 0 0 0 0 1

0 0 0 0 0 0 0 1 0 0

b←sieve 1e9

≢b

1000000000

(10*⍳10) (+⌿↑)⍤0 1 ⊢b

0 4 25 168 1229 9592 78498 664579 5761455 50847534

</pre></div>

The last sequence, the number of primes less than powers of 10, is an initial segment of <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A006880" class="extiw external" title="oeis:A006880">A006880</a>. The last number, 50847534, is the number of primes less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{9}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 10^{9}}</annotation>

</semantics>

</math><img src="./c5f8dddd44eb018f09163e18ba8f19485e38bdb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{9}}" loading="lazy">. It is called Bertelsen's number, memorably described by <a href="MathWorld" title="MathWorld">MathWorld</a> as "an erroneous name erroneously given the erroneous value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (10^{9})=50847478}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi (10^{9})=50847478}</annotation>

</semantics>

</math><img src="./383fa70c48fe4b9fd912fe7bf6a81febb444e255.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.919ex; height:3.176ex;" alt="{\displaystyle \pi (10^{9})=50847478}" loading="lazy">".<a href="#cite_note-MathWorld-12">[12]</a>

<code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">sieve</code> uses two different methods to mark composites with 0s, both effected using local anonymous dfns: The first uses the <a href="Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">sieve of Eratosthenes</a> on an initial mask of 1 and a prefix of the primes 2 3...43, using the insert operator <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⌿</code> (<a href="Fold_(higher-order_function)" title="Fold (higher-order function)">right fold</a>). (The length of the prefix obtains by comparison with the <a href="Primorial" title="Primorial">primorial function</a> <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">×⍀p</code>.) The second finds the smallest new prime <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">q</code> remaining in <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">b</code> (<code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">q←b⍳1</code>), and sets to 0 bit <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">q</code> itself and bits at <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">q</code> times the numbers at remaining 1 bits in an initial segment of <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">b</code> (<code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍸b↑⍨⌈n÷q</code>). This second dfn uses tail recursion.

<div class="mw-heading mw-heading3"><h3 id="Tail_recursion">Tail recursion</h3></div>

Typically, the <a href="Factorial" title="Factorial">factorial</a> function is define recursively (as <a href="#factorial">above</a>), but it can be coded to exploit <a href="Tail_call" title="Tail call">tail recursion</a> by using an accumulator left argument:<a href="#cite_note-fact-13">[13]</a>

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>fac←{⍺←1 ⋄ ⍵=0:⍺ ⋄ (⍺×⍵) ∇ ⍵-1}

</pre></div>

Similarly, the <a href="Determinant" title="Determinant">determinant</a> of a square complex matrix using <a href="Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> can be computed with tail recursion:<a href="#cite_note-det-14">[14]</a>

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>det←{ ⍝ determinant of a square complex matrix

⍺←1 ⍝ product of co-factor coefficients so far

0=≢⍵:⍺ ⍝ result for 0-by-0

(i j)←(⍴⍵)⊤⊃⍒|,⍵ ⍝ row and column index of the maximal element

k←⍳≢⍵

(⍺×⍵[i;j]×¯1*i+j) ∇ ⍵[k~i;k~j] - ⍵[k~i;j] ∘.× ⍵[i;k~j]÷⍵[i;j]

}

</pre></div>

<div class="mw-heading mw-heading3"><h3 id="Multiple_recursion">Multiple recursion</h3></div>

A <a href="Integer_partition" title="Integer partition">partition</a> of a non-negative integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is not significant. For example, <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">2 2</code> and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">2 1 1</code> are partitions of 4, and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">2 1 1</code> and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">1 2 1</code> and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">1 1 2</code> are considered to be the same partition.

The <a href="Partition_function_(number_theory)" title="Partition function (number theory)">partition function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation>

</semantics>

</math><img src="./5e303d2c14cd399b6f52b468c9fd44a542bed422.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}" loading="lazy"> counts the number of partitions. The function is of interest in <a href="Number_theory" title="Number theory">number theory</a>, studied by <a href="Leonhard_Euler" title="Leonhard Euler">Euler</a>, <a href="G._H._Hardy" title="G. H. Hardy">Hardy</a>, <a href="Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Ramanujan</a>, <a href="Paul_Erd%C5%91s" title="Paul Erdős">Erdős</a>, and others. The recurrence relation

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(n)=\sum _{k=1}^{n}(-1)^{k+1}[P(n-{\frac {1}{2}}k(3k-1))+P(n-{\frac {1}{2}}k(3k+1))]}</annotation>

</semantics>

</math><img src="./2468d1bad94400d148e6de56414247554499a1fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.877ex; height:6.843ex;" alt="{\displaystyle P(n)=\sum _{k=1}^{n}(-1)^{k+1}[P(n-{\frac {1}{2}}k(3k-1))+P(n-{\frac {1}{2}}k(3k+1))]}" loading="lazy"></dd></dl>

derived from Euler's <a href="Pentagonal_number_theorem" title="Pentagonal number theorem">pentagonal number theorem</a>.<a href="#cite_note-MathWorldP-15">[15]</a> Written as a dfn:<a href="#cite_note-history50-10">[10]</a>: §16

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> pn ← {1≥⍵:0≤⍵ ⋄ -⌿+⌿∇¨rec ⍵}

rec ← {⍵ - (÷∘2 (×⍤1) ¯1 1 ∘.+ 3∘×) 1+⍳⌈0.5*⍨⍵×2÷3}

pn 10

42

pn¨ ⍳13 ⍝ OEIS A000041

1 1 2 3 5 7 11 15 22 30 42 56 77

</pre></div>

The basis step <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">1≥⍵:0≤⍵</code> states that for <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">1≥⍵</code>, the result of the function is <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">0≤⍵</code>, 1 if ⍵ is 0 or 1 and 0 otherwise. The recursive step is highly multiply recursive. For example, <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">pn 200</code> would result in the function being applied to each element of <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">rec 200</code>, which are:

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> rec 200

199 195 188 178 165 149 130 108 83 55 24 ¯10

198 193 185 174 160 143 123 100 74 45 13 ¯22

</pre></div>

and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">pn 200</code> requires longer than the <a href="Age_of_the_universe" title="Age of the universe">age of the universe</a> to compute (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7.57\times 10^{47}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 7.57\times 10^{47}}</annotation>

</semantics>

</math><img src="./c229aa95f0e45783f163a6209320fedd6f6f27af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.176ex; height:2.676ex;" alt="{\displaystyle 7.57\times 10^{47}}" loading="lazy"> function calls to itself).<a href="#cite_note-history50-10">[10]</a>: §16 The compute time can be reduced by <a href="Memoization" title="Memoization">memoization</a>, here implemented as the direct operator (higher-order function) <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">M</code>:

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>M←{

f←⍺⍺

i←2+'⋄'⍳⍨t←2↓,⎕cr 'f'

⍎'{T←(1+⍵)⍴¯1 ⋄ ',(i↑t),'¯1≢T[⍵]:⊃T[⍵] ⋄ ⊃T[⍵]←⊂',(i↓t),'⍵}⍵'

}

pn M 200

3.973E12

0 ⍕ pn M 200 ⍝ format to 0 decimal places

3972999029388

</pre></div>

This value of <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">pn M 200</code> agrees with that computed by Hardy and Ramanujan in 1918.<a href="#cite_note-16">[16]</a>

The memo operator <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">M</code> defines a variant of its operand function <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺⍺</code> to use a <a href="Cache_(computing)" title="Cache (computing)">cache</a> <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">T</code> and then evaluates it. With the operand <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">pn</code> the variant is:

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>{T←(1+⍵)⍴¯1 ⋄ {1≥⍵:0≤⍵ ⋄ ¯1≢T[⍵]:⊃T[⍵] ⋄ ⊃T[⍵]←⊂-⌿+⌿∇¨rec ⍵}⍵}

</pre></div>

<div class="mw-heading mw-heading3"><h3 id="Direct_operator_(dop)">Direct operator (dop)</h3></div>

<a href="Quicksort" title="Quicksort">Quicksort</a> on an array <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code> works by choosing a "pivot" at random among its major cells, then catenating the sorted major cells which strictly precede the pivot, the major cells equal to the pivot, and the sorted major cells which strictly follow the pivot, as determined by a comparison function <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺⍺</code>. Defined as a direct <a href="Higher-order_function" title="Higher-order function">operator</a> (dop) <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">Q</code>:

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> Q←{1≥≢⍵:⍵ ⋄ (∇ ⍵⌿⍨0>s)⍪(⍵⌿⍨0=s)⍪∇ ⍵⌿⍨0<s←⍵ ⍺⍺ ⍵⌷⍨?≢⍵}

⍝ precedes ⍝ follows ⍝ equals

2 (×-) 8 8 (×-) 2 8 (×-) 8

¯1 1 0

x← 2 19 3 8 3 6 9 4 19 7 0 10 15 14

(×-) Q x

0 2 3 3 4 6 7 8 9 10 14 15 19 19

</pre></div>

<code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">Q3</code> is a variant that catenates the three parts enclosed by the function <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⊂</code> instead of the parts per se. The three parts generated at each recursive step are apparent in the structure of the final result. Applying the function derived from <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">Q3</code> to the same argument multiple times gives different results because the pivots are chosen at random. <a href="Tree_traversal#In-order,_LNR" title="Tree traversal">In-order traversal</a> of the results does yield the same sorted array.

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> Q3←{1≥≢⍵:⍵ ⋄ (⊂∇ ⍵⌿⍨0>s)⍪(⊂⍵⌿⍨0=s)⍪⊂∇ ⍵⌿⍨0<s←⍵ ⍺⍺ ⍵⌷⍨?≢⍵}

(×-) Q3 x

┌────────────────────────────────────────────┬─────┬┐

│┌──────────────┬─┬─────────────────────────┐│19 19││

││┌──────┬───┬─┐│6│┌──────┬─┬──────────────┐││ ││

│││┌┬─┬─┐│3 3│4││ ││┌┬─┬─┐│9│┌┬──┬────────┐│││ ││

│││││0│2││ │ ││ ││││7│8││ │││10│┌──┬──┬┐││││ ││

│││└┴─┴─┘│ │ ││ ││└┴─┴─┘│ │││ ││14│15││││││ ││

││└──────┴───┴─┘│ ││ │ │││ │└──┴──┴┘││││ ││

││ │ ││ │ │└┴──┴────────┘│││ ││

││ │ │└──────┴─┴──────────────┘││ ││

│└──────────────┴─┴─────────────────────────┘│ ││

└────────────────────────────────────────────┴─────┴┘

(×-) Q3 x

┌───────────────────────────┬─┬─────────────────────────────┐

│┌┬─┬──────────────────────┐│7│┌────────────────────┬─────┬┐│

│││0│┌┬─┬─────────────────┐││ ││┌──────┬──┬────────┐│19 19│││

│││ │││2│┌────────────┬─┬┐│││ │││┌┬─┬─┐│10│┌──┬──┬┐││ │││

│││ │││ ││┌───────┬─┬┐│6│││││ │││││8│9││ ││14│15││││ │││

│││ │││ │││┌┬───┬┐│4│││ │││││ │││└┴─┴─┘│ │└──┴──┴┘││ │││

│││ │││ │││││3 3│││ │││ │││││ ││└──────┴──┴────────┘│ │││

│││ │││ │││└┴───┴┘│ │││ │││││ │└────────────────────┴─────┴┘│

│││ │││ ││└───────┴─┴┘│ │││││ │ │

│││ │││ │└────────────┴─┴┘│││ │ │

│││ │└┴─┴─────────────────┘││ │ │

│└┴─┴──────────────────────┘│ │ │

└───────────────────────────┴─┴─────────────────────────────┘

</pre></div>

The above formulation is not new; see for example Figure 3.7 of the classic The Design and Analysis of Computer Algorithms.<a href="#cite_note-AHU-17">[17]</a> However, unlike the <a href="Pidgin_code" title="Pidgin code">pidgin</a> <a href="ALGOL" title="ALGOL">ALGOL</a> program in Figure 3.7, <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">Q</code> is executable, and the partial order used in the sorting is an operand, the <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">(×-)</code> the examples above.<a href="#cite_note-APL1978-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Dfns_with_operators_and_trains">Dfns with operators and trains</h3></div>

Dfns, especially anonymous dfns, work well with operators and trains. The following snippet solves a "Programming Pearls" puzzle:<a href="#cite_note-Bentley1983-18">[18]</a> given a dictionary of English words, here represented as the character matrix <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">a</code>, find all sets of anagrams.

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre> a {⍵[⍋⍵]}⍤1 ⊢a ({⍵[⍋⍵]}⍤1 {⊂⍵}⌸ ⊢) a

pats apst ┌────┬────┬────┐

spat apst │pats│teas│star│

teas aest │spat│sate│ │

sate aest │taps│etas│ │

taps apst │past│seat│ │

etas aest │ │eats│ │

past apst │ │tase│ │

seat aest │ │east│ │

eats aest │ │seta│ │

tase aest └────┴────┴────┘

star arst

east aest

seta aest

</pre></div>

The algorithm works by sorting the rows individually (<code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">{⍵[⍋⍵]}⍤1 ⊢a</code>), and these sorted rows are used as keys ("signature" in the Programming Pearls description) to the key operator <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⌸</code> to group the rows of the matrix.<a href="#cite_note-APL1978-9">[9]</a>: §3.3 The expression on the right is a train, a syntactic form employed by APL to achieve <a href="Tacit_programming" title="Tacit programming">tacit programming</a>. Here, it is an isolated sequence of three functions such that <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">(f g h) ⍵</code> ⇔ <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">(f ⍵) g (h ⍵)</code>, whence the expression on the right is equivalent to <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">({⍵[⍋⍵]}⍤1 ⊢a) {⊂⍵}⌸ a</code>.

<div class="mw-heading mw-heading3"><h3 id="Lexical_scope">Lexical scope</h3></div>

When an inner (nested) dfn refers to a name, it is sought by looking outward through enclosing dfns rather than down the <a href="Call_stack" title="Call stack">call stack</a>. This regime is said to employ <a href="Scope_(computer_science)#Lexical_scoping" title="Scope (computer science)">lexical scope</a> instead of APL's usual <a href="Scope_(computer_science)#Dynamic_scoping" title="Scope (computer science)">dynamic scope</a>. The distinction becomes apparent only if a call is made to a function defined at an outer level. For the more usual inward calls, the two regimes are indistinguishable.<a href="#cite_note-Dyalog17.1-19">[19]</a>: p.137

For example, in the following function <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">which</code>, the variable <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">ty</code> is defined both in <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">which</code> itself and in the inner function <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">f1</code>. When <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">f1</code> calls outward to <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">f2</code> and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">f2</code> refers to <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">ty</code>, it finds the outer one (with value <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">'lexical'</code>) rather than the one defined in <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">f1</code> (with value <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">'dynamic'</code>):

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>which←{

ty←'lexical'

f1←{ty←'dynamic' ⋄ f2 ⍵}

f2←{ty,⍵}

f1 ⍵

}

which ' scope'

lexical scope

</pre></div>

<div class="mw-heading mw-heading3"><h3 id="Error-guard">Error-guard</h3></div>

The following function illustrates use of error guards:<a href="#cite_note-Dyalog17.1-19">[19]</a>: p.139

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>plus←{

tx←'catch all' ⋄ 0::tx

tx←'domain' ⋄ 11::tx

tx←'length' ⋄ 5::tx

⍺+⍵

}

2 plus 3 ⍝ no errors

5

2 3 4 5 plus 'three' ⍝ argument lengths don't match

length

2 3 4 5 plus 'four' ⍝ can't add characters

domain

2 3 plus 3 4⍴5 ⍝ can't add vector to matrix

catch all

</pre></div>

In APL, error number 5 is "length error"; error number 11 is "domain error"; and error number 0 is a "catch all" for error numbers 1 to 999.

The example shows the unwinding of the local environment before an error-guard's expression is evaluated. The local name <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">tx</code> is set to describe the purview of its following error-guard. When an error occurs, the environment is unwound to expose <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">tx</code>'s statically correct value.

<div class="mw-heading mw-heading2"><h2 id="Dfns_versus_tradfns">Dfns versus tradfns</h2></div>

Since direct functions are dfns, APL functions defined in the traditional manner are referred to as tradfns, pronounced "trad funs". Here, dfns and tradfns are compared by consideration of the function <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">sieve</code>: On the left is a dfn (as defined <a href="#sieve">above</a>); in the middle is a tradfn using <a href="Control_flow" title="Control flow">control structures</a>; on the right is a tradfn using <a href="Goto" title="Goto">gotos</a> (<code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">→</code>) and <a href="Label_(computer_science)" title="Label (computer science)">line labels</a>.

<td><div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>sieve←{

}

</pre></div>

</td>

<td><div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>∇ b←sieve1 n;i;m;p;q;r

:If 4≥n ⋄ b←n⍴0 0 1 1 ⋄ :Return ⋄ :EndIf

b←1

:For q :In p ⋄ b←(m⍴b)>m⍴q↑1 ⊣ m←n⌊q×≢b ⋄ :EndFor

b[1]←0

:While r≥q←b⍳1 ⋄ b[q,q×⍸b↑⍨⌈n÷q]←0 ⋄ p⍪←q ⋄ :EndWhile

b[p]←1

∇

</pre></div>

</td>

<td><div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>∇ b←sieve2 n;i;m;p;q;r

→L10 ⍴⍨ 4<n ⋄ b←n⍴0 0 1 1 ⋄ →0

L10:

p←(1+(n≤×\p)⍳1)↑p

i←0 ⋄ b←1

L20:

b←(m⍴b)>m⍴p[i]↑1 ⊣ m←n⌊p[i]×≢b

→L20 ⍴⍨ (≢p)>i←1+i

b[1]←0

L30:

→L40 ⍴⍨ r<q←b⍳1 ⋄ b[q,q×⍸b↑⍨⌈n÷q]←0 ⋄ p⍪←q ⋄ →L30

L40:

b[p]←1

∇

</pre></div>

</td></tr></tbody></table>

<ul><li>A dfn can be <a href="Anonymous_function" title="Anonymous function">anonymous</a>; a tradfn must be named.</li>

<li>A dfn is named by assignment (<code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">←</code>); a tradfn is named by embedding the name in the representation of the function and applying <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⎕fx</code> (a system function) to that representation.</li>

<li>A dfn is handier than a tradfn as an operand (see preceding items: a tradfn must be named; a tradfn is named by embedding ...).</li>

<li>Names <a href="Assignment_(computer_science)" title="Assignment (computer science)">assigned</a> in a dfn are <a href="Local_variable" title="Local variable">local</a> by default; names assigned in a tradfn are <a href="Global_variable" title="Global variable">global</a> unless specified in a locals list.</li>

<li>Locals in a dfn have <a href="Scope_(computer_science)#Lexical_scoping" title="Scope (computer science)">lexical scope</a>; locals in a tradfn have <a href="Scope_(computer_science)#Dynamic_scoping" title="Scope (computer science)">dynamic scope</a>, visible in called functions unless <a href="Variable_shadowing" title="Variable shadowing">shadowed</a> by their locals list.</li>

<li>The arguments of a dfn are named <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺</code> and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code> and the operands of a dop are named <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺⍺</code> and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵⍵</code>; the arguments and operands of a tradfn can have any name, specified on its leading line.</li>

<li>The result (if any) of a dfn is unnamed; the result (if any) of a tradfn is named in its header.</li>

<li>A default value for ⍺ is specified more neatly than for the left argument of a tradfn.</li>

<li><a href="Recursion_(computer_science)" title="Recursion (computer science)">Recursion</a> in a dfn is effected by invoking <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">∇</code> or <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">∇∇</code> or its name; recursion in a tradfn is effected by invoking its name.</li>

<li><a href="Control_flow" title="Control flow">Flow control</a> in a dfn is effected by guards and function calls; that in a tradfn is by control structures and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">→</code> (goto) and line labels.</li>

<li>Evaluating an expression in a dfn not ending in assignment causes return from the dfn; evaluating a line in a tradfn not ending in assignment or goto displays the result of the line.</li>

<li>A dfn returns on evaluating an expression not ending in assignment, on evaluating a guarded expression, or after the last expression; a tradfn returns on <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">→</code> (goto) line 0 or a non-existing line, or on evaluating a <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">:Return</code> control structure, or after the last line.</li>

<li>The simpler flow control in a dfn makes it easier to detect and implement <a href="Tail_call" title="Tail call">tail recursion</a> than in a tradfn.</li>

<li>A dfn may call a tradfn and vice versa; a dfn may be defined in a tradfn, and vice versa.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<a href="Kenneth_E._Iverson" title="Kenneth E. Iverson">Kenneth E. Iverson</a>, the inventor of APL, was dissatisfied with the way user functions (tradfns) were defined. In 1974, he devised "formal function definition" or "direct definition" for use in exposition.<a href="#cite_note-directdef-20">[20]</a> A direct definition has two or four parts, separated by colons:

<div class="mw-highlight mw-highlight-lang-apl mw-content-ltr" dir="ltr"><pre>name : expression

name : expression0 : proposition : expression1

</pre></div>

Within a direct definition, <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍺</code> denotes the left argument and <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">⍵</code> the right argument. In the first instance, the result of <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">expression</code> is the result of the function; in the second instance, the result of the function is that of <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">expression0</code> if <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">proposition</code> evaluates to 0, or <code class="mw-highlight mw-highlight-lang-apl mw-content-ltr" style="" dir="ltr">expression1</code> if it evaluates to 1. Assignments within a direct definition are <a href="Scope_(computer_science)#Dynamic_scoping" title="Scope (computer science)">dynamically local</a>. Examples of using direct definition are found in the 1979 <a href="Turing_Award" title="Turing Award">Turing Award</a> Lecture<a href="#cite_note-TOT-21">[21]</a> and in books and application papers.<a href="#cite_note-Iverson1976-22">[22]</a><a href="#cite_note-Orth1976-23">[23]</a><a href="#cite_note-Hui1987-24">[24]</a><a href="#cite_note-McDonnell1987-25">[25]</a><a href="#cite_note-APL1978-9">[9]</a>

Direct definition was too limited for use in larger systems. The ideas were further developed by multiple authors in multiple works<a href="#cite_note-opfns-26">[26]</a>: §8 <a href="#cite_note-IversonWooster1981-27">[27]</a><a href="#cite_note-Cheney-28">[28]</a>: §4.17 <a href="#cite_note-ratapl-29">[29]</a><a href="#cite_note-dictionary-30">[30]</a><a href="#cite_note-Bunda1987-31">[31]</a><a href="#cite_note-J-32">[32]</a> but the results were unwieldy. Of these, the "alternative APL function definition" of Bunda in 1987<a href="#cite_note-Bunda1987-31">[31]</a> came closest to current facilities, but is flawed in conflicts with existing symbols and in error handling which would have caused practical difficulties, and was never implemented. The main distillates from the different proposals were that (a) the function being defined is anonymous, with subsequent naming (if required) being effected by assignment; (b) the function is denoted by a symbol and thereby enables <a href="Anonymous_recursion" title="Anonymous recursion">anonymous recursion</a>.<a href="#cite_note-APL1978-9">[9]</a>

In 1996, <a href="John_M._Scholes" title="John M. Scholes">John Scholes</a> of Dyalog Limited invented direct functions (dfns).<a href="#cite_note-Scholes1996-1">[1]</a><a href="#cite_note-Scholes2018v-6">[6]</a><a href="#cite_note-Scholes2018t-7">[7]</a> The ideas originated in 1989 when he read a special issue of <a href="The_Computer_Journal" title="The Computer Journal">The Computer Journal</a> on functional programming.<a href="#cite_note-Wadler-33">[33]</a> He then proceeded to study functional programming and became strongly motivated ("sick with desire", like <a href="W._B._Yeats" title="W. B. Yeats">Yeats</a>) to bring these ideas to APL.<a href="#cite_note-Scholes2018v-6">[6]</a><a href="#cite_note-Scholes2018t-7">[7]</a> He initially operated in stealth because he was concerned the changes might be judged too radical and an unnecessary complication of the language; other observers say that he operated in stealth because Dyalog colleagues were not so enamored and thought he was wasting his time and causing trouble for people. Dfns were first presented in the Dyalog Vendor Forum at the APL '96 Conference and released in Dyalog APL in early 1997.<a href="#cite_note-Scholes1996-1">[1]</a> Acceptance and recognition were slow in coming. As late as 2008, in Dyalog at 25,<a href="#cite_note-Dyalog@25-34">[34]</a> a publication celebrating the 25th anniversary of Dyalog Limited, dfns were barely mentioned (mentioned twice as "dynamic functions" and without elaboration). As of 2019, dfns are implemented in Dyalog APL,<a href="#cite_note-Dyalog17.1-19">[19]</a> NARS2000,<a href="#cite_note-NARS2000-35">[35]</a> and ngn/apl.<a href="#cite_note-Nickolov2013-36">[36]</a> They also play a key role in efforts to exploit the computing abilities of a <a href="Graphics_processing_unit" title="Graphics processing unit">graphics processing unit</a> (GPU).<a href="#cite_note-Hsu2019-37">[37]</a><a href="#cite_note-APL1978-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-TOT-21"><a href="#cite_ref-TOT_21-0">^</a> <cite id="CITEREFIverson1980" class="citation journal cs1"><a href="Kenneth_E._Iverson" title="Kenneth E. Iverson">Iverson, Kenneth E.</a> (August 1980). <a rel="nofollow" class="external text" href="https://www.jsoftware.com/papers/tot.htm">"Notation as a Tool of Thought"</a>. <a href="Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a>. 23 (8): 444–465. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F358896.358899">10.1145/358896.358899</a>. Retrieved 8 April 2016.</cite>

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<li id="cite_note-Iverson1976-22"><a href="#cite_ref-Iverson1976_22-0">^</a> <cite id="CITEREFIverson1976" class="citation book cs1"><a href="Kenneth_E._Iverson" title="Kenneth E. Iverson">Iverson, Kenneth E.</a> (1976). Elementary Analysis. APL Press.</cite>

</li>

<li id="cite_note-Orth1976-23"><a href="#cite_ref-Orth1976_23-0">^</a> <cite id="CITEREFOrth1976" class="citation book cs1">Orth, D.L. (1976). Calculus in a New Key. APL Press.</cite>

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<li id="cite_note-Hui1987-24"><a href="#cite_ref-Hui1987_24-0">^</a> <cite id="CITEREFHui1987" class="citation journal cs1"><a href="Roger_Hui" title="Roger Hui">Hui, Roger</a> (May 1987). <a rel="nofollow" class="external text" href="http://www.jsoftware.com/papers/from.htm">"Some Uses of { and }"</a>. APL 87 Conference Proceedings. Retrieved 15 April 2016.</cite>

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<li id="cite_note-McDonnell1987-25"><a href="#cite_ref-McDonnell1987_25-0">^</a> <cite id="CITEREFMcDonnell1987" class="citation cs2">McDonnell, E.E. (May 1987), <a rel="nofollow" class="external text" href="http://www.jsoftware/papers/EEM/life.htm">"Life: Nasty, Brutish, and Short"</a>, APL 87 Conference Proceedings, retrieved 6 October 2019</cite>

</li>

<li id="cite_note-opfns-26"><a href="#cite_ref-opfns_26-0">^</a> <cite id="CITEREFIverson1978" class="citation cs2"><a href="Kenneth_E._Iverson" title="Kenneth E. Iverson">Iverson, Kenneth E.</a> (26 April 1978), <a rel="nofollow" class="external text" href="http://www.jsoftware.com/papers/opfns.htm">"Operators and Functions"</a>, Research Report Number #RC7091, IBM Corporation, retrieved 2019-09-19</cite>

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<li id="cite_note-IversonWooster1981-27"><a href="#cite_ref-IversonWooster1981_27-0">^</a> <cite id="CITEREFIversonWooster1981" class="citation journal cs1"><a href="Kenneth_E._Iverson" title="Kenneth E. Iverson">Iverson, Kenneth E.</a>; Wooster, Peter (September 1981). "A Function Definition Operator". APL81 Conference Proceedings, APL Quote Quad. 12 (1).</cite>

</li>

<li id="cite_note-Cheney-28"><a href="#cite_ref-Cheney_28-0">^</a> <cite id="CITEREFCheney1981" class="citation cs2">Cheney, Carl M. (March 1981), <a rel="nofollow" class="external text" href="http://www.sudleyplace.com/APL/Nested%20Arrays%20System.pdf">APL*Plus Nested Array System Reference Manual</a> (PDF), <a href="Scientific_Time_Sharing_Corporation" title="Scientific Time Sharing Corporation">STSC, Inc.</a>, retrieved 18 September 2019</cite>

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<li id="cite_note-ratapl-29"><a href="#cite_ref-ratapl_29-0">^</a> <cite id="CITEREFIverson1983" class="citation cs2"><a href="Kenneth_E._Iverson" title="Kenneth E. Iverson">Iverson, Kenneth E.</a> (6 January 1983), <a rel="nofollow" class="external text" href="http://www.jsoftware.com/papers/RationalizedAPL.htm">Rationalized APL</a>, <a href="I._P._Sharp_Associates" title="I. P. Sharp Associates">I. P. Sharp Associates</a>, retrieved 2019-09-19</cite>

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<li id="cite_note-dictionary-30"><a href="#cite_ref-dictionary_30-0">^</a> <cite id="CITEREFIverson1987" class="citation journal cs1"><a href="Kenneth_E._Iverson" title="Kenneth E. Iverson">Iverson, Kenneth E.</a> (September 1987). <a rel="nofollow" class="external text" href="http://www.jsoftware.com/papers/APLDictionary.htm">"A Dictionary of APL"</a>. APL Quote Quad. 18 (1): 5–40. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F36983.36984">10.1145/36983.36984</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18301178">18301178</a>. Retrieved 19 September 2019.</cite>

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<li id="cite_note-Bunda1987-31">^ <a href="#cite_ref-Bunda1987_31-0">a</a> <a href="#cite_ref-Bunda1987_31-1">b</a> <cite id="CITEREFBunda1987" class="citation journal cs1">Bunda, John (May 1987). "APL Function Definition Notation". APL87 Conference Proceedings, APL Quote Quad. 17 (4).</cite>

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<li id="cite_note-J-32"><a href="#cite_ref-J_32-0">^</a> <cite id="CITEREFHui1990" class="citation book cs1"><a href="Roger_Hui" title="Roger Hui">Hui, Roger</a>; et al. (July 1990). <a rel="nofollow" class="external text" href="http://www.jsoftware.com/papers/J1990.htm">"APL\?"</a>. Conference proceedings on APL 90: For the future. Vol. 20. pp. 192–200. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F97808.97845">10.1145/97808.97845</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>089791371X</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:235453656">235453656</a>. Retrieved 2019-09-10.</cite>

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<li id="cite_note-Wadler-33"><a href="#cite_ref-Wadler_33-0">^</a> <cite id="CITEREFWadler1989" class="citation journal cs1">Wadler, Philip L.; et al. (1 January 1989). "Special Issue on Functional Programming". <a href="The_Computer_Journal" title="The Computer Journal">The Computer Journal</a>. 32 (2).</cite>

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<li id="cite_note-Dyalog@25-34"><a href="#cite_ref-Dyalog@25_34-0">^</a> <cite id="CITEREFDyalog2008" class="citation journal cs1">Dyalog (September 2008). <a rel="nofollow" class="external text" href="http://www.dyalog.com/uploads/documents/dyalog_25.pdf">"Dyalog at 25"</a> (PDF). Vector. Retrieved 2019-09-20.</cite>

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<li id="cite_note-NARS2000-35"><a href="#cite_ref-NARS2000_35-0">^</a> <cite id="CITEREFSmith2006–2019" class="citation cs2">Smith, Bob (2006–2019), <a rel="nofollow" class="external text" href="http://www.nars2000.org/">NARS2000</a>, retrieved 18 September 2019</cite>

</li>

<li id="cite_note-Nickolov2013-36"><a href="#cite_ref-Nickolov2013_36-0">^</a> <cite id="CITEREFNickolov2013" class="citation journal cs1">Nickolov, Nick (September 2013). <a rel="nofollow" class="external text" href="http://archive.vector.org.uk/art10501160">"Compiling APL to JavaScript"</a>. Vector. 26 (1). Retrieved 19 September 2019.</cite>

</li>

<li id="cite_note-Hsu2019-37"><a href="#cite_ref-Hsu2019_37-0">^</a> <cite id="CITEREFHsu2019" class="citation thesis cs1">Hsu, Aaron (2019). <a rel="nofollow" class="external text" href="http://scholarworks.iu.edu/dspace/bitstream/handle/2022/24749/Hsu%20Dissertation.pdf">A Data Parallel Compiler Hosted on a GPU</a> (PDF) (Ph.D. thesis). <a href="Indiana_University" title="Indiana University">Indiana University</a>. Retrieved 25 December 2019.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://dfns.dyalog.com">Official website</a>, Dyalog</li></ul>

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</style><div id="APL_programming493" style="font-size:114%;margin:0 4em"><a href="APL_(programming_language)" title="APL (programming language)">APL programming</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Features</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="APL_syntax_and_symbols" title="APL syntax and symbols">APL syntax and symbols</a></li>

<li><a href="Digital_encoding_of_APL_symbols" title="Digital encoding of APL symbols">Code pages</a></li>

<li><a href="Iverson_bracket" title="Iverson bracket">Iverson bracket</a></li>

<li><a href="Shared_Variables" title="Shared Variables">Shared Variables</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Programming_language_implementation" title="Programming language implementation">Implementations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Major</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Dyalog APL</li>

<li>SHARP APL</li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Dialect_(computing)" class="mw-redirect" title="Dialect (computing)">Dialects</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li>APLNext → VisualAPL</li>

<li><a href="Polymorphic_Programming_Language" title="Polymorphic Programming Language">Polymorphic Programming Language</a> (PPL)</li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Community_of_practice" title="Community of practice">Community</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Professional_association" title="Professional association">Professional associations</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Association_for_Computing_Machinery" title="Association for Computing Machinery">Association for Computing Machinery</a>: SIGAPL</li>

<li>British APL Association</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Business</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ampere_WS-1" title="Ampere WS-1">Ampere</a></li>

<li>Analogic Corporation</li>

<li><a href="CompuServe" title="CompuServe">CompuServe</a></li>

<li><a href="Digital_Equipment_Corporation" title="Digital Equipment Corporation">Digital Equipment Corporation</a> (DEC)</li>

<li>DNA Systems</li>

<li>Dyalog Ltd.</li>

<li><a href="I._P._Sharp_Associates" title="I. P. Sharp Associates">I. P. Sharp Associates</a></li>

<li><a href="Kx_Systems" class="mw-redirect" title="Kx Systems">Kx Systems</a></li>

<li><a href="MCM/70" title="MCM/70">Micro Computer Machines</a> (MCM)</li>

<li><a href="Science_Research_Associates" title="Science Research Associates">Science Research Associates</a></li>

<li><a href="Scientific_Time_Sharing_Corporation" title="Scientific Time Sharing Corporation">Scientific Time Sharing Corporation</a> (STSC)</li>

<li><a href="Soliton_Incorporated" title="Soliton Incorporated">Soliton Incorporated</a></li>

<li>Telecompute Integrated Systems, Inc.</li>

<li>Time Sharing Resources (TSR)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Education</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carnegie_Mellon_University" title="Carnegie Mellon University">Carnegie Mellon University</a> (CMU)</li>

<li><a href="University_of_Maryland%2C_College_Park" title="University of Maryland, College Park">University of Maryland</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Philip_S._Abrams" title="Philip S. Abrams">Phil Abrams</a></li>

<li><a href="Robert_Bernecky" title="Robert Bernecky">Bob Bernecky</a></li>

<li><a href="Lawrence_M._Breed" title="Lawrence M. Breed">Larry Breed</a></li>

<li><a href="Charles_Brenner_(mathematician)" title="Charles Brenner (mathematician)">Charles Brenner</a></li>

<li><a href="Fred_Brooks" title="Fred Brooks">Fred Brooks</a></li>

<li><a href="Jim_Brown_(computer_scientist)" title="Jim Brown (computer scientist)">Jim Brown</a></li>

<li><a href="Adin_Falkoff" title="Adin Falkoff">Adin Falkoff</a></li>

<li>Patrick E. Hagerty</li>

<li>Herbert Hellerman</li>

<li><a href="Roger_Hui" title="Roger Hui">Roger Hui</a></li>

<li><a href="Kenneth_E._Iverson" title="Kenneth E. Iverson">Kenneth E. Iverson</a></li>

<li><a href="Richard_H._Lathwell" title="Richard H. Lathwell">Dick Lathwell</a></li>

<li><a href="Eugene_McDonnell" title="Eugene McDonnell">Eugene McDonnell</a></li>

<li>Robert Metzger</li>

<li><a href="Roger_Moore_(computer_scientist)" title="Roger Moore (computer scientist)">Roger Moore</a></li>

<li><a href="Alan_Perlis" title="Alan Perlis">Alan Perlis</a></li>

<li><a href="John_M._Scholes" title="John M. Scholes">John Scholes</a></li>

<li>J. Henri Schueler</li>

<li>Bob Smith</li>

<li><a href="Edward_H._Sussenguth" title="Edward H. Sussenguth">Edward H. Sussenguth</a> Jr.</li>

<li><a href="Arthur_Whitney_(computer_scientist)" title="Arthur Whitney (computer scientist)">Arthur Whitney</a></li>

<li>William Yerazunis</li>

<li><a href="Rodnay_Zaks" title="Rodnay Zaks">Rodnay Zaks</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Iverson_Award" title="Iverson Award">Iverson Award</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li>Italics = discontinued</li>

<li>° = <a href="Open-source_software" title="Open-source software">Open-source software</a> <a href="https://commons.wikimedia.org/wiki/Category:APL_(programming_language)" class="extiw external" title="commons:Category:APL (programming language)">Commons</a> Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-28" href="https://en.wikipedia.org/wiki/?title=Direct_function&oldid=1292720546">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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