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== Description ==
The rules for dfns are summarized by the following "reference card":<ref name=refcard>{{citation|last=Scholes|first=John|title=Direct Functions Reference Card|date=
{| class="wikitable"
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== Examples ==
The examples here illustrate different aspects of dfns. Additional examples are found in the cited articles.<ref name=dfnsWS>{{citation|last=Scholes|first=John|title=Direct Functions Workspace|date=
=== Default left argument ===
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=== Tail recursion ===
Typically, the [[factorial]] function is define recursively (as [[#factorial|above]]), but it can be coded to exploit [[tail call|tail recursion]] by using an accumulator left argument:<ref name=fact>{{citation|last=Scholes|first=John|title=Factorial|date=
<syntaxhighlight lang=apl>
fac←{⍺←1 ⋄ ⍵=0:⍺ ⋄ (⍺×⍵) ∇ ⍵-1}
</syntaxhighlight>
Similarly, the [[determinant]] of a square complex matrix using [[Gaussian elimination]] can be computed with tail recursion:<ref name=det>{{citation|last=Scholes|first=John|title=Determinant|date=
<syntaxhighlight lang=apl>
det←{ ⍝ determinant of a square complex matrix
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{r<q←b⍳1:b⊣b[⍵]←1 ⋄ b[q,q×⍸b↑⍨⌈n÷q]←0 ⋄ ∇ ⍵,q}p
}
</pre>
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b[p]←1
∇
</pre>
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Direct definition was too limited for use in larger systems. The ideas were further developed by multiple authors in multiple works<ref name=opfns>{{citation|last=Iverson|first=Kenneth E.|author-link=Kenneth E. Iverson|title=Operators and Functions|journal=Research Report Number #RC7091|date=26 April 1978|publisher=IBM Corporation|url=http://www.jsoftware.com/papers/opfns.htm|access-date=2019-09-19}}</ref>{{rp|§8}}<ref name=IversonWooster1981>{{cite journal|last1=Iverson|first1=Kenneth E.|author-link=Kenneth E. Iverson|last2=Wooster|first2=Peter|title=A Function Definition Operator|journal=APL81 Conference Proceedings, APL Quote Quad|volume=12|number=1|date=September 1981}}</ref><ref name=Cheney>{{citation|last=Cheney|first=Carl M.|title=APL*Plus Nested Array System Reference Manual|date=March 1981|publisher=[[Scientific Time Sharing Corporation|STSC, Inc.]]|url=http://www.sudleyplace.com/APL/Nested%20Arrays%20System.pdf|access-date=18 September 2019}}</ref>{{rp|§4.17}}<ref name=ratapl>{{citation|last=Iverson|first=Kenneth E.|author-link=Kenneth E. Iverson|title=Rationalized APL|date=6 January 1983|publisher=[[I. P. Sharp Associates]]|url=http://www.jsoftware.com/papers/RationalizedAPL.htm|access-date=2019-09-19}}</ref><ref name=dictionary>{{cite journal|last=Iverson|first=Kenneth E.|author-link=Kenneth E. Iverson|title=A Dictionary of APL|journal=APL Quote Quad|volume=18|number=1|date=September 1987|pages=5–40|doi=10.1145/36983.36984|s2cid=18301178|url=http://www.jsoftware.com/papers/APLDictionary.htm|access-date=19 September 2019}}</ref><ref name=Bunda1987>{{cite journal|last=Bunda|first=John|title=APL Function Definition Notation|journal=APL87 Conference Proceedings, APL Quote Quad|volume=17|number=4|date=May 1987}}</ref><ref name=J>{{cite journal|last=Hui|first=Roger|author-link=Roger Hui|display-authors=etal|title=APL\?|journal=APL90 Conference Proceedings, APL Quote Quad|volume=20|number=4|date=July 1990|url=http://www.jsoftware.com/papers/J1990.htm|access-date=2019-09-10}}</ref> but the results were unwieldy. Of these, the "alternative APL function definition" of Bunda in 1987<ref name=Bunda1987/> 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 [[anonymous recursion]].<ref name=APL1978/>
In 1996, [[John M. Scholes|John Scholes]] of Dyalog Limited invented direct functions (dfns).<ref name=Scholes1996/><ref name=Scholes2018v/><ref name=Scholes2018t/> The ideas originated in 1989 when he read a special issue of ''[[The Computer Journal]]'' on functional programming.<ref name=Wadler>{{cite journal|last=Wadler|first=Philip L.|display-authors=etal|title=Special Issue on Functional Programming|journal=[[The Computer Journal]]|volume=32|number=2|date=1 January 1989}}</ref> He then proceeded to study functional programming and became strongly motivated ("sick with desire", like [[W.B. Yeats|Yeats]]) to bring these ideas to APL.<ref name=Scholes2018v/><ref name=Scholes2018t/> 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.<ref name=Scholes1996/> Acceptance and recognition were slow in coming. As late as 2008, in ''Dyalog at 25'',<ref name=Dyalog@25>{{cite journal|last=Dyalog|title=Dyalog at 25|journal=Vector|date=September 2008|url=http://www.dyalog.com/uploads/documents/dyalog_25.pdf|access-date=2019-09-20}}</ref> 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,<ref name=Dyalog17.1/> NARS2000,<ref name=NARS2000>{{citation|last=Smith|first=Bob|title=NARS2000|date=
== References ==
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