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29. Secondaries

It is convenient to supplement the primitives or primaries provided in a language by secondaries whose names belong to an easily recognized class. The following examples use names beginning with a capital letter:

Ad=: [ 0:}-@>:@$@]{.] Append diagonal scalar
Ai=: >:@i. Augmented integers
Area=: [: Det ] ,. %@!@#"1 Area (Vol) try Area tet=:0,=i.3
Bc=: i. !/ i. Binomial coefficients
Bca=: %.@Bc Binomial coefficients (alternating)
By=: ' '&;@,.@[ ,. ] By (format; see Ta)
Cpa=: ]%.i.@#@]^/Ei@[ Coeffs of poly approx
CPA=: (@])%.i.@#@]^/Ei@[ Coeffs of poly approx (adverb)
Det=: -/ . * Determinant
Dpc=: 1: }. ] * i.@# Differentiate poly coeffs
D1=: ("0)(D.1) Derivative (scalar, first)
Ei=: i.@(+*+0&=) Extended integers
Epc=: Bc@# X ] Expand poly coeffs
Ipc=: 0: , ] % Ai@# Integrate poly coeffs
Inv=: ^:_1 Inverse
Id=: =@i. Identity matrix
Mat=: -: /:~ Monotone ascending test
Mdt=: -: \:~ Monotone descending test
Mrg=: +&$ {. ,@(|:@,:) Merge
Over=:   ({.;}.)@":@, Over (format; see Ta)
Pad=: 2 :'x.%.]^/Ei@(y."_)' Polynomial approx of degree
Pp=: +//.@(*/) Polynomial coeffs product
Si=: (Ei@+: - |) : (-/ i.) Symmetric and subsiding int
Span=: 2 : 'y."_ x.\ ]' Span of apply of left arg
S1=: |:@|@(^!._1/~%.^/~)@i.   Stirling numbers (1st kind)
S2=: |:@ (^/~%.^!._1/~)@i. Stirling numbers (2nd kind)
Ta=: 1 : '[By]Over x./' Table adverb
Thr=: ] * 0.1&^@[ <: |@] Threshold for non-zero
Tile=: $@]{.[$~$@]+2:|1:+$@] Tile (try 0 1 Tile i. 2 3 4)
X=: +/ . * Times (matrix product)
XA=: -/ . * Times (alternating)


Exercises

29.1   Enter the definitions of the secondaries (or at least those used in these exercises), and then enter the following expressions:
(Ai 2 3);(Ai 2 _3);(Ei 2 3);(Ei 2 _3)
(i.;i.@-;Ai;Ai@-;Ei;Ei@-) 4
(Si 4);(7 4 Si 4)
+Ta~@i. 4
(S1;S2) 7
(];X/;%/;%./;(%./%{.)) y=: (Bc ,: Bca) 5
(0 1&Cb;1 _1&Cb) i. 2 3 4
29.2   Perform further experiments with the secondaries.
 


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