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22. Recursion
The factorial function ! is commonly defined
by the statement that factorial n is n times
factorial n1 , and by the further
statement that factorial 0 is 1 .
Such a definition is called recursive, because the
function being defined recurs in its definition.
A case statement can be used to make a recursive definition,
the case that employs the function under definition being
chosen repeatedly until the terminating case is encountered.
For example:
factorial=: 1:`(]*factorial@<:) @. *
factorial "0 i.6
1 1 2 6 24 120
Note that 1: denotes the constant function whose result
is 1 .
In the sentence (sum=: +/) i.5 the verb defined by the
phrase +/ is assigned a name before being used,
but in the sentence +/ i.5 it is used anonymously.
In the definition of factorial above, it was essential
to assign a name to make it possible to refer to it within the definition.
However, the word $: provides selfreference
that permits anonymous recursive definition. For example:
1:`(]*$:@<:) @. * "0 i. 6
1 1 2 6 24 120
In the Tower of Hanoi puzzle, a set of n discs (each
of a different size) is to be moved from post A to post B
using a third post C and under the restriction that a larger
disc is never to be placed on a smaller. The following is a
recursive definition of the process:
h=: b`(p,.q,.r)@.c
c=: 1: < [
b=: 2&,@[ $ ]
p=: <:@[ h 1: A. ]
q=: 1: h ]
r=: <:@[ h 5: A. ]
3 h x=: 'ABC'
AABACCA
BCCBABB
0 1 2 3 4 <@h"0 1 x
++++++
AAACAABACCAAACABBAACCBCAAC
BCBBBCCBABBCBBCACCBBAABCBB
++++++
Exercises
22.1 
Use the following as exercises in reading and writing:
f=:1:`(+//.@(,:~)@($:@<:))@.* Binomial Coeffs
<@f"0 i.6 Boxed binomials
g=:1:`((],+/@(_2&{.))@$:@<:)@.* Fibonacci

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