nvec ← from ##.to unto ⍝ Sequence ⍺ .. ⍵
Returns a vector of numbers ⍺ to ⍵ inclusive. ⍺ is the _first_ value in the seq-
uence, and ⍵ the _last_. More generally, ⍺ may be a 2-item vector, which determ-
ines the first two items of the sequence and hence the _step_ size.
Notice that any of _first_, _step_ and _last_ may be positive, negative and/or
fractional. If _last_ is not a whole number of _step_s from _first_, the
sequence will stop short of _last_.
Technical notes:
The following simple function suffices for the ascending sequence ⍺, ⍺+1, ··· ⍵.
Notice how (·-⎕io) makes it independent of index origin:
to←{(⍺-⎕io)+⍳1+⍵-⍺}
Useful refinements to the function allow _descending_ sequences and optional
_step_ sizes other than 1:
to←{⎕io←0 ⍝ Sequence ⍺ .. ⍵
from step←1 ¯1×-\2↑⍺,⍺+×⍵-⍺ ⍝ step default is +/- 1.
from+step×⍳0⌈1+⌊(⍵-from)÷step+step=0 ⍝ ⍺ thru ⍵ inclusive.
}
Attempts to re-cast the function as a one-liner, include:
to←{From+0,+\Step⍴⍨0⌈⌊Inc÷Step←--/2⍴⍺,From+×Inc←⍵-From←1⍴⍺}
to←{↑+/⍵{⍵×{⍵-⎕IO}⍳1+0⌈⌊(⍺⍺-⍺)÷⍵+⍵=0}\1 ¯1×-\2↑⍺,⍺+×⍵-⍺}
to←{(⍬⍴⍺)+(|-/⌽2↑⍺,⍺+1){⍺×(×⍵)×(-⎕IO)+⍳1+⌊|⍵÷⍺}⍵-⍬⍴⍺}
VMJ suggests the following extension, which accepts an optional explicit _step_
size as a second item of its _right_ argument. _step_ defers to _next_ if both
are supplied.
to←{
⎕IO←0 ⋄ ⍺←1 ⋄ z←0.0000000001
(a n)←2↑⍺,⍺ ⋄ (b s)←2↑⍵,1
d←|s{⍵≠0:⍵ ⋄ ⍺≠0:⍺ ⋄ 1}n-a
{⍵×z<|0-⍵}a+d{(⍺××⍵)×⍳1+⌊|⍵÷⍺}b-a
}
1 3 to 5 ⍝ using: _next_
1 3 5
1 to 5 2 ⍝ using: _step_
1 3 5
Bob Smith suggests this extension for nested sequences:
to←{ ⍝ Sequence ⍺ .. ⍵
from step←⊂¨1 ¯1×-\2↑⍺,⍺+×⍵-⍺ ⍝ step default is +/- 1.
size←0⌈1+⌊⊃(⍵-from)÷step+step=0 ⍝ shape of result
from+step×(⍳size)-⎕io ⍝ ⍺ thru ⍵ inclusive.
}
1 disp (¯1 6)(0 4) to ⊂3 ¯2
┌→───┬────┬────┬────┬─────┐
↓¯1 6│¯1 4│¯1 2│¯1 0│¯1 ¯2│
├~──→┼~──→┼~──→┼~──→┼~───→┤
│0 6 │0 4 │0 2 │0 0 │0 ¯2 │
├~──→┼~──→┼~──→┼~──→┼~───→┤
│1 6 │1 4 │1 2 │1 0 │1 ¯2 │
├~──→┼~──→┼~──→┼~──→┼~───→┤
│2 6 │2 4 │2 2 │2 0 │2 ¯2 │
├~──→┼~──→┼~──→┼~──→┼~───→┤
│3 6 │3 4 │3 2 │3 0 │3 ¯2 │
└~──→┴~──→┴~──→┴~──→┴~───→┘
We could extend Bob's idea to accommodate character and complex arguments:
to←{ ⍝ Sequence ⍺ .. ⍵
type←10|⎕DR ⍺,⍵ ⍝ item type.
type∊0 2:⎕UCS(⎕UCS ⍺)∇ ⎕UCS ⍵ ⍝ char sequences.
type=9:⍺ ∇{ ⍝ complex arg
j←1 0J1∘(+.×) ⍝ number from (re im) pair
p←9 11∘○¨ ⍝ (re im) pair from number
j¨(p ⍺)⍺⍺ p ⍵ ⍝ complex sequence via nested pairs.
}⍵ ⍝
from step←⊂¨1 ¯1×-\2↑⍺,⍺+×⍵-⍺ ⍝ step default is +/- 1.
size←0⌈1+⌊⊃(⍵-from)÷step+step=0 ⍝ shape of result
from+step×(⍳size)-⎕IO ⍝ ⍺ thru ⍵ inclusive.
}
'a' to 'z'
abcdefghijklmnopqrstuvwxyz
'AC' to 'Z'
ACEGIKMOQSUWY
0 to 3j4
0 0J1 0J2 0J3 0J4
1 1J1 1J2 1J3 1J4
2 2J1 2J2 2J3 2J4
3 3J1 3J2 3J3 3J4
Adám Brudzewsky suggests the following version. Adám says:
Fills in the sequence gap between the right end of ⍺ and the left end of ⍵.
Depends on the data type of the first element of ⍵, and on the number of trail-
ing elements in ⍺ with the same data type. For a numeric first element of ⍵ and
with two or more trailing numbers in ⍺, it extrapolates from the last two elem-
ents of ⍺, letting their difference determine the step size. With one (for
character first elements of ⍵: or more) trailing compatible element(s) in ⍺, it
fills the gap by using a step size of ±1 from the last element of ⍺. If there
are no trailing compatible elements in ⍺ or if called monadically, it begins the
sequence at 0, using a step size of ±1.
to←{⎕IO←0 ⍝ Fill sequence gap
Char←0 2∊⍨10|⎕DR ⍝ character?
end←⊃⍵ ⍝ of sub-sequence
tail←1↓⍵ ⍝ to be appended
charend←Char end ⍝ character ⍵?
default←⎕UCS⍣charend⊢0 ⍝ default begin from 0
⍺←default ⍝ default if monadic
charbegins←Char¨¯2↑⍺ ⍝ character ⍺?
lead←-(2-charend)⌊(≢⍺)⌊+/charend=charbegins ⍝ to be considered
head←lead↓⍺ ⍝ to be prepended
begin←(¯1⌊lead)↑¯2↑default,lead↑⍺ ⍝ first one/two items
charend:head,tail,⍨⎕UCS(⎕UCS begin)∇ ⎕UCS end ⍝ char sequences
from step←-⍨\2↑begin,begin+×end-begin ⍝ step default is +/- 1
head,tail,⍨from+step×⍳0⌈1+⌊(end-from)÷step+step=0 ⍝ ⍺ thru ⍵ inclusive
}
'a' to 'zA' to 'Z' ⍝ English alphabet
abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ
'a'to'zæøåA'to'ZÆØÅ' ⍝ Danish alphabet
abcdefghijklmnopqrstuvwxyzæøåABCDEFGHIJKLMNOPQRSTUVWXYZÆØÅ
to 10 20 to 100 200 300 400 ⍝ Numbers
0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400
0 ¯1J2 to ¯4J8 ⍝ Complex numbers
0 ¯1J2 ¯2J4 ¯3J6 ¯4J8
'='¯2 to 3 9 8.5 to 6,'+-×÷À' to 'Å' to 3 ⍝ Mixed data types
= ¯2 ¯1 0 1 2 3 9 8.5 8 7.5 7 6.5 6 +-×÷ÀÁÂÃÄÅ 0 1 2 3
Examples:
3 to 10 ⍝ Inclusive ascending sequence.
3 4 5 6 7 8 9 10
10 to 3 ⍝ Descending sequence.
10 9 8 7 6 5 4 3
7 to 7 ⍝ Single-item sequence.
7
5 7 to 13 ⍝ 2-item start determines step.
5 7 9 11 13
¯10 ¯15 to ¯25 ⍝ Negative start and step.
¯10 ¯15 ¯20 ¯25
1.5 1.7 to 2.5 ⍝ Fractional start and step.
1.5 1.7 1.9 2.1 2.3 2.5
0 3 to 10 ⍝ Sequence may stop short of _last_.
0 3 6 9
See also: sieve
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