P-Adic numbers, basic: Difference between revisions
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</pre>
=={{header|Julia}}==
Uses the Nemo abstract algebra library. The Nemo library's rational reconstruction function gives up quite easily,
so another alternative to FreeBasic's crat() using determinants is below.
<lang Julia>using Nemo, LinearAlgebra
import Base.Rational, Base.digits
set_printing_mode(FlintPadicField, :terse)
""" convert to Rational (rational reconstruction) """
function toRational(pa::padic)
rat = lift(QQ, pa)
r, den = BigInt(numerator(rat)), Int(denominator(rat))
p, k = Int(prime(parent(pa))), Int(precision(pa))
N = BigInt(p^k)
a1, a2 = [N, 0], [r, 1]
while dot(a1, a1) > dot(a2, a2)
q = dot(a1, a2) // dot(a2, a2)
a1, a2 = a2, a1 - BigInt(round(q)) * a2
end
if dot(a1, a1) < N
return (Rational{Int}(a1[1]) // Rational{Int}(a1[2])) // Int(den)
else
return Int(r) // den
end
end
function dstring(pa::padic)
u, v, n, p, k = pa.u, pa.v, pa.N, pa.parent.p, pa.parent.prec_max
d = digits(v > 0 ? u * p^v : u, base=pa.parent.p, pad=k)
return prod([i == k + v && v != 0 ? "$x . " : "$x " for (i, x) in enumerate(reverse(d))])
end
const DATA = [
[2, 1, 2, 4, 1, 1],
[4, 1, 2, 4, 3, 1],
[4, 1, 2, 5, 3, 1],
[4, 9, 5, 4, 8, 9],
[26, 25, 5, 4, -109, 125],
[49, 2, 7, 6, -4851, 2],
[-9, 5, 3, 8, 27, 7],
[5, 19, 2, 12, -101, 384],
# Base 10 10-adic p-adics are not allowed by Nemo library -- p must be a prime
# familiar digits
[11, 4, 2, 43, 679001, 207],
[-8, 9, 23, 9, 302113, 92],
[-22, 7, 3, 23, 46071, 379],
[-22, 7, 32749, 3, 46071, 379],
[35, 61, 5, 20, 9400, 109],
[-101, 109, 61, 7, 583376, 6649],
[-25, 26, 7, 13, 5571, 137],
[1, 4, 7, 11, 9263, 2837],
[122, 407, 7, 11, -517, 1477],
# more subtle
[5, 8, 7, 11, 353, 30809],
]
for (num1, den1, P, K, num2, den2) in DATA
Qp = PadicField(P, K)
a = Qp(QQ(num1 // den1))
b = Qp(QQ(num2 // den2))
c = a + b
r = toRational(c)
println(a, "\n", dstring(a), "\n", b, "\n", dstring(b), "\n+ =\n", c, "\n", dstring(c), " $r\n")
end
</lang>{{out}}
<pre>
2 + O(2^5)
0 0 1 0
1 + O(2^4)
0 0 0 1
+ =
3 + O(2^4)
0 0 1 1 3//1
4 + O(2^6)
0 1 0 0
3 + O(2^4)
0 0 1 1
+ =
7 + O(2^4)
0 1 1 1 -1//1
4 + O(2^7)
0 0 1 0 0
3 + O(2^5)
0 0 0 1 1
+ =
7 + O(2^5)
0 0 1 1 1 7//1
556 + O(5^4)
4 2 1 1
487 + O(5^4)
3 4 2 2
+ =
418 + O(5^4)
3 1 3 3 4//3
26/25 + O(5^2)
0 1 . 0 1
516/125 + O(5^1)
4 . 0 3 1
+ =
21/125 + O(5^1)
0 . 0 4 1 21//125
58849 + O(7^6)
3 3 3 4 0 0
56399 + O(7^6)
3 2 3 3 0 0
+ =
115248 + O(7^6)
6 6 0 0 0 0 0//1
5247 + O(3^8)
2 1 0 1 2 1 0 0
3753 + O(3^8)
1 2 0 1 1 0 0 0
+ =
2439 + O(3^8)
1 0 1 0 0 1 0 0 72//35
647 + O(2^12)
0 0 1 0 1 0 0 0 0 1 1 1
2697/128 + O(2^5)
1 0 1 0 1 . 0 0 0 1 0 0 1
+ =
3593/128 + O(2^5)
1 1 1 0 0 . 0 0 0 1 0 0 1 3593//128
11/4 + O(2^41)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 . 1 1
2379619371607 + O(2^43)
0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 1 1
+ =
722384464231/4 + O(2^41)
0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 1 . 1 1 330311//1618052
200128073495 + O(23^9)
2 12 17 20 10 5 2 12 17
450288240894/23 + O(23^8)
5 17 5 17 6 0 10 12 . 2
+ =
1450928608353/23 + O(23^8)
18 12 3 4 11 3 0 6 . 2 1450928608353//23
40347076637 + O(3^23)
1 0 2 1 2 0 1 0 2 1 2 0 1 0 2 1 2 0 1 0 2 0 2
69303290076 + O(3^23)
2 0 1 2 1 2 1 2 2 1 2 1 0 0 2 2 0 1 1 2 1 0 0
+ =
15507187886 + O(3^23)
0 1 1 1 1 0 0 0 2 0 1 1 1 1 2 0 2 2 0 0 0 0 2 15507187886//1
30105603673496 + O(32749^3)
28070 18713 23389
4819014836161 + O(32749^3)
4493 8727 10145
+ =
34924618509657 + O(32749^3)
32563 27441 785 314159//2653
51592217117060 + O(5^20)
2 3 2 3 0 2 4 1 3 3 0 0 4 0 2 2 1 2 2 0
64744861847850 + O(5^20)
3 1 4 4 1 2 3 4 4 3 4 1 1 3 1 1 2 4 0 0
+ =
20969647324285 + O(5^20)
1 0 2 2 2 0 3 1 3 1 4 2 0 3 3 3 4 1 2 0 577215//6649
1701117681882 + O(61^7)
33 1 7 16 48 7 50
1701117687235/61 + O(61^6)
33 1 7 16 49 34 . 35
+ =
1758782693344/61 + O(61^6)
34 8 24 3 57 23 . 35 1758782693344//61
40991504402 + O(7^13)
2 6 5 0 5 4 4 0 1 6 1 2 2
46676457609 + O(7^13)
3 2 4 1 4 5 4 2 2 5 5 3 5
+ =
87667962011 + O(7^13)
6 2 2 2 3 3 1 2 4 4 6 6 0 141421//3562
494331686 + O(7^11)
1 5 1 5 1 5 1 5 1 5 2
1936205041 + O(7^11)
6 5 6 6 0 3 2 0 4 4 1
+ =
453209984 + O(7^11)
1 4 1 4 2 1 3 5 6 2 3 39889//11348
1778136580 + O(7^11)
6 2 0 3 0 6 2 4 4 4 3
384219886/7 + O(7^10)
1 2 3 4 3 5 4 6 4 1 . 1
+ =
967215488/7 + O(7^10)
3 2 6 5 3 1 2 4 1 4 . 1 967215488//7
1235829215 + O(7^11)
4 2 4 2 4 2 4 2 4 2 5
726006041 + O(7^11)
2 3 6 6 3 6 4 3 4 5 5
+ =
1961835256 + O(7^11)
6 6 4 2 1 2 1 6 2 1 3 -25145//36122
</pre>
=={{header|Phix}}==
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