Easy to use and nicely formatted p-adic numbers. Everything rides on Nemo.
This module provides the type Padic{p} to represent p-adic numbers (for prime p).
Numbers of this sort are created either with Integer or Rational arguments:
julia> using SimplePadics
julia> Padic{5}(82)
…312.0_{5}
julia> Padic{5}(1//15)
…313131313.2_{5}
For p equal to 2 we have the abbreviation Dyadic that is exacly equivalent to Padic{2}.
julia> Dyadic(1//10)
…1100110.1_{2}
The function base returns p for numbers of type Padic{p}:
julia> a = Padic{7}(100//49)
…2.02_{7}
julia> base(a)
7
The standard arithmetic operations (addition, subtraction, multiplication, division) may be performed with arguments that are both of the same p-adic type, or a mixture of a p-adic and an Integer or Rational. Some examples:
julia> a = Padic{7}(10)
…13.0_{7}
julia> a+1
…14.0_{7}
julia> 2a
…26.0_{7}
julia> a/a
…1.0_{7}
julia> inv(a)
…462046205.0_{7}
When the prime p is greater than 7, we use letters to stand for digits beyond 9 (that is, a for ten, b for eleven, and so forth).
julia> a = Padic{17}(1000)
…37e.0_{17}
julia> 3 * 17^2 + 7 * 17 + 14
1000
To get the digits as a list of integers, use digits(a). The first entry in the result is the most significant digit (rightmost).
julia> a = Padic{7}(124//49)
…2.35_{7}
julia> digits(a)
3-element Vector{Int64}:
5
3
2
See valuation to determine the location of the radix point.
julia> valuation(a)
-2
The precision of Padic numbers is globally controlled using the functions set_precision and get_precision.
Use set_precision(n::Int) to set the precision to n digits. Use set_precision() to
set the precision to the default value (10).
The function get_precision() returns the current setting.
Warning:
Padicnumbers created with different precisions cannot be compared for equality or combined in operations.
julia> a = Padic{5}(-1)
…4444444444.0_{5}
julia> set_precision(20)
20
julia> b = Padic{5}(-1)
…44444444444444444444.0_{5}
julia> a == b
ERROR: Incompatible padic rings in padic operation
The functions sqrt, exp, log, and valuation from Nemo are imported into this module. For example:
julia> a = Padic{5}(-1)
…4444444444.0_{5}
julia> sqrt(a)
…3032431212.0_{5}
julia> ans^2
…4444444444.0_{5}
julia> a = Padic{5}(500)
…4000.0_{5}
julia> valuation(a)
3
julia> 1/a
…3333333.334_{5}
julia> valuation(1/a)
-3
julia> abs(a)
0.008
julia> abs(1/a)
125.0
Let F be a SimplePolynomial with integer or rational coefficients. The function p_root(F,p) returns a p-adic root of F (if one exists) or throws an error otherwise.
julia> using SimplePadics, SimplePolynomials
julia> x = getx()
x
julia> F = x^2 + 1
1 + x^2
julia> t = p_root(F,5)
…3032431212.0_{5}
julia> t^2 + 1
…0.0_{5}
julia> sqrt(Padic{5}(-1))
…3032431212.0_{5}
The function has_p_root(F,p) tests if a polynomial has a p-adic root.
Returns true if F has a root and false otherwise.
Matrices/vectors populated with Padic numbers can be used in the LinearAlgebraX
module. Some examples here:
julia> using LinearAlgebraX
julia> A = rand(Int,5,5) .% 10 # create a random matrix with entries between -9 and 9.
5×5 Matrix{Int64}:
-9 -5 -4 -9 -8
0 -1 9 -3 8
2 -9 0 0 -4
0 7 -6 -5 8
0 -5 6 -2 -6
julia> A = Padic{5}.(A) # convert those values to 5-adic numbers
5×5 Matrix{Padic{5}}:
…4444444431.0_{5} …44444444440.0_{5} …4444444441.0_{5} …4444444431.0_{5} …4444444432.0_{5}
…0.0_{5} …4444444444.0_{5} …14.0_{5} …4444444442.0_{5} …13.0_{5}
…2.0_{5} …4444444431.0_{5} …0.0_{5} …0.0_{5} …4444444441.0_{5}
…0.0_{5} …12.0_{5} …4444444434.0_{5} …44444444440.0_{5} …13.0_{5}
…0.0_{5} …44444444440.0_{5} …11.0_{5} …4444444443.0_{5} …4444444434.0_{5}
julia> detx(A)
…4441202132.0_{5}
julia> invx(A)
5×5 Matrix{Padic{5}}:
…4344224002.0_{5} …4441024033.0_{5} …1242400242.0_{5} …2100330003.0_{5} …211103034.0_{5}
…4333241114.0_{5} …2024323204.0_{5} …1141013103.0_{5} …3103232201.0_{5} …4420230301.0_{5}
…1011004214.0_{5} …3224241133.0_{5} …4324312303.0_{5} …404043040.0_{5} …2430224440.0_{5}
…1031224011.0_{5} …2022011034.0_{5} …2244123102.0_{5} …1312340431.0_{5} …23100104.0_{5}
…1324102422.0_{5} …2324440300.0_{5} …101223004.0_{5} …21033133.0_{5} …220320101.0_{5}
julia> A * ans
5×5 Matrix{Any}:
…1.0_{5} …0.0_{5} …0.0_{5} …0.0_{5} …0.0_{5}
…0.0_{5} …1.0_{5} …0.0_{5} …0.0_{5} …0.0_{5}
…0.0_{5} …0.0_{5} …1.0_{5} …0.0_{5} …0.0_{5}
…0.0_{5} …0.0_{5} …0.0_{5} …1.0_{5} …0.0_{5}
…0.0_{5} …0.0_{5} …0.0_{5} …0.0_{5} …1.0_{5}
julia> nullspacex(A[1:3,:])
5×2 Matrix{Padic{5}}:
…3414203434.0_{5} …2234314433.0_{5}
…3020203312.0_{5} …241004443.0_{5}
…3314221230.0_{5} …2120313420.0_{5}
…1.0_{5} …0.0_{5}
…0.0_{5} …1.0_{5}