Goal -- Write new classes of p-adic elements, analogous to IntegerMod_int and IntegerMod_int64 for the case that self.unit or self.value fits into machine precision.
Type -- speed improvements
Priority -- Medium-Low
Difficulty -- Easy
Prerequisites -- None
Background -- understand the inheritance structure for p-adic base rings, take a look at sage.rings.finite_rings.integer_mod
Contributors -- David Roe
Progress - not started
Related Tickets --
Discussion
This will be useful in the following cases:
p |
max precision on 32-bit |
max precision on 64-bit |
2 |
15 |
30 |
3 |
9 |
19 |
5 |
6 |
13 |
7 |
5 |
11 |
11,13 |
4 |
8 |
17,19 |
3 |
7 |
23,29,31 |
3 |
6 |
37-73 |
2 |
5 |
79-211 |
2 |
4 |
223-1289 |
1 |
3 |
1291-46337 |
1 |
2 |
To get an idea of the speed improvements available:
sage: R19 = Zmod(3^19)
sage: type((R19)(1))
<type 'sage.rings.finite_rings.integer_mod.IntegerMod_int64'>
sage: R20 = Zmod(3^20)
sage: type((R20)(1))
<type 'sage.rings.finite_rings.integer_mod.IntegerMod_gmp'>
sage: a19 = R19.random_element(); b19 = R19.random_element(); a20 = R20.random_element(); b20 = R20.random_element()
sage: timeit("c19 = a19*b19")
625 loops, best of 3: 197 ns per loop
sage: timeit("c20 = a20*b20")
625 loops, best of 3: 454 ns per loop
sage: timeit("c19 = a19 + b19")
625 loops, best of 3: 178 ns per loop
sage: timeit("c20 = a20+b20")
625 loops, best of 3: 389 ns per loopThis may be better done as an interface to p-adics in FLINT 2: see https://github.com/SPancratz/flint2/blob/trunk/padic.h
Tasks
Using sage/rings/finite_rings/integer_mod.pyx and sage/rings/padics/padic_(capped_relative_element.pyx AND capped_absolute_element.pyx AND fixed_modulus_element.pyx) as a model, implement Zp and Qp using machine arithmetic.