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""" `p`-Adic Capped Relative Elements
Elements of `p`-Adic Rings with Capped Relative Precision
AUTHORS:
- David Roe: initial version, rewriting to use templates (2012-3-1) - Genya Zaytman: documentation - David Harvey: doctests """
#***************************************************************************** # Copyright (C) 2007-2013 David Roe <roed.math@gmail.com> # William Stein <wstein@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # # http://www.gnu.org/licenses/ #*****************************************************************************
include "sage/libs/linkages/padics/mpz.pxi" include "CR_template.pxi"
from sage.libs.pari import pari from sage.libs.pari.convert_gmp cimport new_gen_from_padic from sage.rings.finite_rings.integer_mod import Mod from sage.rings.padics.pow_computer cimport PowComputer_class
cdef extern from "sage/rings/padics/transcendantal.c": cdef void padiclog(mpz_t ans, const mpz_t a, unsigned long p, unsigned long prec, const mpz_t modulo) cdef void padicexp(mpz_t ans, const mpz_t a, unsigned long p, unsigned long prec, const mpz_t modulo) cdef void padicexp_Newton(mpz_t ans, const mpz_t a, unsigned long p, unsigned long prec, unsigned long precinit, const mpz_t modulo)
cdef class PowComputer_(PowComputer_base): """ A PowComputer for a capped-relative padic ring or field. """ def __init__(self, Integer prime, long cache_limit, long prec_cap, long ram_prec_cap, bint in_field): """ Initialization.
EXAMPLES::
sage: R = ZpCR(5) sage: type(R.prime_pow) <type 'sage.rings.padics.padic_capped_relative_element.PowComputer_'> sage: R.prime_pow._prec_type 'capped-rel' """
cdef class pAdicCappedRelativeElement(CRElement): """ Constructs new element with given parent and value.
INPUT:
- ``x`` -- value to coerce into a capped relative ring or field
- ``absprec`` -- maximum number of digits of absolute precision
- ``relprec`` -- maximum number of digits of relative precision
EXAMPLES::
sage: R = Zp(5, 10, 'capped-rel')
Construct from integers::
sage: R(3) 3 + O(5^10) sage: R(75) 3*5^2 + O(5^12) sage: R(0) 0 sage: R(-1) 4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10) sage: R(-5) 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + O(5^11) sage: R(-7*25) 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + 4*5^11 + O(5^12)
Construct from rationals::
sage: R(1/2) 3 + 2*5 + 2*5^2 + 2*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + 2*5^7 + 2*5^8 + 2*5^9 + O(5^10) sage: R(-7875/874) 3*5^3 + 2*5^4 + 2*5^5 + 5^6 + 3*5^7 + 2*5^8 + 3*5^10 + 3*5^11 + 3*5^12 + O(5^13) sage: R(15/425) Traceback (most recent call last): ... ValueError: p divides the denominator
Construct from IntegerMod::
sage: R(Integers(125)(3)) 3 + O(5^3) sage: R(Integers(5)(3)) 3 + O(5) sage: R(Integers(5^30)(3)) 3 + O(5^10) sage: R(Integers(5^30)(1+5^23)) 1 + O(5^10) sage: R(Integers(49)(3)) Traceback (most recent call last): ... TypeError: p does not divide modulus 49
::
sage: R(Integers(48)(3)) Traceback (most recent call last): ... TypeError: p does not divide modulus 48
Some other conversions::
sage: R(R(5)) 5 + O(5^11)
Construct from Pari objects::
sage: R = Zp(5) sage: x = pari(123123) ; R(x) 3 + 4*5 + 4*5^2 + 4*5^3 + 5^4 + 4*5^5 + 2*5^6 + 5^7 + O(5^20) sage: R(pari(R(5252))) 2 + 2*5^3 + 3*5^4 + 5^5 + O(5^20) sage: R = Zp(5,prec=5) sage: R(pari(-1)) 4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5) sage: pari(R(-1)) 4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5) sage: pari(R(0)) 0 sage: R(pari(R(0,5))) O(5^5)
# todo: doctests for converting from other types of p-adic rings
""" def lift(self): """ Return an integer or rational congruent to ``self`` modulo ``self``'s precision. If a rational is returned, its denominator will equal ``p^ordp(self)``.
EXAMPLES::
sage: R = Zp(7,4,'capped-rel'); a = R(8); a.lift() 8 sage: R = Qp(7,4); a = R(8); a.lift() 8 sage: R = Qp(7,4); a = R(8/7); a.lift() 8/7 """
cdef lift_c(self): """ Implementation of lift.
TESTS::
sage: O(5^5).lift() #indirect doctest 0 sage: R = Qp(5); R(0).lift() 0 sage: R(5/9).lift() 264909532335070 sage: R(9/5).lift() 9/5 """ cdef Integer ans cdef Rational ansr else: else: mpq_set_si(ansr.value, 0, 1) return self else:
def __pari__(self): """ Converts this element to an equivalent pari element.
EXAMPLES::
sage: R = Zp(17, 10); a = ~R(14); pari(a) #indirect doctest 11 + 3*17 + 17^2 + 6*17^3 + 13*17^4 + 15*17^5 + 10*17^6 + 3*17^7 + 17^8 + 6*17^9 + O(17^10) sage: pari(R(0)) 0 sage: pari(R(0,5)) O(17^5) """
cdef pari_gen _to_gen(self): """ Converts this element to an equivalent pari element.
EXAMPLES::
sage: R = Zp(5, 10); a = R(17); pari(a) #indirect doctest 2 + 3*5 + O(5^10) sage: pari(R(0)) 0 sage: pari(R(0,5)) O(5^5) sage: pari(R(0,5)).debug() [&=...] PADIC(lg=5):... (precp=0,valp=5):... ... ... ... p : [&=...] INT(lg=3):... (+,lgefint=3):... ... p^l : [&=...] INT(lg=3):... (+,lgefint=3):... ... I : gen_0 """ else: self.prime_pow.prime.value, self.unit) def _integer_(self, Z=None): """ Returns an integer congruent to this element modulo ``p^self.absolute_precision()``.
EXAMPLES::
sage: R = Zp(5); a = R(-1); a._integer_() 95367431640624 """ raise ValueError("Cannot form an integer out of a p-adic field element with negative valuation")
def residue(self, absprec=1, field=None, check_prec=True): """ Reduces this element modulo `p^{\mathrm{absprec}}`.
INPUT:
- ``absprec`` -- a non-negative integer (default: ``1``)
- ``field`` -- boolean (default ``None``). Whether to return an element of GF(p) or Zmod(p).
- ``check_prec`` -- boolean (default ``True``). Whether to raise an error if this element has insufficient precision to determine the reduction.
OUTPUT:
This element reduced modulo `p^\mathrm{absprec}` as an element of `\ZZ/p^\mathrm{absprec}\ZZ`
EXAMPLES::
sage: R = Zp(7,4) sage: a = R(8) sage: a.residue(1) 1
This is different from applying ``% p^n`` which returns an element in the same ring::
sage: b = a.residue(2); b 8 sage: b.parent() Ring of integers modulo 49 sage: c = a % 7^2; c 1 + 7 + O(7^4) sage: c.parent() 7-adic Ring with capped relative precision 4
For elements in a field, application of ``% p^n`` always returns zero, the remainder of the division by ``p^n``::
sage: K = Qp(7,4) sage: a = K(8) sage: a.residue(2) 8 sage: a % 7^2 0
sage: b = K(1/7) sage: b.residue() Traceback (most recent call last): ... ValueError: element must have non-negative valuation in order to compute residue.
TESTS::
sage: R = Zp(7,4) sage: a = R(8) sage: a.residue(0) 0 sage: a.residue(-1) Traceback (most recent call last): ... ValueError: cannot reduce modulo a negative power of p. sage: a.residue(5) Traceback (most recent call last): ... PrecisionError: not enough precision known in order to compute residue. sage: a.residue(5, check_prec=False) 8
sage: a.residue(field=True).parent() Finite Field of size 7
.. SEEALSO::
:meth:`_mod_`
""" cdef Integer selfvalue, modulus cdef long aprec raise ValueError("field keyword may only be set at precision 1") else: # Need to do this better. else:
def _log_binary_splitting(self, aprec, mina=0): r""" Return ``\log(self)`` for ``self`` equal to 1 in the residue field
This is a helper method for :meth:`log`. It uses a fast binary splitting algorithm.
INPUT:
- ``aprec`` -- an integer, the precision to which the result is correct. ``aprec`` must not exceed the precision cap of the ring over which this element is defined. - ``mina`` -- an integer (default: 0), the series will check `n` up to this valuation (and beyond) to see if they can contribute to the series.
NOTE::
The function does not check that its argument ``self`` is 1 in the residue field. If this assumption is not fullfiled the behaviour of the function is not specified.
ALGORITHM:
1. Raise `u` to the power `p^v` for a suitable `v` in order to make it closer to 1. (`v` is chosen such that `p^v` is close to the precision.)
2. Write
.. MATH::
u^{p-1} = \prod_{i=1}^\infty (1 - a_i p^{(v+1)*2^i})
with `0 \leq a_i < p^{(v+1)*2^i}` and compute `\log(1 - a_i p^{(v+1)*2^i})` using the standard Taylor expansion
.. MATH::
\log(1 - x) = -x - 1/2 x^2 - 1/3 x^3 - 1/4 x^4 - 1/5 x^5 - \cdots
together with a binary splitting method.
3. Divide the result by `p^v`
The complexity of this algorithm is quasi-linear.
EXAMPLES::
sage: r = Qp(5,prec=4)(6) sage: r._log_binary_splitting(2) 5 + O(5^2) sage: r._log_binary_splitting(4) 5 + 2*5^2 + 4*5^3 + O(5^4) sage: r._log_binary_splitting(100) 5 + 2*5^2 + 4*5^3 + O(5^4)
sage: r = Zp(5,prec=4,type='fixed-mod')(6) sage: r._log_binary_splitting(5) 5 + 2*5^2 + 4*5^3 + O(5^4)
""" cdef unsigned long p cdef pAdicCappedRelativeElement ans
def _exp_binary_splitting(self, aprec): """ Compute the exponential power series of this element
This is a helper method for :meth:`exp`.
INPUT:
- ``aprec`` -- an integer, the precision to which to compute the exponential
NOTE::
The function does not check that its argument ``self`` is the disk of convergence of ``exp``. If this assumption is not fullfiled the behaviour of the function is not specified.
ALGORITHM:
Write
.. MATH::
self = \sum_{i=1}^\infty a_i p^{2^i}
with `0 \leq a_i < p^{2^i}` and compute `\exp(a_i p^{2^i})` using the standard Taylor expansion
.. MATH::
\exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + \cdots
together with a binary splitting method.
The binary complexity of this algorithm is quasi-linear.
EXAMPLES::
sage: R = Zp(7,5) sage: x = R(7) sage: x.exp(algorithm="binary_splitting") # indirect doctest 1 + 7 + 4*7^2 + 2*7^3 + O(7^5)
""" cdef unsigned long p cdef pAdicCappedRelativeElement ans
def _exp_newton(self, aprec, log_algorithm=None): """ Compute the exponential power series of this element
This is a helper method for :meth:`exp`.
INPUT:
- ``aprec`` -- an integer, the precision to which to compute the exponential
- ``log_algorithm`` (default: None) -- the algorithm used for computing the logarithm. This attribute is passed to the log method. See :meth:`log` for more details about the possible algorithms.
NOTE::
The function does not check that its argument ``self`` is the disk of convergence of ``exp``. If this assumption is not fullfiled the behaviour of the function is not specified.
ALGORITHM:
Solve the equation `\log(x) = self` using the Newton scheme::
.. MATH::
x_{i+1} = x_i \cdot (1 + self - \log(x_i))
The binary complexity of this algorithm is roughly the same than that of the computation of the logarithm.
EXAMPLES::
sage: R.<w> = Zq(7^2,5) sage: x = R(7*w) sage: x.exp(algorithm="newton") # indirect doctest 1 + w*7 + (4*w + 2)*7^2 + (w + 6)*7^3 + 5*7^4 + O(7^5) """ cdef unsigned long p cdef pAdicCappedRelativeElement ans
padicexp_Newton(ans.unit, selfint.value, p, prec, 2, self.prime_pow.pow_mpz_t_tmp(prec)) else:
def unpickle_pcre_v1(R, unit, ordp, relprec): """ Unpickles a capped relative element.
EXAMPLES::
sage: from sage.rings.padics.padic_capped_relative_element import unpickle_pcre_v1 sage: R = Zp(5) sage: a = unpickle_pcre_v1(R, 17, 2, 5); a 2*5^2 + 3*5^3 + O(5^7) """
def base_p_list(Integer n, bint pos, PowComputer_class prime_pow): """ Returns a base-`p` list of digits of ``n``.
INPUT:
- ``n`` -- a positive Integer.
- ``pos`` -- a boolean. If True, then returns the standard base `p` expansion. Otherwise, the digits lie in the range `-p/2` to `p/2`.
- ``prime_pow`` -- A PowComputer giving the prime.
EXAMPLES::
sage: from sage.rings.padics.padic_capped_relative_element import base_p_list sage: base_p_list(192837, True, Zp(5).prime_pow) [2, 2, 3, 2, 3, 1, 2, 2] sage: 2 + 2*5 + 3*5^2 + 2*5^3 + 3*5^4 + 5^5 + 2*5^6 + 2*5^7 192837 sage: base_p_list(192837, False, Zp(5).prime_pow) [2, 2, -2, -2, -1, 2, 2, 2] sage: 2 + 2*5 - 2*5^2 - 2*5^3 - 5^4 + 2*5^5 + 2*5^6 + 2*5^7 192837 """ raise ValueError("n must be nonnegative") # We need a p-adic element to feed to ExpansionIter before resetting its curvalue |