Coverage for local/lib/python2.7/site-packages/sage/rings/big_oh.py : 62%

""" 

Big O for various types (power series, p-adics, etc.) 

.. SEEALSO:: 

- `asymptotic expansions <../../../asymptotic/index.html>`_ 

- `p-adic numbers <../../../padics/index.html>`_ 

- `power series <../../../power_series/index.html>`_ 

- `polynomials <../../../polynomial_rings/index.html>`_ 

""" 

from __future__ import absolute_import 

from six import integer_types 

import sage.arith.all as arith 

from . import laurent_series_ring_element 

import sage.rings.padics.factory as padics_factory 

import sage.rings.padics.padic_generic_element as padic_generic_element 

from . import power_series_ring_element 

from . import integer 

from . import rational 

from sage.rings.polynomial.polynomial_element import Polynomial 

from . import multi_power_series_ring_element 

def O(*x, **kwds): 

""" 

Big O constructor for various types. 

EXAMPLES: 

This is useful for writing power series elements:: 

sage: R.<t> = ZZ[['t']] 

sage: (1+t)^10 + O(t^5) 

1 + 10*t + 45*t^2 + 120*t^3 + 210*t^4 + O(t^5) 

A power series ring is created implicitly if a polynomial 

element is passed:: 

sage: R.<x> = QQ['x'] 

sage: O(x^100) 

O(x^100) 

sage: 1/(1+x+O(x^5)) 

1 - x + x^2 - x^3 + x^4 + O(x^5) 

sage: R.<u,v> = QQ[[]] 

sage: 1 + u + v^2 + O(u, v)^5 

1 + u + v^2 + O(u, v)^5 

This is also useful to create `p`-adic numbers:: 

sage: O(7^6) 

O(7^6) 

sage: 1/3 + O(7^6) 

5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5 + O(7^6) 

It behaves well with respect to adding negative powers of `p`:: 

sage: a = O(11^-32); a 

O(11^-32) 

sage: a.parent() 

11-adic Field with capped relative precision 20 

There are problems if you add a rational with very negative 

valuation to an `O`-Term:: 

sage: 11^-12 + O(11^15) 

11^-12 + O(11^8) 

The reason that this fails is that the constructor doesn't know 

the right precision cap to use. If you cast explicitly or use 

other means of element creation, you can get around this issue:: 

sage: K = Qp(11, 30) 

sage: K(11^-12) + O(11^15) 

11^-12 + O(11^15) 

sage: 11^-12 + K(O(11^15)) 

11^-12 + O(11^15) 

sage: K(11^-12, absprec = 15) 

11^-12 + O(11^15) 

sage: K(11^-12, 15) 

11^-12 + O(11^15) 

We can also work with `asymptotic expansions`_:: 

sage: A.<n> = AsymptoticRing(growth_group='QQ^n * n^QQ * log(n)^QQ', coefficient_ring=QQ); A 

Asymptotic Ring <QQ^n * n^QQ * log(n)^QQ> over Rational Field 

sage: O(n) 

O(n) 

TESTS:: 

sage: var('x, y') 

(x, y) 

sage: O(x) 

Traceback (most recent call last): 

... 

ArithmeticError: O(x) not defined 

sage: O(y) 

Traceback (most recent call last): 

... 

ArithmeticError: O(y) not defined 

sage: O(x, y) 

Traceback (most recent call last): 

... 

ArithmeticError: O(x, y) not defined 

sage: O(4, 2) 

Traceback (most recent call last): 

... 

ArithmeticError: O(4, 2) not defined 

""" 

if len(x) > 1: 

if isinstance(x[0], multi_power_series_ring_element.MPowerSeries): 

return multi_power_series_ring_element.MO(x, **kwds) 

else: 

raise ArithmeticError("O(%s) not defined" % 

(', '.join(str(e) for e in x),)) 

x = x[0] 

if isinstance(x, power_series_ring_element.PowerSeries): 

return x.parent()(0, x.degree(), **kwds) 

elif isinstance(x, Polynomial): 

if x.parent().ngens() != 1: 

raise NotImplementedError("completion only currently defined " 

"for univariate polynomials") 

if not x.is_monomial(): 

raise NotImplementedError("completion only currently defined " 

"for the maximal ideal (x)") 

return x.parent().completion(x.parent().gen())(0, x.degree(), **kwds) 

elif isinstance(x, laurent_series_ring_element.LaurentSeries): 

return laurent_series_ring_element.LaurentSeries(x.parent(), 0).\ 

add_bigoh(x.valuation(), **kwds) 

elif isinstance(x, integer_types + (integer.Integer, rational.Rational)): 

# p-adic number 

if x <= 0: 

raise ArithmeticError("x must be a prime power >= 2") 

F = arith.factor(x) 

if len(F) != 1: 

raise ArithmeticError("x must be prime power") 

p, r = F[0] 

if r >= 0: 

return padics_factory.Zp(p, prec=max(r, 20), 

type='capped-rel')(0, absprec=r, **kwds) 

else: 

return padics_factory.Qp(p, prec=max(r, 20), 

type='capped-rel')(0, absprec=r, **kwds) 

elif isinstance(x, padic_generic_element.pAdicGenericElement): 

return x.parent()(0, absprec=x.valuation(), **kwds) 

elif hasattr(x, 'O'): 

return x.O(**kwds) 

raise ArithmeticError("O(%s) not defined" % (x,))