Power series implemented using PARI¶
EXAMPLES:
This implementation can be selected for any base ring supported by
PARI by passing the keyword implementation='pari' to the
PowerSeriesRing() constructor:
sage: R.<q> = PowerSeriesRing(ZZ, implementation='pari'); R
Power Series Ring in q over Integer Ring
sage: S.<t> = PowerSeriesRing(CC, implementation='pari'); S                         # needs sage.rings.real_mpfr
Power Series Ring in t over Complex Field with 53 bits of precision
>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, implementation='pari', names=('q',)); (q,) = R._first_ngens(1); R
Power Series Ring in q over Integer Ring
>>> S = PowerSeriesRing(CC, implementation='pari', names=('t',)); (t,) = S._first_ngens(1); S                         # needs sage.rings.real_mpfr
Power Series Ring in t over Complex Field with 53 bits of precision
Note that only the type of the elements depends on the implementation, not the type of the parents:
sage: type(R)
<class 'sage.rings.power_series_ring.PowerSeriesRing_domain_with_category'>
sage: type(q)
<class 'sage.rings.power_series_pari.PowerSeries_pari'>
sage: type(S)                                                                       # needs sage.rings.real_mpfr
<class 'sage.rings.power_series_ring.PowerSeriesRing_over_field_with_category'>
sage: type(t)                                                                       # needs sage.rings.real_mpfr
<class 'sage.rings.power_series_pari.PowerSeries_pari'>
>>> from sage.all import *
>>> type(R)
<class 'sage.rings.power_series_ring.PowerSeriesRing_domain_with_category'>
>>> type(q)
<class 'sage.rings.power_series_pari.PowerSeries_pari'>
>>> type(S)                                                                       # needs sage.rings.real_mpfr
<class 'sage.rings.power_series_ring.PowerSeriesRing_over_field_with_category'>
>>> type(t)                                                                       # needs sage.rings.real_mpfr
<class 'sage.rings.power_series_pari.PowerSeries_pari'>
If \(k\) is a finite field implemented using PARI, this is the default implementation for power series over \(k\):
sage: k.<c> = GF(5^12)
sage: type(c)
<class 'sage.rings.finite_rings.element_pari_ffelt.FiniteFieldElement_pari_ffelt'>
sage: A.<x> = k[[]]
sage: type(x)
<class 'sage.rings.power_series_pari.PowerSeries_pari'>
>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(12), names=('c',)); (c,) = k._first_ngens(1)
>>> type(c)
<class 'sage.rings.finite_rings.element_pari_ffelt.FiniteFieldElement_pari_ffelt'>
>>> A = k[['x']]; (x,) = A._first_ngens(1)
>>> type(x)
<class 'sage.rings.power_series_pari.PowerSeries_pari'>
Warning
Because this implementation uses the PARI interface, the PARI variable ordering must be respected in the sense that the variable name of the power series ring must have higher priority than any variable names occurring in the base ring:
sage: R.<y> = QQ[]
sage: S.<x> = PowerSeriesRing(R, implementation='pari'); S
Power Series Ring in x over Univariate Polynomial Ring in y over Rational Field
>>> from sage.all import *
>>> R = QQ['y']; (y,) = R._first_ngens(1)
>>> S = PowerSeriesRing(R, implementation='pari', names=('x',)); (x,) = S._first_ngens(1); S
Power Series Ring in x over Univariate Polynomial Ring in y over Rational Field
Reversing the variable ordering leads to errors:
sage: R.<x> = QQ[]
sage: S.<y> = PowerSeriesRing(R, implementation='pari')
Traceback (most recent call last):
...
PariError: incorrect priority in gtopoly: variable x <= y
>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> S = PowerSeriesRing(R, implementation='pari', names=('y',)); (y,) = S._first_ngens(1)
Traceback (most recent call last):
...
PariError: incorrect priority in gtopoly: variable x <= y
AUTHORS:
- Peter Bruin (December 2013): initial version 
- class sage.rings.power_series_pari.PowerSeries_pari[source]¶
- Bases: - PowerSeries- A power series implemented using PARI. - INPUT: - parent– the power series ring to use as the parent
- f– object from which to construct a power series
- prec– (default: infinity) precision of the element to be constructed
- check– ignored, but accepted for compatibility with- PowerSeries_poly
 - dict(copy=None)[source]¶
- Return a dictionary of coefficients for - self.- This is simply a dict for the underlying polynomial; it need not have keys corresponding to every number smaller than - self.prec().- EXAMPLES: - sage: R.<t> = PowerSeriesRing(ZZ, implementation='pari') sage: f = 1 + t^10 + O(t^12) sage: f.monomial_coefficients() {0: 1, 10: 1} - >>> from sage.all import * >>> R = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = R._first_ngens(1) >>> f = Integer(1) + t**Integer(10) + O(t**Integer(12)) >>> f.monomial_coefficients() {0: 1, 10: 1} - dictis an alias:- sage: f.dict() {0: 1, 10: 1} - >>> from sage.all import * >>> f.dict() {0: 1, 10: 1} 
 - integral(var=None)[source]¶
- Return the formal integral of - self.- By default, the integration variable is the variable of the power series. Otherwise, the integration variable is the optional parameter - var.- Note - The integral is always chosen so the constant term is 0. - EXAMPLES: - sage: k.<w> = PowerSeriesRing(QQ, implementation='pari') sage: (1+17*w+15*w^3+O(w^5)).integral() w + 17/2*w^2 + 15/4*w^4 + O(w^6) sage: (w^3 + 4*w^4 + O(w^7)).integral() 1/4*w^4 + 4/5*w^5 + O(w^8) sage: (3*w^2).integral() w^3 - >>> from sage.all import * >>> k = PowerSeriesRing(QQ, implementation='pari', names=('w',)); (w,) = k._first_ngens(1) >>> (Integer(1)+Integer(17)*w+Integer(15)*w**Integer(3)+O(w**Integer(5))).integral() w + 17/2*w^2 + 15/4*w^4 + O(w^6) >>> (w**Integer(3) + Integer(4)*w**Integer(4) + O(w**Integer(7))).integral() 1/4*w^4 + 4/5*w^5 + O(w^8) >>> (Integer(3)*w**Integer(2)).integral() w^3 
 - list()[source]¶
- Return the list of known coefficients for - self.- This is just the list of coefficients of the underlying polynomial; it need not have length equal to - self.prec().- EXAMPLES: - sage: R.<t> = PowerSeriesRing(ZZ, implementation='pari') sage: f = 1 - 5*t^3 + t^5 + O(t^7) sage: f.list() [1, 0, 0, -5, 0, 1] sage: # needs sage.rings.padics sage: S.<u> = PowerSeriesRing(pAdicRing(5), implementation='pari') sage: (2 + u).list() [2 + O(5^20), 1 + O(5^20)] - >>> from sage.all import * >>> R = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = R._first_ngens(1) >>> f = Integer(1) - Integer(5)*t**Integer(3) + t**Integer(5) + O(t**Integer(7)) >>> f.list() [1, 0, 0, -5, 0, 1] >>> # needs sage.rings.padics >>> S = PowerSeriesRing(pAdicRing(Integer(5)), implementation='pari', names=('u',)); (u,) = S._first_ngens(1) >>> (Integer(2) + u).list() [2 + O(5^20), 1 + O(5^20)] 
 - monomial_coefficients(copy=None)[source]¶
- Return a dictionary of coefficients for - self.- This is simply a dict for the underlying polynomial; it need not have keys corresponding to every number smaller than - self.prec().- EXAMPLES: - sage: R.<t> = PowerSeriesRing(ZZ, implementation='pari') sage: f = 1 + t^10 + O(t^12) sage: f.monomial_coefficients() {0: 1, 10: 1} - >>> from sage.all import * >>> R = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = R._first_ngens(1) >>> f = Integer(1) + t**Integer(10) + O(t**Integer(12)) >>> f.monomial_coefficients() {0: 1, 10: 1} - dictis an alias:- sage: f.dict() {0: 1, 10: 1} - >>> from sage.all import * >>> f.dict() {0: 1, 10: 1} 
 - padded_list(n=None)[source]¶
- Return a list of coefficients of - selfup to (but not including) \(q^n\).- The list is padded with zeroes on the right so that it has length \(n\). - INPUT: - n– nonnegative integer (optional); if \(n\) is not given, it will be taken to be the precision of- self, unless this is- +Infinity, in which case we just return- self.list()
 - EXAMPLES: - sage: R.<q> = PowerSeriesRing(QQ, implementation='pari') sage: f = 1 - 17*q + 13*q^2 + 10*q^4 + O(q^7) sage: f.list() [1, -17, 13, 0, 10] sage: f.padded_list(7) [1, -17, 13, 0, 10, 0, 0] sage: f.padded_list(10) [1, -17, 13, 0, 10, 0, 0, 0, 0, 0] sage: f.padded_list(3) [1, -17, 13] sage: f.padded_list() [1, -17, 13, 0, 10, 0, 0] sage: g = 1 - 17*q + 13*q^2 + 10*q^4 sage: g.list() [1, -17, 13, 0, 10] sage: g.padded_list() [1, -17, 13, 0, 10] sage: g.padded_list(10) [1, -17, 13, 0, 10, 0, 0, 0, 0, 0] - >>> from sage.all import * >>> R = PowerSeriesRing(QQ, implementation='pari', names=('q',)); (q,) = R._first_ngens(1) >>> f = Integer(1) - Integer(17)*q + Integer(13)*q**Integer(2) + Integer(10)*q**Integer(4) + O(q**Integer(7)) >>> f.list() [1, -17, 13, 0, 10] >>> f.padded_list(Integer(7)) [1, -17, 13, 0, 10, 0, 0] >>> f.padded_list(Integer(10)) [1, -17, 13, 0, 10, 0, 0, 0, 0, 0] >>> f.padded_list(Integer(3)) [1, -17, 13] >>> f.padded_list() [1, -17, 13, 0, 10, 0, 0] >>> g = Integer(1) - Integer(17)*q + Integer(13)*q**Integer(2) + Integer(10)*q**Integer(4) >>> g.list() [1, -17, 13, 0, 10] >>> g.padded_list() [1, -17, 13, 0, 10] >>> g.padded_list(Integer(10)) [1, -17, 13, 0, 10, 0, 0, 0, 0, 0] 
 - polynomial()[source]¶
- Convert - selfto a polynomial.- EXAMPLES: - sage: R.<t> = PowerSeriesRing(GF(7), implementation='pari') sage: f = 3 - t^3 + O(t^5) sage: f.polynomial() 6*t^3 + 3 - >>> from sage.all import * >>> R = PowerSeriesRing(GF(Integer(7)), implementation='pari', names=('t',)); (t,) = R._first_ngens(1) >>> f = Integer(3) - t**Integer(3) + O(t**Integer(5)) >>> f.polynomial() 6*t^3 + 3 
 - reverse(precision=None)[source]¶
- Return the reverse of - self.- The reverse of a power series \(f\) is the power series \(g\) such that \(g(f(x)) = x\). This exists if and only if the valuation of - selfis exactly 1 and the coefficient of \(x\) is a unit.- If the optional argument - precisionis given, the reverse is returned with this precision. If- fhas infinite precision and the argument- precisionis not given, then the reverse is returned with the default precision of- f.parent().- EXAMPLES: - sage: R.<x> = PowerSeriesRing(QQ, implementation='pari') sage: f = 2*x + 3*x^2 - x^4 + O(x^5) sage: g = f.reverse() sage: g 1/2*x - 3/8*x^2 + 9/16*x^3 - 131/128*x^4 + O(x^5) sage: f(g) x + O(x^5) sage: g(f) x + O(x^5) sage: A.<t> = PowerSeriesRing(ZZ, implementation='pari') sage: a = t - t^2 - 2*t^4 + t^5 + O(t^6) sage: b = a.reverse(); b t + t^2 + 2*t^3 + 7*t^4 + 25*t^5 + O(t^6) sage: a(b) t + O(t^6) sage: b(a) t + O(t^6) sage: B.<b,c> = PolynomialRing(ZZ) sage: A.<t> = PowerSeriesRing(B, implementation='pari') sage: f = t + b*t^2 + c*t^3 + O(t^4) sage: g = f.reverse(); g t - b*t^2 + (2*b^2 - c)*t^3 + O(t^4) sage: f(g) t + O(t^4) sage: g(f) t + O(t^4) sage: A.<t> = PowerSeriesRing(ZZ, implementation='pari') sage: B.<x> = PowerSeriesRing(A, implementation='pari') sage: f = (1 - 3*t + 4*t^3 + O(t^4))*x + (2 + t + t^2 + O(t^3))*x^2 + O(x^3) sage: g = f.reverse(); g (1 + 3*t + 9*t^2 + 23*t^3 + O(t^4))*x + (-2 - 19*t - 118*t^2 + O(t^3))*x^2 + O(x^3) - >>> from sage.all import * >>> R = PowerSeriesRing(QQ, implementation='pari', names=('x',)); (x,) = R._first_ngens(1) >>> f = Integer(2)*x + Integer(3)*x**Integer(2) - x**Integer(4) + O(x**Integer(5)) >>> g = f.reverse() >>> g 1/2*x - 3/8*x^2 + 9/16*x^3 - 131/128*x^4 + O(x^5) >>> f(g) x + O(x^5) >>> g(f) x + O(x^5) >>> A = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = A._first_ngens(1) >>> a = t - t**Integer(2) - Integer(2)*t**Integer(4) + t**Integer(5) + O(t**Integer(6)) >>> b = a.reverse(); b t + t^2 + 2*t^3 + 7*t^4 + 25*t^5 + O(t^6) >>> a(b) t + O(t^6) >>> b(a) t + O(t^6) >>> B = PolynomialRing(ZZ, names=('b', 'c',)); (b, c,) = B._first_ngens(2) >>> A = PowerSeriesRing(B, implementation='pari', names=('t',)); (t,) = A._first_ngens(1) >>> f = t + b*t**Integer(2) + c*t**Integer(3) + O(t**Integer(4)) >>> g = f.reverse(); g t - b*t^2 + (2*b^2 - c)*t^3 + O(t^4) >>> f(g) t + O(t^4) >>> g(f) t + O(t^4) >>> A = PowerSeriesRing(ZZ, implementation='pari', names=('t',)); (t,) = A._first_ngens(1) >>> B = PowerSeriesRing(A, implementation='pari', names=('x',)); (x,) = B._first_ngens(1) >>> f = (Integer(1) - Integer(3)*t + Integer(4)*t**Integer(3) + O(t**Integer(4)))*x + (Integer(2) + t + t**Integer(2) + O(t**Integer(3)))*x**Integer(2) + O(x**Integer(3)) >>> g = f.reverse(); g (1 + 3*t + 9*t^2 + 23*t^3 + O(t^4))*x + (-2 - 19*t - 118*t^2 + O(t^3))*x^2 + O(x^3) - The optional argument - precisionsets the precision of the output:- sage: R.<x> = PowerSeriesRing(QQ, implementation='pari') sage: f = 2*x + 3*x^2 - 7*x^3 + x^4 + O(x^5) sage: g = f.reverse(precision=3); g 1/2*x - 3/8*x^2 + O(x^3) sage: f(g) x + O(x^3) sage: g(f) x + O(x^3) - >>> from sage.all import * >>> R = PowerSeriesRing(QQ, implementation='pari', names=('x',)); (x,) = R._first_ngens(1) >>> f = Integer(2)*x + Integer(3)*x**Integer(2) - Integer(7)*x**Integer(3) + x**Integer(4) + O(x**Integer(5)) >>> g = f.reverse(precision=Integer(3)); g 1/2*x - 3/8*x^2 + O(x^3) >>> f(g) x + O(x^3) >>> g(f) x + O(x^3) - If the input series has infinite precision, the precision of the output is automatically set to the default precision of the parent ring: - sage: R.<x> = PowerSeriesRing(QQ, default_prec=20, implementation='pari') sage: (x - x^2).reverse() # get some Catalan numbers x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 + 208012*x^13 + 742900*x^14 + 2674440*x^15 + 9694845*x^16 + 35357670*x^17 + 129644790*x^18 + 477638700*x^19 + O(x^20) sage: (x - x^2).reverse(precision=3) x + x^2 + O(x^3) - >>> from sage.all import * >>> R = PowerSeriesRing(QQ, default_prec=Integer(20), implementation='pari', names=('x',)); (x,) = R._first_ngens(1) >>> (x - x**Integer(2)).reverse() # get some Catalan numbers x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 + 208012*x^13 + 742900*x^14 + 2674440*x^15 + 9694845*x^16 + 35357670*x^17 + 129644790*x^18 + 477638700*x^19 + O(x^20) >>> (x - x**Integer(2)).reverse(precision=Integer(3)) x + x^2 + O(x^3) 
 - valuation()[source]¶
- Return the valuation of - self.- EXAMPLES: - sage: R.<t> = PowerSeriesRing(QQ, implementation='pari') sage: (5 - t^8 + O(t^11)).valuation() 0 sage: (-t^8 + O(t^11)).valuation() 8 sage: O(t^7).valuation() 7 sage: R(0).valuation() +Infinity - >>> from sage.all import * >>> R = PowerSeriesRing(QQ, implementation='pari', names=('t',)); (t,) = R._first_ngens(1) >>> (Integer(5) - t**Integer(8) + O(t**Integer(11))).valuation() 0 >>> (-t**Integer(8) + O(t**Integer(11))).valuation() 8 >>> O(t**Integer(7)).valuation() 7 >>> R(Integer(0)).valuation() +Infinity