Power Series Methods

The class PowerSeries_poly provides additional methods for univariate power series.

class sage.rings.power_series_poly.BaseRingFloorDivAction[source]

Bases: Action

The floor division action of the base ring on a formal power series.

class sage.rings.power_series_poly.PowerSeries_poly[source]

Bases: PowerSeries

EXAMPLES:

sage: R.<q> = PowerSeriesRing(CC); R                                        # needs sage.rings.real_mpfr
Power Series Ring in q over Complex Field with 53 bits of precision
sage: loads(q.dumps()) == q                                                 # needs sage.rings.real_mpfr
True

sage: R.<t> = QQ[[]]
sage: f = 3 - t^3 + O(t^5)
sage: a = f^3; a
27 - 27*t^3 + O(t^5)
sage: b = f^-3; b
1/27 + 1/27*t^3 + O(t^5)
sage: a*b
1 + O(t^5)

>>> from sage.all import *
>>> R = PowerSeriesRing(CC, names=('q',)); (q,) = R._first_ngens(1); R                                        # needs sage.rings.real_mpfr
Power Series Ring in q over Complex Field with 53 bits of precision
>>> loads(q.dumps()) == q                                                 # needs sage.rings.real_mpfr
True

>>> R = QQ[['t']]; (t,) = R._first_ngens(1)
>>> f = Integer(3) - t**Integer(3) + O(t**Integer(5))
>>> a = f**Integer(3); a
27 - 27*t^3 + O(t^5)
>>> b = f**-Integer(3); b
1/27 + 1/27*t^3 + O(t^5)
>>> a*b
1 + O(t^5)

Check that Issue #22216 is fixed:

sage: R.<T> = PowerSeriesRing(QQ)
sage: R(pari('1 + O(T)'))                                                   # needs sage.libs.pari
1 + O(T)
sage: R(pari('1/T + O(T)'))                                                 # needs sage.libs.pari
Traceback (most recent call last):
...
ValueError: series has negative valuation

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('T',)); (T,) = R._first_ngens(1)
>>> R(pari('1 + O(T)'))                                                   # needs sage.libs.pari
1 + O(T)
>>> R(pari('1/T + O(T)'))                                                 # needs sage.libs.pari
Traceback (most recent call last):
...
ValueError: series has negative valuation
degree()[source]

Return the degree of the underlying polynomial of self.

That is, if self is of the form \(f(x) + O(x^n)\), we return the degree of \(f(x)\). Note that if \(f(x)\) is \(0\), we return \(-1\), just as with polynomials.

EXAMPLES:

sage: R.<t> = ZZ[[]]
sage: (5 + t^3 + O(t^4)).degree()
3
sage: (5 + O(t^4)).degree()
0
sage: O(t^4).degree()
-1

>>> from sage.all import *
>>> R = ZZ[['t']]; (t,) = R._first_ngens(1)
>>> (Integer(5) + t**Integer(3) + O(t**Integer(4))).degree()
3
>>> (Integer(5) + O(t**Integer(4))).degree()
0
>>> O(t**Integer(4)).degree()
-1
dict()[source]

Return a dictionary of coefficients for self.

This is simply a dict for the underlying polynomial, so need not have keys corresponding to every number smaller than self.prec().

EXAMPLES:

sage: R.<t> = ZZ[[]]
sage: f = 1 + t^10 + O(t^12)
sage: f.monomial_coefficients()
{0: 1, 10: 1}

>>> from sage.all import *
>>> R = ZZ[['t']]; (t,) = R._first_ngens(1)
>>> f = Integer(1) + t**Integer(10) + O(t**Integer(12))
>>> f.monomial_coefficients()
{0: 1, 10: 1}

dict is 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 integral of this power series.

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> = QQ[[]]
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 = QQ[['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, so in particular, need not have length equal to self.prec().

EXAMPLES:

sage: R.<t> = ZZ[[]]
sage: f = 1 - 5*t^3 + t^5 + O(t^7)
sage: f.list()
[1, 0, 0, -5, 0, 1]

>>> from sage.all import *
>>> R = ZZ[['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]
monomial_coefficients()[source]

Return a dictionary of coefficients for self.

This is simply a dict for the underlying polynomial, so need not have keys corresponding to every number smaller than self.prec().

EXAMPLES:

sage: R.<t> = ZZ[[]]
sage: f = 1 + t^10 + O(t^12)
sage: f.monomial_coefficients()
{0: 1, 10: 1}

>>> from sage.all import *
>>> R = ZZ[['t']]; (t,) = R._first_ngens(1)
>>> f = Integer(1) + t**Integer(10) + O(t**Integer(12))
>>> f.monomial_coefficients()
{0: 1, 10: 1}

dict is an alias:

sage: f.dict()
{0: 1, 10: 1}

>>> from sage.all import *
>>> f.dict()
{0: 1, 10: 1}
pade(m, n)[source]

Return the Padé approximant of self of index \((m, n)\).

The Padé approximant of index \((m, n)\) of a formal power series \(f\) is the quotient \(Q/P\) of two polynomials \(Q\) and \(P\) such that \(\deg(Q)\leq m\), \(\deg(P)\leq n\) and

\[f(z) - Q(z)/P(z) = O(z^{m+n+1}).\]

The formal power series \(f\) must be known up to order \(n + m\).

See Wikipedia article Padé_approximant

INPUT:

  • m, n – integers, describing the degrees of the polynomials

OUTPUT: a ratio of two polynomials

ALGORITHM:

This method uses the formula as a quotient of two determinants.

See also

EXAMPLES:

sage: z = PowerSeriesRing(QQ, 'z').gen()
sage: exp(z).pade(4, 0)
1/24*z^4 + 1/6*z^3 + 1/2*z^2 + z + 1
sage: exp(z).pade(1, 1)
(-z - 2)/(z - 2)
sage: exp(z).pade(3, 3)
(-z^3 - 12*z^2 - 60*z - 120)/(z^3 - 12*z^2 + 60*z - 120)
sage: log(1-z).pade(4, 4)
(25/6*z^4 - 130/3*z^3 + 105*z^2 - 70*z)/(z^4 - 20*z^3 + 90*z^2
- 140*z + 70)
sage: sqrt(1+z).pade(3, 2)
(1/6*z^3 + 3*z^2 + 8*z + 16/3)/(z^2 + 16/3*z + 16/3)
sage: exp(2*z).pade(3, 3)
(-z^3 - 6*z^2 - 15*z - 15)/(z^3 - 6*z^2 + 15*z - 15)

>>> from sage.all import *
>>> z = PowerSeriesRing(QQ, 'z').gen()
>>> exp(z).pade(Integer(4), Integer(0))
1/24*z^4 + 1/6*z^3 + 1/2*z^2 + z + 1
>>> exp(z).pade(Integer(1), Integer(1))
(-z - 2)/(z - 2)
>>> exp(z).pade(Integer(3), Integer(3))
(-z^3 - 12*z^2 - 60*z - 120)/(z^3 - 12*z^2 + 60*z - 120)
>>> log(Integer(1)-z).pade(Integer(4), Integer(4))
(25/6*z^4 - 130/3*z^3 + 105*z^2 - 70*z)/(z^4 - 20*z^3 + 90*z^2
- 140*z + 70)
>>> sqrt(Integer(1)+z).pade(Integer(3), Integer(2))
(1/6*z^3 + 3*z^2 + 8*z + 16/3)/(z^2 + 16/3*z + 16/3)
>>> exp(Integer(2)*z).pade(Integer(3), Integer(3))
(-z^3 - 6*z^2 - 15*z - 15)/(z^3 - 6*z^2 + 15*z - 15)
polynomial()[source]

Return the underlying polynomial of self.

EXAMPLES:

sage: R.<t> = GF(7)[[]]
sage: f = 3 - t^3 + O(t^5)
sage: f.polynomial()
6*t^3 + 3

>>> from sage.all import *
>>> R = GF(Integer(7))[['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 \(f\), i.e., the series \(g\) such that \(g(f(x)) = x\).

Given an optional argument precision, return the reverse with given precision (note that the reverse can have precision at most f.prec()). If \(f\) has infinite precision, and the argument precision is not given, then the precision of the reverse defaults to the default precision of f.parent().

Note that this is only possible if the valuation of self is exactly 1.

ALGORITHM:

We first attempt to pass the computation to pari; if this fails, we use Lagrange inversion. Using sage: set_verbose(1) will print a message if passing to pari fails.

If the base ring has positive characteristic, then we attempt to lift to a characteristic zero ring and perform the reverse there. If this fails, an error is raised.

EXAMPLES:

sage: R.<x> = PowerSeriesRing(QQ)
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)
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)
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)
sage: B.<s> = A[[]]
sage: f = (1 - 3*t + 4*t^3 + O(t^4))*s + (2 + t + t^2 + O(t^3))*s^2 + O(s^3)
sage: from sage.misc.verbose import set_verbose
sage: set_verbose(1)
sage: g = f.reverse(); g
verbose 1 (<module>) passing to pari failed; trying Lagrange inversion
(1 + 3*t + 9*t^2 + 23*t^3 + O(t^4))*s + (-2 - 19*t - 118*t^2 + O(t^3))*s^2 + O(s^3)
sage: set_verbose(0)
sage: f(g) == g(f) == s
True

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, 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, 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, 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, names=('t',)); (t,) = A._first_ngens(1)
>>> B = A[['s']]; (s,) = B._first_ngens(1)
>>> f = (Integer(1) - Integer(3)*t + Integer(4)*t**Integer(3) + O(t**Integer(4)))*s + (Integer(2) + t + t**Integer(2) + O(t**Integer(3)))*s**Integer(2) + O(s**Integer(3))
>>> from sage.misc.verbose import set_verbose
>>> set_verbose(Integer(1))
>>> g = f.reverse(); g
verbose 1 (<module>) passing to pari failed; trying Lagrange inversion
(1 + 3*t + 9*t^2 + 23*t^3 + O(t^4))*s + (-2 - 19*t - 118*t^2 + O(t^3))*s^2 + O(s^3)
>>> set_verbose(Integer(0))
>>> f(g) == g(f) == s
True

If the leading coefficient is not a unit, we pass to its fraction field if possible:

sage: A.<t> = PowerSeriesRing(ZZ)
sage: a = 2*t - 4*t^2 + t^4 - t^5 + O(t^6)
sage: a.reverse()
1/2*t + 1/2*t^2 + t^3 + 79/32*t^4 + 437/64*t^5 + O(t^6)

sage: B.<b> = PolynomialRing(ZZ)
sage: A.<t> = PowerSeriesRing(B)
sage: f = 2*b*t + b*t^2 + 3*b^2*t^3 + O(t^4)
sage: g = f.reverse(); g
1/(2*b)*t - 1/(8*b^2)*t^2 + ((-3*b + 1)/(16*b^3))*t^3 + O(t^4)
sage: f(g)
t + O(t^4)
sage: g(f)
t + O(t^4)

>>> from sage.all import *
>>> A = PowerSeriesRing(ZZ, names=('t',)); (t,) = A._first_ngens(1)
>>> a = Integer(2)*t - Integer(4)*t**Integer(2) + t**Integer(4) - t**Integer(5) + O(t**Integer(6))
>>> a.reverse()
1/2*t + 1/2*t^2 + t^3 + 79/32*t^4 + 437/64*t^5 + O(t^6)

>>> B = PolynomialRing(ZZ, names=('b',)); (b,) = B._first_ngens(1)
>>> A = PowerSeriesRing(B, names=('t',)); (t,) = A._first_ngens(1)
>>> f = Integer(2)*b*t + b*t**Integer(2) + Integer(3)*b**Integer(2)*t**Integer(3) + O(t**Integer(4))
>>> g = f.reverse(); g
1/(2*b)*t - 1/(8*b^2)*t^2 + ((-3*b + 1)/(16*b^3))*t^3 + O(t^4)
>>> f(g)
t + O(t^4)
>>> g(f)
t + O(t^4)

We can handle some base rings of positive characteristic:

sage: A8.<t> = PowerSeriesRing(Zmod(8))
sage: a = t - 15*t^2 - 2*t^4 + t^5 + O(t^6)
sage: b = a.reverse(); b
t + 7*t^2 + 2*t^3 + 5*t^4 + t^5 + O(t^6)
sage: a(b)
t + O(t^6)
sage: b(a)
t + O(t^6)

>>> from sage.all import *
>>> A8 = PowerSeriesRing(Zmod(Integer(8)), names=('t',)); (t,) = A8._first_ngens(1)
>>> a = t - Integer(15)*t**Integer(2) - Integer(2)*t**Integer(4) + t**Integer(5) + O(t**Integer(6))
>>> b = a.reverse(); b
t + 7*t^2 + 2*t^3 + 5*t^4 + t^5 + O(t^6)
>>> a(b)
t + O(t^6)
>>> b(a)
t + O(t^6)

The optional argument precision sets the precision of the output:

sage: R.<x> = PowerSeriesRing(QQ)
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, 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)
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), 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)
truncate(prec='infinity')[source]

The polynomial obtained from power series by truncation at precision prec.

EXAMPLES:

sage: R.<I> = GF(2)[[]]
sage: f = 1/(1+I+O(I^8)); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
sage: f.truncate(5)
I^4 + I^3 + I^2 + I + 1

>>> from sage.all import *
>>> R = GF(Integer(2))[['I']]; (I,) = R._first_ngens(1)
>>> f = Integer(1)/(Integer(1)+I+O(I**Integer(8))); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
>>> f.truncate(Integer(5))
I^4 + I^3 + I^2 + I + 1
truncate_powerseries(prec)[source]

Given input prec = \(n\), returns the power series of degree \(< n\) which is equivalent to self modulo \(x^n\).

EXAMPLES:

sage: R.<I> = GF(2)[[]]
sage: f = 1/(1+I+O(I^8)); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
sage: f.truncate_powerseries(5)
1 + I + I^2 + I^3 + I^4 + O(I^5)

>>> from sage.all import *
>>> R = GF(Integer(2))[['I']]; (I,) = R._first_ngens(1)
>>> f = Integer(1)/(Integer(1)+I+O(I**Integer(8))); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
>>> f.truncate_powerseries(Integer(5))
1 + I + I^2 + I^3 + I^4 + O(I^5)
valuation()[source]

Return the valuation of self.

EXAMPLES:

sage: R.<t> = QQ[[]]
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 = QQ[['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
sage.rings.power_series_poly.make_powerseries_poly_v0(parent, f, prec, is_gen)[source]

Return the power series specified by f, prec, and is_gen.

This function exists for the purposes of pickling. Do not delete this function – if you change the internal representation, instead make a new function and make sure that both kinds of objects correctly unpickle as the new type.

EXAMPLES:

sage: R.<t> = QQ[[]]
sage: sage.rings.power_series_poly.make_powerseries_poly_v0(R, t, infinity, True)
t

>>> from sage.all import *
>>> R = QQ[['t']]; (t,) = R._first_ngens(1)
>>> sage.rings.power_series_poly.make_powerseries_poly_v0(R, t, infinity, True)
t