Power Series
Sage provides an implementation of dense and sparse power series over any Sage base ring. This is the base class of the implementations of univariate and multivariate power series ring elements in Sage (see also Power Series Methods, Multivariate Power Series.).
AUTHORS:
EXAMPLE:
sage: R.<x> = PowerSeriesRing(ZZ)
sage: TestSuite(R).run()
sage: R([1,2,3])
1 + 2*x + 3*x^2
sage: R([1,2,3], 10)
1 + 2*x + 3*x^2 + O(x^10)
sage: f = 1 + 2*x - 3*x^3 + O(x^4); f
1 + 2*x - 3*x^3 + O(x^4)
sage: f^10
1 + 20*x + 180*x^2 + 930*x^3 + O(x^4)
sage: g = 1/f; g
1 - 2*x + 4*x^2 - 5*x^3 + O(x^4)
sage: g * f
1 + O(x^4)
In Python (as opposed to Sage) create the power series ring and its generator as follows:
sage: R, x = objgen(PowerSeriesRing(ZZ, 'x'))
sage: parent(x)
Power Series Ring in x over Integer Ring
EXAMPLE:
This example illustrates that coercion for power series rings is consistent with coercion for polynomial rings.
sage: poly_ring1.<gen1> = PolynomialRing(QQ)
sage: poly_ring2.<gen2> = PolynomialRing(QQ)
sage: huge_ring.<x> = PolynomialRing(poly_ring1)
The generator of the first ring gets coerced in as itself, since it is the base ring.
sage: huge_ring(gen1)
gen1
The generator of the second ring gets mapped via the natural map sending one generator to the other.
sage: huge_ring(gen2)
x
With power series the behavior is the same.
sage: power_ring1.<gen1> = PowerSeriesRing(QQ)
sage: power_ring2.<gen2> = PowerSeriesRing(QQ)
sage: huge_power_ring.<x> = PowerSeriesRing(power_ring1)
sage: huge_power_ring(gen1)
gen1
sage: huge_power_ring(gen2)
x
Bases: sage.structure.element.AlgebraElement
A power series. Base class of univariate and multivariate power series. The following methods are available with both types of objects.
Return this series plus . Does not change
self.
EXAMPLES:
sage: R.<x> = PowerSeriesRing(ZZ)
sage: p = 1 + x^2 + x^10; p
1 + x^2 + x^10
sage: p.O(15)
1 + x^2 + x^10 + O(x^15)
sage: p.O(5)
1 + x^2 + O(x^5)
sage: p.O(-5)
Traceback (most recent call last):
...
ValueError: n must be at least 0
If , then this function returns
.
EXAMPLES:
sage: R.<x> = PowerSeriesRing(ZZ)
sage: p = 1 + x^2 + x^10; p
1 + x^2 + x^10
sage: p.V(3)
1 + x^6 + x^30
sage: (p+O(x^20)).V(3)
1 + x^6 + x^30 + O(x^60)
Return the power series of precision at most prec got by adding
to
, where
is the variable.
EXAMPLES:
sage: R.<A> = RDF[[]]
sage: f = (1+A+O(A^5))^5; f
1.0 + 5.0*A + 10.0*A^2 + 10.0*A^3 + 5.0*A^4 + O(A^5)
sage: f.add_bigoh(3)
1.0 + 5.0*A + 10.0*A^2 + O(A^3)
sage: f.add_bigoh(5)
1.0 + 5.0*A + 10.0*A^2 + 10.0*A^3 + 5.0*A^4 + O(A^5)
Return a copy of this power series but with coefficients in R.
The following coercion uses base_extend implicitly:
sage: R.<t> = ZZ[['t']]
sage: (t - t^2) * Mod(1, 3)
t + 2*t^2
Return the base ring that this power series is defined over.
EXAMPLES:
sage: R.<t> = GF(49,'alpha')[[]]
sage: (t^2 + O(t^3)).base_ring()
Finite Field in alpha of size 7^2
Change if possible the coefficients of self to lie in R.
EXAMPLES:
sage: R.<T> = QQ[[]]; R
Power Series Ring in T over Rational Field
sage: f = 1 - 1/2*T + 1/3*T^2 + O(T^3)
sage: f.base_extend(GF(5))
Traceback (most recent call last):
...
TypeError: no base extension defined
sage: f.change_ring(GF(5))
1 + 2*T + 2*T^2 + O(T^3)
sage: f.change_ring(GF(3))
Traceback (most recent call last):
...
ZeroDivisionError: Inverse does not exist.
We can only change the ring if there is a __call__ coercion defined. The following succeeds because ZZ(K(4)) is defined.
sage: K.<a> = NumberField(cyclotomic_polynomial(3), 'a')
sage: R.<t> = K[['t']]
sage: (4*t).change_ring(ZZ)
4*t
This does not succeed because ZZ(K(a+1)) is not defined.
sage: K.<a> = NumberField(cyclotomic_polynomial(3), 'a')
sage: R.<t> = K[['t']]
sage: ((a+1)*t).change_ring(ZZ)
Traceback (most recent call last):
...
TypeError: Unable to coerce a + 1 to an integer
Return the nonzero coefficients of self.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ)
sage: f = t + t^2 - 10/3*t^3
sage: f.coefficients()
[1, 1, -10/3]
Return minimum precision of and self.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ)
sage: f = t + t^2 + O(t^3)
sage: g = t + t^3 + t^4 + O(t^4)
sage: f.common_prec(g)
3
sage: g.common_prec(f)
3
sage: f = t + t^2 + O(t^3)
sage: g = t^2
sage: f.common_prec(g)
3
sage: g.common_prec(f)
3
sage: f = t + t^2
sage: f = t^2
sage: f.common_prec(g)
+Infinity
Return the degree of this power series, which is by definition the degree of the underlying polynomial.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ, sparse=True)
sage: f = t^100000 + O(t^10000000)
sage: f.degree()
100000
The formal derivative of this power series, with respect to variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.
See also
_derivative()
EXAMPLES:
sage: R.<x> = PowerSeriesRing(QQ)
sage: g = -x + x^2/2 - x^4 + O(x^6)
sage: g.derivative()
-1 + x - 4*x^3 + O(x^5)
sage: g.derivative(x)
-1 + x - 4*x^3 + O(x^5)
sage: g.derivative(x, x)
1 - 12*x^2 + O(x^4)
sage: g.derivative(x, 2)
1 - 12*x^2 + O(x^4)
Deprecated: Use ogf_to_egf() instead. See trac ticket #15705 for details.
Returns the ordinary generating function power series, assuming self is an exponential generating function power series.
This function is known as serlaplace in PARI/GP.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ)
sage: f = t + t^2/factorial(2) + 2*t^3/factorial(3)
sage: f.egf_to_ogf()
t + t^2 + 2*t^3
Return exp of this power series to the indicated precision.
INPUT:
ALGORITHM: See solve_linear_de().
Note
AUTHORS:
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ, default_prec=10)
Check that is, well,
:
sage: (t + O(t^10)).exp()
1 + t + 1/2*t^2 + 1/6*t^3 + 1/24*t^4 + 1/120*t^5 + 1/720*t^6 + 1/5040*t^7 + 1/40320*t^8 + 1/362880*t^9 + O(t^10)
Check that is
:
sage: (sum([-(-t)^n/n for n in range(1, 10)]) + O(t^10)).exp()
1 + t + O(t^10)
Check that is whatever it is:
sage: (2*t + t^2 - t^5 + O(t^10)).exp()
1 + 2*t + 3*t^2 + 10/3*t^3 + 19/6*t^4 + 8/5*t^5 - 7/90*t^6 - 538/315*t^7 - 425/168*t^8 - 30629/11340*t^9 + O(t^10)
Check requesting lower precision:
sage: (t + t^2 - t^5 + O(t^10)).exp(5)
1 + t + 3/2*t^2 + 7/6*t^3 + 25/24*t^4 + O(t^5)
Can’t get more precision than the input:
sage: (t + t^2 + O(t^3)).exp(10)
1 + t + 3/2*t^2 + O(t^3)
Check some boundary cases:
sage: (t + O(t^2)).exp(1)
1 + O(t)
sage: (t + O(t^2)).exp(0)
O(t^0)
Handle nonzero constant term (fixes trac ticket #4477):
sage: R.<x> = PowerSeriesRing(RR)
sage: (1 + x + x^2 + O(x^3)).exp()
2.71828182845905 + 2.71828182845905*x + 4.07742274268857*x^2 + O(x^3)
sage: R.<x> = PowerSeriesRing(ZZ)
sage: (1 + x + O(x^2)).exp()
Traceback (most recent call last):
...
ArithmeticError: exponential of constant term does not belong to coefficient ring (consider working in a larger ring)
sage: R.<x> = PowerSeriesRing(GF(5))
sage: (1 + x + O(x^2)).exp()
Traceback (most recent call last):
...
ArithmeticError: constant term of power series does not support exponentiation
Return the exponents appearing in self with nonzero coefficients.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ)
sage: f = t + t^2 - 10/3*t^3
sage: f.exponents()
[1, 2, 3]
EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ)
sage: t.is_dense()
True
sage: R.<t> = PowerSeriesRing(ZZ, sparse=True)
sage: t.is_dense()
False
Return True if this is the generator (the variable) of the power series ring.
EXAMPLES:
sage: R.<t> = QQ[[]]
sage: t.is_gen()
True
sage: (1 + 2*t).is_gen()
False
Note that this only returns True on the actual generator, not on something that happens to be equal to it.
sage: (1*t).is_gen()
False
sage: 1*t == t
True
Return True if this element is a monomial. That is, if self is
for some non-negative integer
.
EXAMPLES:
sage: k.<z> = PowerSeriesRing(QQ, 'z')
sage: z.is_monomial()
True
sage: k(1).is_monomial()
True
sage: (z+1).is_monomial()
False
sage: (z^2909).is_monomial()
True
sage: (3*z^2909).is_monomial()
False
EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ)
sage: t.is_sparse()
False
sage: R.<t> = PowerSeriesRing(ZZ, sparse=True)
sage: t.is_sparse()
True
Return True if this function has a square root in this ring, e.g.,
there is an element in self.parent()
such that
equals self.
ALGORITHM: If the base ring is a field, this is true whenever the power series has even valuation and the leading coefficient is a perfect square.
For an integral domain, it attempts the square root in the fraction field and tests whether or not the result lies in the original ring.
EXAMPLES:
sage: K.<t> = PowerSeriesRing(QQ, 't', 5)
sage: (1+t).is_square()
True
sage: (2+t).is_square()
False
sage: (2+t.change_ring(RR)).is_square()
True
sage: t.is_square()
False
sage: K.<t> = PowerSeriesRing(ZZ, 't', 5)
sage: (1+t).is_square()
False
sage: f = (1+t)^100
sage: f.is_square()
True
Return True if this power series is invertible.
A power series is invertible precisely when the constant term is invertible.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(ZZ)
sage: (-1 + t - t^5).is_unit()
True
sage: (3 + t - t^5).is_unit()
False
AUTHORS:
Return the Laurent series associated to this power series, i.e., this series considered as a Laurent series.
EXAMPLES:
sage: k.<w> = QQ[[]]
sage: f = 1+17*w+15*w^3+O(w^5)
sage: parent(f)
Power Series Ring in w over Rational Field
sage: g = f.laurent_series(); g
1 + 17*w + 15*w^3 + O(w^5)
See this method in derived classes:
Implementations MUST override this in the derived class.
EXAMPLES:
sage: R.<x> = PowerSeriesRing(ZZ)
sage: PowerSeries.list(1+x^2)
Traceback (most recent call last):
...
NotImplementedError
Return log of this power series to the indicated precision.
This works only if the constant term of the power series is 1.
INPUT:
ALGORITHM: See solve_linear_de().
Warning
Screwy things can happen if the coefficient ring is not a field of characteristic zero. See solve_linear_de().
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ, default_prec=10)
sage: (1 + t + O(t^10)).log()
t - 1/2*t^2 + 1/3*t^3 - 1/4*t^4 + 1/5*t^5 - 1/6*t^6 + 1/7*t^7 - 1/8*t^8 + 1/9*t^9 + O(t^10)
sage: t.exp().log()
t + O(t^10)
sage: (1+t).log().exp()
1 + t + O(t^10)
sage: (-1 + t + O(t^10)).log()
Traceback (most recent call last):
...
ArithmeticError: constant term of power series is not 1
Deprecated: Use egf_to_ogf() instead. See trac ticket #15705 for details.
Returns the exponential generating function power series, assuming self is an ordinary generating function power series.
This can also be computed as serconvol(f,exp(t)) in PARI/GP.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ)
sage: f = t + t^2 + 2*t^3
sage: f.ogf_to_egf()
t + 1/2*t^2 + 1/3*t^3
Return a list of coefficients of self up to (but not including)
.
Includes 0’s in the list on the right so that the list has length
.
INPUT:
EXAMPLES:
sage: R.<q> = PowerSeriesRing(QQ)
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]
See this method in derived classes:
Implementations MUST override this in the derived class.
EXAMPLES:
sage: R.<x> = PowerSeriesRing(ZZ)
sage: PowerSeries.polynomial(1+x^2)
Traceback (most recent call last):
...
NotImplementedError
The precision of is by definition
.
EXAMPLES:
sage: R.<t> = ZZ[[]]
sage: (t^2 + O(t^3)).prec()
3
sage: (1 - t^2 + O(t^100)).prec()
100
Return the absolute precision of this series.
By definition, the absolute precision of
is
.
EXAMPLES:
sage: R.<t> = ZZ[[]]
sage: (t^2 + O(t^3)).precision_absolute()
3
sage: (1 - t^2 + O(t^100)).precision_absolute()
100
Return the relative precision of this series, that is the difference between its absolute precision and its valuation.
By convension, the relative precision of (or
for any
) is
.
EXAMPLES:
sage: R.<t> = ZZ[[]]
sage: (t^2 + O(t^3)).precision_relative()
1
sage: (1 - t^2 + O(t^100)).precision_relative()
100
sage: O(t^4).precision_relative()
0
Return this power series multiplied by the power . If
is negative, terms below
will be
discarded. Does not change this power series.
Note
Despite the fact that higher order terms are printed to the
right in a power series, right shifting decreases the
powers of , while left shifting increases
them. This is to be consistent with polynomials, integers,
etc.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ['y'], 't', 5)
sage: f = ~(1+t); f
1 - t + t^2 - t^3 + t^4 + O(t^5)
sage: f.shift(3)
t^3 - t^4 + t^5 - t^6 + t^7 + O(t^8)
sage: f >> 2
1 - t + t^2 + O(t^3)
sage: f << 10
t^10 - t^11 + t^12 - t^13 + t^14 + O(t^15)
sage: t << 29
t^30
AUTHORS:
Obtain a power series solution to an inhomogeneous linear differential equation of the form:
INPUT:
OUTPUT: the power series , to indicated precision
ALGORITHM: A divide-and-conquer strategy; see the source code.
Running time is approximately , where
is the time required for a polynomial multiplication
of length
over the coefficient ring. (If you’re working
over something like
, running time analysis can be a
little complicated because the coefficients tend to explode.)
Note
AUTHORS:
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ, default_prec=10)
sage: a = 2 - 3*t + 4*t^2 + O(t^10)
sage: b = 3 - 4*t^2 + O(t^7)
sage: f = a.solve_linear_de(prec=5, b=b, f0=3/5)
sage: f
3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)
sage: f.derivative() - a*f - b
O(t^4)
sage: a = 2 - 3*t + 4*t^2
sage: b = b = 3 - 4*t^2
sage: f = a.solve_linear_de(b=b, f0=3/5)
Traceback (most recent call last):
...
ValueError: cannot solve differential equation to infinite precision
sage: a.solve_linear_de(prec=5, b=b, f0=3/5)
3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)
Return a square root of self.
INPUT:
- prec - integer (default: None): if not None and the series has infinite precision, truncates series at precision prec.
- extend - bool (default: False); if True, return a square root in an extension ring, if necessary. Otherwise, raise a ValueError if the square root is not in the base power series ring. For example, if extend is True the square root of a power series with odd degree leading coefficient is defined as an element of a formal extension ring.
- name - string; if extend is True, you must also specify the print name of the formal square root.
- all - bool (default: False); if True, return all square roots of self, instead of just one.
ALGORITHM: Newton’s method
EXAMPLES:
sage: K.<t> = PowerSeriesRing(QQ, 't', 5)
sage: sqrt(t^2)
t
sage: sqrt(1+t)
1 + 1/2*t - 1/8*t^2 + 1/16*t^3 - 5/128*t^4 + O(t^5)
sage: sqrt(4+t)
2 + 1/4*t - 1/64*t^2 + 1/512*t^3 - 5/16384*t^4 + O(t^5)
sage: u = sqrt(2+t, prec=2, extend=True, name = 'alpha'); u
alpha
sage: u^2
2 + t
sage: u.parent()
Univariate Quotient Polynomial Ring in alpha over Power Series Ring in t over Rational Field with modulus x^2 - 2 - t
sage: K.<t> = PowerSeriesRing(QQ, 't', 50)
sage: sqrt(1+2*t+t^2)
1 + t
sage: sqrt(t^2 +2*t^4 + t^6)
t + t^3
sage: sqrt(1 + t + t^2 + 7*t^3)^2
1 + t + t^2 + 7*t^3 + O(t^50)
sage: sqrt(K(0))
0
sage: sqrt(t^2)
t
sage: K.<t> = PowerSeriesRing(CDF, 5)
sage: v = sqrt(-1 + t + t^3, all=True); v
[1.0*I - 0.5*I*t - 0.125*I*t^2 - 0.5625*I*t^3 - 0.2890625*I*t^4 + O(t^5),
-1.0*I + 0.5*I*t + 0.125*I*t^2 + 0.5625*I*t^3 + 0.2890625*I*t^4 + O(t^5)]
sage: [a^2 for a in v]
[-1.0 + 1.0*t + 0.0*t^2 + 1.0*t^3 + O(t^5), -1.0 + 1.0*t + 0.0*t^2 + 1.0*t^3 + O(t^5)]
A formal square root:
sage: K.<t> = PowerSeriesRing(QQ, 5)
sage: f = 2*t + t^3 + O(t^4)
sage: s = f.sqrt(extend=True, name='sqrtf'); s
sqrtf
sage: s^2
2*t + t^3 + O(t^4)
sage: parent(s)
Univariate Quotient Polynomial Ring in sqrtf over Power Series Ring in t over Rational Field with modulus x^2 - 2*t - t^3 + O(t^4)
TESTS:
sage: R.<x> = QQ[[]]
sage: (x^10/2).sqrt()
Traceback (most recent call last):
...
ValueError: unable to take the square root of 1/2
AUTHORS:
Return the square root of self in this ring. If this cannot be done then an error will be raised.
This function succeeds if and only if self. is_square()
EXAMPLES:
sage: K.<t> = PowerSeriesRing(QQ, 't', 5)
sage: (1+t).square_root()
1 + 1/2*t - 1/8*t^2 + 1/16*t^3 - 5/128*t^4 + O(t^5)
sage: (2+t).square_root()
Traceback (most recent call last):
...
ValueError: Square root does not live in this ring.
sage: (2+t.change_ring(RR)).square_root()
1.41421356237309 + 0.353553390593274*t - 0.0441941738241592*t^2 + 0.0110485434560398*t^3 - 0.00345266983001244*t^4 + O(t^5)
sage: t.square_root()
Traceback (most recent call last):
...
ValueError: Square root not defined for power series of odd valuation.
sage: K.<t> = PowerSeriesRing(ZZ, 't', 5)
sage: f = (1+t)^20
sage: f.square_root()
1 + 10*t + 45*t^2 + 120*t^3 + 210*t^4 + O(t^5)
sage: f = 1+t
sage: f.square_root()
Traceback (most recent call last):
...
ValueError: Square root does not live in this ring.
AUTHORS:
The polynomial obtained from power series by truncation.
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
Return the valuation of this power series.
This is equal to the valuation of the underlying polynomial.
EXAMPLES:
Sparse examples:
sage: R.<t> = PowerSeriesRing(QQ, sparse=True)
sage: f = t^100000 + O(t^10000000)
sage: f.valuation()
100000
sage: R(0).valuation()
+Infinity
Dense examples:
sage: R.<t> = PowerSeriesRing(ZZ)
sage: f = 17*t^100 +O(t^110)
sage: f.valuation()
100
sage: t.valuation()
1
Factor self as as with
nonzero. Then this function returns
.
Note
This valuation zero part need not be a unit if, e.g.,
is not invertible in the base ring.
EXAMPLES:
sage: R.<t> = PowerSeriesRing(QQ)
sage: ((1/3)*t^5*(17-2/3*t^3)).valuation_zero_part()
17/3 - 2/9*t^3
In this example the valuation 0 part is not a unit:
sage: R.<t> = PowerSeriesRing(ZZ, sparse=True)
sage: u = (-2*t^5*(17-t^3)).valuation_zero_part(); u
-34 + 2*t^3
sage: u.is_unit()
False
sage: u.valuation()
0
Return a string with the name of the variable of this power series.
EXAMPLES:
sage: R.<x> = PowerSeriesRing(Rationals())
sage: f = x^2 + 3*x^4 + O(x^7)
sage: f.variable()
'x'
AUTHORS:
Return True if x is an instance of a univariate or multivariate power series.
EXAMPLES:
sage: R.<x> = PowerSeriesRing(ZZ)
sage: from sage.rings.power_series_ring_element import is_PowerSeries
sage: is_PowerSeries(1+x^2)
True
sage: is_PowerSeries(x-x)
True
sage: is_PowerSeries(0)
False
sage: var('x')
x
sage: is_PowerSeries(1+x^2)
False