# Lazy Power Series¶

This file provides an implementation of lazy univariate power series, which uses the stream class for its internal data structure. The lazy power series keep track of their approximate order as much as possible without forcing the computation of any additional coefficients. This is required for recursively defined power series.

This code is based on the work of Ralf Hemmecke and Martin Rubey’s Aldor-Combinat, which can be found at http://www.risc.uni-linz.ac.at/people/hemmecke/aldor/combinat/index.html. In particular, the relevant section for this file can be found at http://www.risc.uni-linz.ac.at/people/hemmecke/AldorCombinat/combinatse9.html.

class sage.combinat.species.series.LazyPowerSeries(A, stream=None, order=None, aorder=None, aorder_changed=True, is_initialized=False, name=None)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: f = L()
Uninitialized lazy power series


EXAMPLES: Test Plus 1

sage: from sage.combinat.species.series import *
sage: from sage.combinat.species.stream import Stream
sage: L = LazyPowerSeriesRing(QQ)
sage: gs0 = L([0])
sage: gs1 = L([1])
sage: sum1 = gs0 + gs1
sage: sum2 = gs1 + gs1
sage: sum3 = gs1 + gs0
sage: [gs0.coefficient(i) for i in range(11)]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
sage: [gs1.coefficient(i) for i in range(11)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: [sum1.coefficient(i) for i in range(11)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: [sum2.coefficient(i) for i in range(11)]
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
sage: [sum3.coefficient(i) for i in range(11)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]


Test Plus 2

sage: gs1 = L([1,2,4,8,0])
sage: gs2 = L([-1, 0,-1,-9,22,0])
sage: sum = gs1 + gs2
sage: sum2 = gs2 + gs1
sage: [ sum.coefficient(i) for i in range(5) ]
[0,  2, 3, -1, 22]
sage: [ sum.coefficient(i) for i in range(5, 11) ]
[0, 0, 0, 0, 0, 0]
sage: [ sum2.coefficient(i) for i in range(5) ]
[0,  2, 3, -1, 22]
sage: [ sum2.coefficient(i) for i in range(5, 11) ]
[0, 0, 0, 0, 0, 0]

coefficient(n)

Returns the coefficient of xn in self.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: f = L(ZZ)
sage: [f.coefficient(i) for i in range(5)]
[0, 1, -1, 2, -2]

coefficients(n)

Returns the first n coefficients of self.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: f = L([1,2,3,0])
sage: f.coefficients(5)
[1, 2, 3, 0, 0]

compose(y)

Returns the composition of this power series and the power series y.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: s = L([1])
sage: t = L([0,0,1])
sage: u = s(t)
sage: u.coefficients(11)
[1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34]


Test Compose 2

sage: s = L([1])
sage: t = L([0,0,1,0])
sage: u = s(t)
sage: u.aorder
0
sage: u.order
Unknown series order
sage: u.coefficients(10)
[1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
sage: u.aorder
0
sage: u.order
0


Test Compose 3 s = 1/(1-x), t = x/(1-x) s(t) = (1-x)/(1-2x)

sage: s = L([1])
sage: t = L([0,1])
sage: u = s(t)
sage: u.coefficients(14)
[1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096]

composition(y)

Returns the composition of this power series and the power series y.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: s = L([1])
sage: t = L([0,0,1])
sage: u = s(t)
sage: u.coefficients(11)
[1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34]


Test Compose 2

sage: s = L([1])
sage: t = L([0,0,1,0])
sage: u = s(t)
sage: u.aorder
0
sage: u.order
Unknown series order
sage: u.coefficients(10)
[1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
sage: u.aorder
0
sage: u.order
0


Test Compose 3 s = 1/(1-x), t = x/(1-x) s(t) = (1-x)/(1-2x)

sage: s = L([1])
sage: t = L([0,1])
sage: u = s(t)
sage: u.coefficients(14)
[1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096]

compute_aorder(*args, **kwargs)

The default compute_aorder does nothing.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: a = L(1)
sage: a.compute_aorder() is None
True

compute_coefficients(i)

Computes all the coefficients of self up to i.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: a = L([1,2,3])
sage: a.compute_coefficients(5)
sage: a
1 + 2*x + 3*x^2 + 3*x^3 + 3*x^4 + 3*x^5 + ...

define(x)

EXAMPLES: Test Recursive 0

sage: L = LazyPowerSeriesRing(QQ)
sage: one = L(1)
sage: monom = L.gen()
sage: s = L()
sage: s._name = 's'
sage: s.define(one+monom*s)
sage: s.aorder
0
sage: s.order
Unknown series order
sage: [s.coefficient(i) for i in range(6)]
[1, 1, 1, 1, 1, 1]


Test Recursive 1

sage: s = L()
sage: s._name = 's'
sage: s.define(one+monom*s*s)
sage: s.aorder
0
sage: s.order
Unknown series order
sage: [s.coefficient(i) for i in range(6)]
[1, 1, 2, 5, 14, 42]


Test Recursive 1b

sage: s = L()
sage: s._name = 's'
sage: s.define(monom + s*s)
sage: s.aorder
1
sage: s.order
Unknown series order
sage: [s.coefficient(i) for i in range(7)]
[0, 1, 1, 2, 5, 14, 42]


Test Recursive 2

sage: s = L()
sage: s._name = 's'
sage: t = L()
sage: t._name = 't'
sage: s.define(one+monom*t*t*t)
sage: t.define(one+monom*s*s)
sage: [s.coefficient(i) for i in range(9)]
[1, 1, 3, 9, 34, 132, 546, 2327, 10191]
sage: [t.coefficient(i) for i in range(9)]
[1, 1, 2, 7, 24, 95, 386, 1641, 7150]


Test Recursive 2b

sage: s = L()
sage: s._name = 's'
sage: t = L()
sage: t._name = 't'
sage: s.define(monom + t*t*t)
sage: t.define(monom + s*s)
sage: [s.coefficient(i) for i in range(9)]
[0, 1, 0, 1, 3, 3, 7, 30, 63]
sage: [t.coefficient(i) for i in range(9)]
[0, 1, 1, 0, 2, 6, 7, 20, 75]


Test Recursive 3

sage: s = L()
sage: s._name = 's'
sage: s.define(one+monom*s*s*s)
sage: [s.coefficient(i) for i in range(10)]
[1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675]

derivative()

EXAMPLES:

sage: from sage.combinat.species.stream import Stream
sage: L = LazyPowerSeriesRing(QQ)
sage: one = L(1)
sage: monom = L.gen()
sage: s = L([1])
sage: u = s.derivative()
sage: u.coefficients(10)
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

sage: s = L()
sage: s._name = 's'
sage: s.define(one+monom*s*s)
sage: u = s.derivative()
sage: u.coefficients(5) #[1*1, 2*2, 3*5, 4*14, 5*42]
[1, 4, 15, 56, 210]

sage: s = L([1])
sage: t = L([0,1])
sage: u = s(t).derivative()
sage: v = (s.derivative().compose(t))*t.derivative()
sage: u.coefficients(11)
[1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120, 11264]
sage: v.coefficients(11)
[1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120, 11264]

sage: s = L(); s._name='s'
sage: t = L(); t._name='t'
sage: s.define(monom+t*t*t)
sage: t.define(monom+s*s)
sage: u = (s*t).derivative()
sage: v = s.derivative()*t + s*t.derivative()
sage: u.coefficients(10)
[0, 2, 3, 4, 30, 72, 133, 552, 1791, 4260]
sage: v.coefficients(10)
[0, 2, 3, 4, 30, 72, 133, 552, 1791, 4260]
sage: u.coefficients(10) == v.coefficients(10)
True

sage: f = L._new_initial(2, Stream([0,0,4,5,6,0]))
sage: d = f.derivative()
sage: d.get_aorder()
1
sage: d.coefficients(5)
[0, 8, 15, 24, 0]

exponential()

TESTS:

sage: def inv_factorial():
...       q = 1
...       yield 0
...       yield q
...       n = 2
...       while True:
...           q = q / n
...           yield q
...           n += 1
sage: L = LazyPowerSeriesRing(QQ)
sage: f = L(inv_factorial()) #e^(x)-1
sage: u = f.exponential()
sage: g = inv_factorial()
sage: z1 = [1,1,2,5,15,52,203,877,4140,21147,115975]
sage: l1 = [z*next(g) for z in z1]
sage: l1 = [1] + l1[1:]
sage: u.coefficients(11)
[1, 1, 1, 5/6, 5/8, 13/30, 203/720, 877/5040, 23/224, 1007/17280, 4639/145152]
sage: l1 == u.coefficients(11)
True

get_aorder()

Returns the approximate order of self.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: a = L.gen()
sage: a.get_aorder()
1

get_order()

Returns the order of self.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: a = L.gen()
sage: a.get_order()
1

get_stream()

Returns self’s underlying Stream object.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: a = L.gen()
sage: s = a.get_stream()
sage: [s[i] for i in range(5)]
[0, 1, 0, 0, 0]

initialize_coefficient_stream(compute_coefficients)

Initializes the coefficient stream.

INPUT: compute_coefficients

TESTS:

sage: from sage.combinat.species.series_order import inf, unk
sage: L = LazyPowerSeriesRing(QQ)
sage: f = L()
sage: compute_coefficients = lambda ao: iter(ZZ)
sage: f.order = inf
sage: f.aorder = inf
sage: f.initialize_coefficient_stream(compute_coefficients)
sage: f.coefficients(5)
[0, 0, 0, 0, 0]

sage: f = L()
sage: compute_coefficients = lambda ao: iter(ZZ)
sage: f.order = 1
sage: f.aorder = 1
sage: f.initialize_coefficient_stream(compute_coefficients)
sage: f.coefficients(5)
[0, 1, -1, 2, -2]

integral(integration_constant=0)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: zero = L(0)
sage: s = zero
sage: t = s.integral()
sage: t.is_zero()
True

sage: s = zero
sage: t = s.integral(1)
sage: t.coefficients(6)
[1, 0, 0, 0, 0, 0]
sage: t._stream.is_constant()
True

sage: s = L.term(1, 0)
sage: t = s.integral()
sage: t.coefficients(6)
[0, 1, 0, 0, 0, 0]
sage: t._stream.is_constant()
True

sage: s = L.term(1,0)
sage: t = s.integral(1)
sage: t.coefficients(6)
[1, 1, 0, 0, 0, 0]
sage: t._stream.is_constant()
True

sage: s = L.term(1, 4)
sage: t = s.integral()
sage: t.coefficients(10)
[0, 0, 0, 0, 0, 1/5, 0, 0, 0, 0]

sage: s = L.term(1,4)
sage: t = s.integral(1)
sage: t.coefficients(10)
[1, 0, 0, 0, 0, 1/5, 0, 0, 0, 0]


TESTS:

sage: from sage.combinat.species.stream import Stream
sage: f = L._new_initial(2, Stream([0,0,4,5,6,0]))
sage: i = f.derivative().integral()
sage: i.get_aorder()
2
sage: i.coefficients(5)
[0, 0, 4, 5, 6]
sage: i = f.derivative().integral(1)
sage: i.get_aorder()
0
sage: i.coefficients(5)
[1, 0, 4, 5, 6]

is_finite(n=None)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: a = L([0,0,1,0,0]); a
O(1)
sage: a.is_finite()
False
sage: c = a[4]
sage: a.is_finite()
False
sage: a.is_finite(4)
False
sage: c = a[5]
sage: a.is_finite()
True
sage: a.is_finite(4)
True

is_zero()

Returns True if and only if self is zero.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: s = L([0,2,3,0])
sage: s.is_zero()
False

sage: s = L(0)
sage: s.is_zero()
True

sage: s = L([0])
sage: s.is_zero()
False
sage: s.coefficient(0)
0
sage: s.coefficient(1)
0
sage: s.is_zero()
True

iterator(n=0, initial=None)

Returns an iterator for the coefficients of self starting at n.

EXAMPLES:

sage: from sage.combinat.species.stream import Stream
sage: L = LazyPowerSeriesRing(QQ)
sage: f = L(range(10))
sage: g = f.iterator(2)
sage: [next(g) for i in range(5)]
[2, 3, 4, 5, 6]
sage: g = f.iterator(2, initial=[0,0])
sage: [next(g) for i in range(5)]
[0, 0, 2, 3, 4]

refine_aorder()

Refines the approximate order of self as much as possible without computing any coefficients.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: a = L([0,0,0,0,1])
sage: a.aorder
0
sage: a.coefficient(2)
0
sage: a.aorder
0
sage: a.refine_aorder()
sage: a.aorder
3

sage: a = L([0,0])
sage: a.aorder
0
sage: a.coefficient(5)
0
sage: a.refine_aorder()
sage: a.aorder
Infinite series order

sage: a = L([0,0,1,0,0,0])
sage: a[4]
0
sage: a.refine_aorder()
sage: a.aorder
2

restricted(min=None, max=None)

Returns the power series restricted to the coefficients starting at min and going up to, but not including max. If min is not specified, then it is assumed to be zero. If max is not specified, then it is assumed to be infinity.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: a = L([1])
sage: a.restricted().coefficients(10)
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: a.restricted(min=2).coefficients(10)
[0, 0, 1, 1, 1, 1, 1, 1, 1, 1]
sage: a.restricted(max=5).coefficients(10)
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0]
sage: a.restricted(min=2, max=6).coefficients(10)
[0, 0, 1, 1, 1, 1, 0, 0, 0, 0]

set_approximate_order(new_order)

Sets the approximate order of self and returns True if the approximate order has changed otherwise it will return False.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: f = L([0,0,0,3,2,1,0])
sage: f.get_aorder()
0
sage: f.set_approximate_order(3)
True
sage: f.set_approximate_order(3)
False

tail()

Returns the power series whose coefficients obtained by subtracting the constant term from this series and then dividing by x.

EXAMPLES:

sage: from sage.combinat.species.stream import Stream
sage: L = LazyPowerSeriesRing(QQ)
sage: f = L(range(20))
sage: g = f.tail()
sage: g.coefficients(10)
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

times(y)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: gs0 = L(0)
sage: gs1 = L([1])

sage: prod0 = gs0 * gs1
sage: [prod0.coefficient(i) for i in range(11)]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

sage: prod1 = gs1 * gs0
sage: [prod1.coefficient(i) for i in range(11)]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

sage: prod2 = gs1 * gs1
sage: [prod2.coefficient(i) for i in range(11)]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

sage: gs1 = L([1,2,4,8,0])
sage: gs2 = L([-1, 0,-1,-9,22,0])

sage: prod1 = gs1 * gs2
sage: [prod1.coefficient(i) for i in range(11)]
[-1, -2, -5, -19, 0, 0, 16, 176, 0, 0, 0]

sage: prod2 = gs2 * gs1
sage: [prod2.coefficient(i) for i in range(11)]
[-1, -2, -5, -19, 0, 0, 16, 176, 0, 0, 0]

class sage.combinat.species.series.LazyPowerSeriesRing(R, element_class=None, names=None)

Bases: sage.rings.ring.Algebra

TESTS:

sage: from sage.combinat.species.series import LazyPowerSeriesRing
sage: L = LazyPowerSeriesRing(QQ)
Lazy Power Series Ring over Rational Field

gen(i=0)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: L.gen().coefficients(5)
[0, 1, 0, 0, 0]

identity_element()

Returns the one power series.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: L.identity_element()
1

ngens()

EXAMPLES:

sage: LazyPowerSeriesRing(QQ).ngens()
1

product_generator(g)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: s1 = L([1,1,0])
sage: s2 = L([1,0,1,0])
sage: s3 = L([1,0,0,1,0])
sage: s4 = L([1,0,0,0,1,0])
sage: s5 = L([1,0,0,0,0,1,0])
sage: s6 = L([1,0,0,0,0,0,1,0])
sage: s = [s1, s2, s3, s4, s5, s6]
sage: def g():
...       for a in s:
...           yield a
sage: p = L.product_generator(g())
sage: p.coefficients(26)
[1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0]

sage: def m(n):
...       yield 1
...       while True:
...           for i in range(n-1):
...               yield 0
...           yield 1
...
sage: def s(n):
...       q = 1/n
...       yield 0
...       while True:
...           for i in range(n-1):
...               yield 0
...           yield q
...

sage: def lhs_gen():
...       n = 1
...       while True:
...           yield L(m(n))
...           n += 1
...

sage: def rhs_gen():
...       n = 1
...       while True:
...           yield L(s(n))
...           n += 1
...
sage: lhs = L.product_generator(lhs_gen())
sage: rhs = L.sum_generator(rhs_gen()).exponential()
sage: lhs.coefficients(10)
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]
sage: rhs.coefficients(10)
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]

sum(a)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: l = [L(ZZ)]*3
sage: L.sum(l).coefficients(10)
[0, 3, -3, 6, -6, 9, -9, 12, -12, 15]

sum_generator(g)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: g = [L([1])]*6 + [L(0)]
sage: t = L.sum_generator(g)
sage: t.coefficients(10)
[1, 2, 3, 4, 5, 6, 6, 6, 6, 6]

sage: s = L([1])
sage: def g():
...       while True:
...           yield s
sage: t = L.sum_generator(g())
sage: t.coefficients(9)
[1, 2, 3, 4, 5, 6, 7, 8, 9]

term(r, n)

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: L.term(0,0)
0
sage: L.term(3,2).coefficients(5)
[0, 0, 3, 0, 0]

zero()

Returns the zero power series.

EXAMPLES:

sage: L = LazyPowerSeriesRing(QQ)
sage: L.zero()
0


TESTS:

Check that the method $$zero_element$$ raises a warning (trac ticket #17694):

sage: L.zero_element()
See http://trac.sagemath.org/17694 for details.
0

zero_element(*args, **kwds)

Deprecated: Use zero() instead. See trac ticket #17694 for details.

sage.combinat.species.series.uninitialized()

EXAMPLES:

sage: from sage.combinat.species.series import uninitialized
sage: uninitialized()
Traceback (most recent call last):
...
RuntimeError: we should never be here


#### Previous topic

Recursive Species

Series Order