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.
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class
sage.combinat.species.series.
LazyPowerSeries
(A, stream=None, order=None, aorder=None, aorder_changed=True, is_initialized=False, name=None)¶ Bases:
sage.structure.element.AlgebraElement
EXAMPLES:
sage: L = LazyPowerSeriesRing(QQ) sage: f = L() sage: loads(dumps(f)) Uninitialized lazy power series
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add
(y)¶ 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]
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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]
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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]
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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]
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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]
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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
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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 + ...
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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]
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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]
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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
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get_aorder
()¶ Returns the approximate order of self.
EXAMPLES:
sage: L = LazyPowerSeriesRing(QQ) sage: a = L.gen() sage: a.get_aorder() 1
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get_order
()¶ Returns the order of self.
EXAMPLES:
sage: L = LazyPowerSeriesRing(QQ) sage: a = L.gen() sage: a.get_order() 1
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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]
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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]
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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]
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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
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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
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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]
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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
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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]
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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
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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]
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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]
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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) sage: loads(dumps(L)) Lazy Power Series Ring over Rational Field
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gen
(i=0)¶ EXAMPLES:
sage: L = LazyPowerSeriesRing(QQ) sage: L.gen().coefficients(5) [0, 1, 0, 0, 0]
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identity_element
()¶ Returns the one power series.
EXAMPLES:
sage: L = LazyPowerSeriesRing(QQ) sage: L.identity_element() 1
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ngens
()¶ EXAMPLES:
sage: LazyPowerSeriesRing(QQ).ngens() 1
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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]
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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]
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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]
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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]
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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() doctest:...: DeprecationWarning: zero_element is deprecated. Please use zero instead. See http://trac.sagemath.org/17694 for details. 0
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zero_element
(*args, **kwds)¶ Deprecated: Use
zero()
instead. See trac ticket #17694 for details.
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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