Lazy series may be created in a number of ways. Arithmetic for ring elements is available as well as a few predicates. Coefficients of the monomials in the series can be calculated.
Return the series in the lazy series ring R with constant term c and every other coefficient 0 where c is any ring element coercible into the coefficient ring of R.
Return the lazy series in the lazy series ring R whose coefficients are those of the lazy series s coerced into the coefficient ring of R.
Return the series in the lazy series ring R whose coefficients are the elements of S where S is a sequence of elements each coercible into the coefficient ring of R. The coefficients are taken to be given in the order which Coefficients will return them and PrintToPrecision will print the terms of the series. Any coefficient not given is assumed to be zero so that all coefficients are calculable. The resulting series can only have finitely many non zero terms and all such non zero coefficients must be given in S.
Create the lazy series in the lazy series ring R with finitely many non zero terms which are the terms of the polynomial f. The number of variables of the parent ring of f must be the same as the number of variables of R. The coefficients of f must be coercible into the coefficient ring of R.It is also possible for f to be a rational function, p/q. The series created from f will be LazySeries(R, p)*LazySeries(R, q) - 1.
> L := LazyPowerSeriesRing(AlgebraicClosure(), 3); > LR := LazyPowerSeriesRing(Rationals(), 3); > s := L!1; > s; Lazy power series > LR!s; Lazy power series > P<x, y, z> := RationalFunctionField(Rationals(), 3); > LazySeries(L, (x + y + 8*z)^7/(1 + 5*x*y*z + x^8)^3); Lazy power series
Creating a lazy series from a map simulates the existence of lazy series as series for which coefficients are not known on creation of the series but calculated by some rule given when creating the series.
This rule can be coded in a map where the input to the map is the exponents of the variables in a term and the output is the coefficient of the term described by the input. So for a lazy series ring with variables x, y and z, a general term of a series will look like cijk * xir * yjs * zkt. The coefficient of this term, cijk, in a series defined by a map m is the result of m(< i, j, k >).
Creates a series in the lazy series ring R from the map m. The map m must take as input either an integer (when R is univariate only) or a tuple of integers (of length the number of variables of R) and return an element of the coefficient ring of R. This element will be taken as the coefficient of the term of the series whose variables have the exponents given in the input tuple, as described above.
> L<x> := LazyPowerSeriesRing(MaximalOrder(QuadraticField(5)), 1);
> Z := Integers();
> m := map<Z -> CoefficientRing(L) | t :-> 2*t>;
> s := elt<L | m>;
> PrintToPrecision(s, 10);
2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + 18*x^9 +
20*x^10
> Coefficient(s, 34);
68
> m(34);
68
> Coefficient(s, 2^30 + 10);
2147483668
> m(2^30 + 10);
2147483668
> L<x, y, z> := LazyPowerSeriesRing(AlgebraicClosure(), 3);
> Z := Integers();
> m := map<car<Z, Z, Z> -> CoefficientRing(L) | t :-> t[1]*t[2]*t[3]>;
> s := elt<L | m>;
> PrintToPrecision(s, 5);
x*y*z + 2*x^2*y*z + 2*x*y^2*z + 2*x*y*z^2 + 3*x^3*y*z + 4*x^2*y^2*z +
4*x^2*y*z^2 + 3*x*y^3*z + 4*x*y^2*z^2 + 3*x*y*z^3
> Coefficient(s, [1, 1, 1]);
1
> m(<1, 1, 1>);
1
> Coefficient(s, [3, 1, 2]);
6
> m(<3, 1, 2>);
6
All the usual arithmetic operations are possible for lazy series.
The sum of the two lazy series s and t.
The negation of the lazy series s.
The difference between lazy series s and t.
The product of the lazy series s and t.
The sum of the lazy series s and the element r of the coefficient ring of the parent ring of s.
The product of the lazy series s and the element c of the coefficient ring of the parent ring of s.
The product of the lazy series s and the monomial xn, where xn is x1n1 x ... x xrnr where r is the number of variables of the parent ring of s, n is the sequence [n1, ..., nr] and x = x1, ..., xr are the series variables of the parent ring of s.
Given a lazy series s and an integer n, return the nth power of s. It is allowed for n to be negative and inverses will be taken where possible.
> L<a, b, c, d> := LazyPowerSeriesRing(Rationals(), 4); > (a + 4*b + (-c)*[8, 9, 2^30 + 10, 2] - d*b + 5)^-8; Lazy power series
In theory, all the coefficients of most series can be calculated. In practice it is not possible to compute infinitely many. Some problems arise however when a series has been created using multiplication, inversion, evaluation or by taking square roots. For such series the jth coefficient for all j < i must be known for the ith coefficient to be calculated. In such cases the exponents specified for each variable in the monomial whose coefficient is required must be small integers (< 230).
The default ordering of coefficients is by total degree of the corresponding monomials. This is the same order on multivariate polynomials by default. When drawn on paper or imagined in 3 or more dimensions this looks like a spiral if the coordinates representing the monomial exponents are joined. This is the same ordering which is used to compute the valuation of a series.
Once computed, coefficients are stored with the series so any subsequent call to these functions will be faster in non--trivial cases than the first call. It is possible to interrupt any of these functions and return to the prompt.
Returns the coefficient in the univariate lazy series s of xi where x is the series variable of the parent of s and i is a non negative integer.
Returns the coefficient in the multivariate lazy series s of the monomial x1T1 * ... * xrTr where T is the sequence [T1, ..., Tr], x1, ..., xr are the variables of the parent ring of s and r is the rank of the parent of s.
The coefficients of the lazy series s whose monomials have total degree at least l (0 if not given) and at most n where the monomials are ordered using the default "spiral" degree order. The bounds l and n must be non negative integers.
The valuation of the lazy series s. This is the exponent of the monomial with the first non--zero coefficient as returned by Coefficients above. The return value will be either Infty, an integer (univariate case) or a sequence (multivariate case).
Print the sum of all terms of the lazy series s whose degree is no more than n where n is a non negative integer. The series is printed using the "spiral" ordering.
Print the sum of the terms of the lazy series s whose degree is at least l and at most n. The terms are printed using the spiral ordering. The bounds l and n must be non negative integers.
The coefficient in the lazy series s whose monomial exponent is the valuation of s. That is, the first non--zero coefficient of s where the ordering which determines "first" is the "spiral" ordering used by Coefficients and Valuation.
The term in the lazy series s whose monomial exponent is the valuation of s. That is, the first non--zero term of s where the ordering which determines "first" is the "spiral" ordering used by Coefficients and Valuation.
> L<a, b, c, d> := LazyPowerSeriesRing(Rationals(), 4); > s := (1 + 2*a + 3*b + 4*d)^-5;Find the coefficient of a * b * c * d.
> Coefficient(s, [1, 1, 1, 1]); 0Find the coefficients of all monomials with total degree at most 6.
> time Coefficients(s, 6); [ 1, -10, -15, 0, -20, 60, 180, 0, 240, 135, 0, 360, 0, 0, 240, -280, -1260, 0, -1680, -1890, 0, -5040, 0, 0, -3360, -945, 0, -3780, 0, 0, -5040, 0, 0, 0, -2240, 1120, 6720, 0, 8960, 15120, 0, 40320, 0, 0, 26880, 15120, 0, 60480, 0, 0, 80640, 0, 0, 0, 35840, 5670, 0, 30240, 0, 0, 60480, 0, 0, 0, 53760, 0, 0, 0, 0, 17920, -4032, -30240, 0, -40320, -90720, 0, -241920, 0, 0, -161280, -136080, 0, -544320, 0, 0, -725760, 0, 0, 0, -322560, -102060, 0, -544320, 0, 0, -1088640, 0, 0, 0, -967680, 0, 0, 0, 0, -322560, -30618, 0, -204120, 0, 0, -544320, 0, 0, 0, -725760, 0, 0, 0, 0, -483840, 0, 0, 0, 0, 0, -129024, 13440, 120960, 0, 161280, 453600, 0, 1209600, 0, 0, 806400, 907200, 0, 3628800, 0, 0, 4838400, 0, 0, 0, 2150400, 1020600, 0, 5443200, 0, 0, 10886400, 0, 0, 0, 9676800, 0, 0, 0, 0, 3225600, 612360, 0, 4082400, 0, 0, 10886400, 0, 0, 0, 14515200, 0, 0, 0, 0, 9676800, 0, 0, 0, 0, 0, 2580480, 153090, 0, 1224720, 0, 0, 4082400, 0, 0, 0, 7257600, 0, 0, 0, 0, 7257600, 0, 0, 0, 0, 0, 3870720, 0, 0, 0, 0, 0, 0, 860160 ] Time: 0.140 > #$1; 210PrintToPrecision will display the monomials to which these coefficients correspond.
> time PrintToPrecision(s, 6);
1 - 10*a - 15*b - 20*d + 60*a^2 + 180*a*b + 240*a*d + 135*b^2 + 360*b*d +
240*d^2 - 280*a^3 - 1260*a^2*b - 1680*a^2*d - 1890*a*b^2 - 5040*a*b*d -
3360*a*d^2 - 945*b^3 - 3780*b^2*d - 5040*b*d^2 - 2240*d^3 + 1120*a^4 +
6720*a^3*b + 8960*a^3*d + 15120*a^2*b^2 + 40320*a^2*b*d + 26880*a^2*d^2 +
15120*a*b^3 + 60480*a*b^2*d + 80640*a*b*d^2 + 35840*a*d^3 + 5670*b^4 +
30240*b^3*d + 60480*b^2*d^2 + 53760*b*d^3 + 17920*d^4 - 4032*a^5 -
30240*a^4*b - 40320*a^4*d - 90720*a^3*b^2 - 241920*a^3*b*d - 161280*a^3*d^2
- 136080*a^2*b^3 - 544320*a^2*b^2*d - 725760*a^2*b*d^2 - 322560*a^2*d^3 -
102060*a*b^4 - 544320*a*b^3*d - 1088640*a*b^2*d^2 - 967680*a*b*d^3 -
322560*a*d^4 - 30618*b^5 - 204120*b^4*d - 544320*b^3*d^2 - 725760*b^2*d^3 -
483840*b*d^4 - 129024*d^5 + 13440*a^6 + 120960*a^5*b + 161280*a^5*d +
453600*a^4*b^2 + 1209600*a^4*b*d + 806400*a^4*d^2 + 907200*a^3*b^3 +
3628800*a^3*b^2*d + 4838400*a^3*b*d^2 + 2150400*a^3*d^3 + 1020600*a^2*b^4 +
5443200*a^2*b^3*d + 10886400*a^2*b^2*d^2 + 9676800*a^2*b*d^3 +
3225600*a^2*d^4 + 612360*a*b^5 + 4082400*a*b^4*d + 10886400*a*b^3*d^2 +
14515200*a*b^2*d^3 + 9676800*a*b*d^4 + 2580480*a*d^5 + 153090*b^6 +
1224720*b^5*d + 4082400*b^4*d^2 + 7257600*b^3*d^3 + 7257600*b^2*d^4 +
3870720*b*d^5 + 860160*d^6
Time: 0.010
The valuation of s can be obtained as follows. The valuation of zero is
a special case.
> Valuation(s); [ 0, 0, 0, 0 ] > Valuation(s*0); Infinity
Returns the coefficients of the monomials in the lazy series s whose exponents are given by [i1, ..., ir] where each ij ≤nj. The argument n may either be a non negative integer (univariate case) or a sequence of non negative integers of length r where r is the rank of the parent ring of s. The index of the [i1, ..., ir]-th coefficient in the return sequence is given by ∑j=1r ij * ( ∏k=j + 1r (nk + 1) ).
Return the index in the return value of CoefficientsNonSpiral(s, n) of the monomial in the lazy series s whose exponents are given by the (trivial in the univariate case) sequence i.
> L<a, b, c, d> := LazyPowerSeriesRing(Rationals(), 4); > s := (1 + 2*a + 3*b + 4*d)^-5; > time CoefficientsNonSpiral(s, [3, 3, 3, 2]); [ 1, -20, 240, 0, 0, 0, 0, 0, 0, 0, 0, 0, -15, 360, -5040, 0, 0, 0, 0, 0, 0, 0, 0, 0, 135, -3780, 60480, 0, 0, 0, 0, 0, 0, 0, 0, 0, -945, 30240, -544320, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 240, -3360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 180, -5040, 80640, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1890, 60480, -1088640, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15120, -544320, 10886400, 0, 0, 0, 0, 0, 0, 0, 0, 0, 60, -1680, 26880, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1260, 40320, -725760, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15120, -544320, 10886400, 0, 0, 0, 0, 0, 0, 0, 0, 0, -136080, 5443200, -119750400, 0, 0, 0, 0, 0, 0, 0, 0, 0, -280, 8960, -161280, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6720, -241920, 4838400, 0, 0, 0, 0, 0, 0, 0, 0, 0, -90720, 3628800, -79833600, 0, 0, 0, 0, 0, 0, 0, 0, 0, 907200, -39916800, 958003200, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] Time: 0.370 > #$1; 192It appears that the "spiral" algorithm may be faster since it computed more coefficients in less time. This may be because of the operations (inversion) involved in calculating s and that CoefficientsNonSpiral computed coefficients of larger degree monomials than Coefficients. In calculating these coefficients for larger degrees, coefficients which were not asked for may have been calculated. Coefficients will only need to calculate the coefficients which have been asked for --- all intermediate calculations would have been asked for.
Return true if the lazy series s and t are exactly the same series.
Return true if the lazy series s was created as the zero series.
Return true if the lazy series s was created as the one series.
Return true if the lazy series s was created as the minus one series.
Return true if the lazy series s is a unit.
Return true if all the terms of the lazy series s with degree at most n are zero. Calling this function without the second argument returns IsZero(s).
Return true if the terms of the lazy series s and t with degree at most n are the same. Calling this function without the third argument returns s eq t.
Return the (nth) derivative of the lazy series s with respect to the vth variable of the parent ring of s. If v is not given, the parent ring of s must be univariate and the derivative with respect to the unique variable is returned. It is only allowed to give values of v from 1 to the rank of the parent of s and for n to be a positive integer.
Return the integral of the lazy series s with respect to the vth variable of the parent ring of s. If v is not given, the parent ring of s must be univariate and the integral with respect to the unique variable is returned. It is only allowed to give values of v from 1 to the rank of the parent of s.
Return the lazy series s evaluated at the lazy series t or the sequence T of lazy series. The series t or the series in T must have zero constant term so that every coefficient of the result can be finitely calculated.
> R := LazyPowerSeriesRing(Rationals(), 2);
> AssignNames(~R, ["x","y"]);
> m := map<car<Integers(), Integers()> -> Rationals() | t :-> 1>;
> s := elt<R | m>;
> PrintToPrecision(s, 3);
1 + x + y + x^2 + x*y + y^2 + x^3 + x^2*y + x*y^2 + y^3
> R1 := LazyPowerSeriesRing(Rationals(), 1);
> AssignNames(~R1, ["z"]);
> m1 := map<car<Integers()> -> Rationals() | t :-> t[1]>;
> s1 := elt<R1 | m1>;
> PrintToPrecision(s1, 3);
z + 2*z^2 + 3*z^3
> e := Evaluate(s, [s1,s1]);
> PrintToPrecision(e, 10);
1 + 2*z + 7*z^2 + 22*z^3 + 67*z^4 + 200*z^5 + 588*z^6 + 1708*z^7 + 4913*z^8 +
14018*z^9 + 39725*z^10
> Parent(e);
Lazy power series ring in 1 variable over Rational Field
> f := Evaluate(s1, s - 1);
> PrintToPrecision(f, 10);
x + y + 3*x^2 + 5*x*y + 3*y^2 + 8*x^3 + 18*x^2*y + 18*x*y^2 + 8*y^3 + 20*x^4 +
56*x^3*y + 75*x^2*y^2 + 56*x*y^3 + 20*y^4 + 48*x^5 + 160*x^4*y + 264*x^3*y^2
+ 264*x^2*y^3 + 160*x*y^4 + 48*y^5 + 112*x^6 + 432*x^5*y + 840*x^4*y^2 +
1032*x^3*y^3 + 840*x^2*y^4 + 432*x*y^5 + 112*y^6 + 256*x^7 + 1120*x^6*y +
2496*x^5*y^2 + 3600*x^4*y^3 + 3600*x^3*y^4 + 2496*x^2*y^5 + 1120*x*y^6 +
256*y^7 + 576*x^8 + 2816*x^7*y + 7056*x^6*y^2 + 11616*x^5*y^3 +
13620*x^4*y^4 + 11616*x^3*y^5 + 7056*x^2*y^6 + 2816*x*y^7 + 576*y^8 +
1280*x^9 + 6912*x^8*y + 19200*x^7*y^2 + 35392*x^6*y^3 + 47280*x^5*y^4 +
47280*x^4*y^5 + 35392*x^3*y^6 + 19200*x^2*y^7 + 6912*x*y^8 + 1280*y^9 +
2816*x^10 + 16640*x^9*y + 50688*x^8*y^2 + 103168*x^7*y^3 + 154000*x^6*y^4 +
175344*x^5*y^5 + 154000*x^4*y^6 + 103168*x^3*y^7 + 50688*x^2*y^8 +
16640*x*y^9 + 2816*y^10
> f := Evaluate(s1 + 1, s - 1);
> PrintToPrecision(f, 10);
1 + x + y + 3*x^2 + 5*x*y + 3*y^2 + 8*x^3 + 18*x^2*y + 18*x*y^2 + 8*y^3 + 20*x^4
+ 56*x^3*y + 75*x^2*y^2 + 56*x*y^3 + 20*y^4 + 48*x^5 + 160*x^4*y +
264*x^3*y^2 + 264*x^2*y^3 + 160*x*y^4 + 48*y^5 + 112*x^6 + 432*x^5*y +
840*x^4*y^2 + 1032*x^3*y^3 + 840*x^2*y^4 + 432*x*y^5 + 112*y^6 + 256*x^7 +
1120*x^6*y + 2496*x^5*y^2 + 3600*x^4*y^3 + 3600*x^3*y^4 + 2496*x^2*y^5 +
1120*x*y^6 + 256*y^7 + 576*x^8 + 2816*x^7*y + 7056*x^6*y^2 + 11616*x^5*y^3 +
13620*x^4*y^4 + 11616*x^3*y^5 + 7056*x^2*y^6 + 2816*x*y^7 + 576*y^8 +
1280*x^9 + 6912*x^8*y + 19200*x^7*y^2 + 35392*x^6*y^3 + 47280*x^5*y^4 +
47280*x^4*y^5 + 35392*x^3*y^6 + 19200*x^2*y^7 + 6912*x*y^8 + 1280*y^9 +
2816*x^10 + 16640*x^9*y + 50688*x^8*y^2 + 103168*x^7*y^3 + 154000*x^6*y^4 +
175344*x^5*y^5 + 154000*x^4*y^6 + 103168*x^3*y^7 + 50688*x^2*y^8 +
16640*x*y^9 + 2816*y^10
The square root of the lazy series s.
Return true if the lazy series s is a square and the square root if so.
Given a series s in a lazy series ring whose coefficient ring is a polynomial ring (either univariate or multivariate), consider the polynomial which would be formed if this series was written as a polynomial with series as the coefficients. For i a non negative integer and the coefficient ring of the series ring a univariate polynomial ring this function returns the series which is the ith coefficient of the polynomial resulting from the rewriting of the series.When the coefficient ring of the parent of s is a multivariate polynomial ring i should be a sequence of non negative integers of length the rank of the coefficient ring. The series returned will be the series which is the coefficient of x1i1 * ... * xrir in the rewritten series where r is the rank of the coefficient ring and xj are the indeterminates of the coefficient ring.