Dear all, in the past few months, Travis and I have worked on a reimplementation of lazy formal power series. With https://trac.sagemath.org/ticket/32367 its design and implementation is now well-developed, and powerful enough so that I'd like to briefly present it here.
If you notice any shortcomings or have further feature requests, I am happy to hear of them! Note that there are several other tickets on the same topic (all with keyword "LazyPowerSeries"), which are not yet merged, but almost ready. What is it all about? For us, a 'lazy' series ring is an 'exact' implementation of the concept of formal power series. The adjective 'lazy' means that a coefficient is only computed on demand, whereas 'exact' means that you do not have to worry about precision (up to the exactness of the base ring) - in principle, all the coefficients can be accessed. By default, the first few coefficients will be printed, and the number of coefficients printed can be set by a global option. We have implemented several such rings: sage: L.<z> = LazyLaurentSeriesRing(QQ) sage: P.<q,t> = LazyPowerSeriesRing(QQ) sage: D = LazyDirichletSeriesRing(QQ, "s") sage: s = SymmetricFunctions(QQ).s(); S = LazySymmetricFunctions(s) sage: nc = NonCommutativeSymmetricFunctions(QQ).Complete(); NC = nc.formal_series_ring() # or anything in GradedAlgebrasWithBasis ---- Please note that, in particular, the LazyPowerSeriesRing in sage/combinat/species/series.py and the LazyLaurentSeriesRing in sage/rings/lazy_laurent_series.py have been replaced. ---- For all of these, all the operations you might expect should be available. For example: sage: sin(z)^2 + cos(z)^2 1 + O(z^7) sage: (1/sinh(z) + 1/cosh(z))*exp(-z) z^-1 - 2/3*z + 14/45*z^3 - 124/945*z^5 + O(z^6) sage: f = L(lambda n: Partitions(n).cardinality(), valuation=0) sage: f 1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 7*z^5 + 11*z^6 + O(z^7) sage: ~f # the Euler function (see ticket #34381) 1 - z - z^2 + z^5 + O(z^7) We can efficiently compute with multivariate power series: sage: exp(q*log(1/(1-t))) 1 + q*t + 1/2*q*t^2 + (1/2*q^2*t^2+1/3*q*t^3) + (1/2*q^2*t^3+1/4*q*t^4) + (1/6*q^3*t^3+11/24*q^2*t^4+1/5*q*t^5) + O(q,t)^7 Dirichlet series look and behave quite differently, but the machinery behind is very similar: sage: D(moebius)^-1 1 + 1/(2^s) + 1/(3^s) + 1/(4^s) + 1/(5^s) + 1/(6^s) + 1/(7^s) + O(1/(8^s)) Finally, we can compute with symmetric functions of unbounded degree. In particular, plethysms are supported: sage: H = S(lambda n: s[n]) sage: H(H-1) s[] + s[1] + 2*s[2] + (s[2,1]+3*s[3]) + (2*s[2,2]+2*s[3,1]+5*s[4]) + (s[2,2,1]+4*s[3,2]+5*s[4,1]+7*s[5]) + (2*s[2,2,2]+2*s[3,2,1]+2*s[3,3]+s[4,1,1]+10*s[4,2]+8*s[5,1]+11*s[6]) + O^7 One important feature, which works for all of these, is the possibility to define series via implicit equations: sage: b = P.undefined(valuation=1) sage: b.define(q + t/(1-b)) sage: b[5] q^4*t + 10*q^3*t^2 + 30*q^2*t^3 + 35*q*t^4 + 14*t^5 This feature also works with several series that depend on each other: sage: f = L.undefined(0) sage: g = L.undefined(0) sage: f.define(1 + z*g^3) sage: g.define(1 + z*f^2) sage: f 1 + z + 3*z^2 + 9*z^3 + 34*z^4 + 132*z^5 + 546*z^6 + O(z^7) sage: g 1 + z + 2*z^2 + 7*z^3 + 24*z^4 + 95*z^5 + 386*z^6 + O(z^7) Again, comments are very welcome! Best wishes, Martin -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/071fb7d5-0865-4778-bfa5-ed17f204609an%40googlegroups.com.
