Add function-like behavior for Taylor1, HomogeneousPolynomial and TaylorN?

[PerezHz]

Hi! I was reading the Function-like objects section of the Julia language documentation, and also @dpsanders's recent blog post, and thought that perhaps it would be nice to add function-like behavior for Taylor1, TaylorN and HomogeneousPolynomial variables? My idea is to implement something like:

julia> using TaylorSeries

julia> t = Taylor1(16)
 1.0 t + 𝒪(t¹⁷)

julia> p = sin(t)
 1.0 t - 0.16666666666666666 t³ + 0.008333333333333333 t⁵ - 0.0001984126984126984 t⁷ + 2.7557319223985893e-6 t⁹ - 2.505210838544172e-8 t¹¹ + 1.6059043836821616e-10 t¹³ - 7.647163731819817e-13 t¹⁵ + 𝒪(t¹⁷)

julia> p(0.1)
ERROR: MethodError: objects of type TaylorSeries.Taylor1{Float64} are not callable

julia> (p::Taylor1)(x) = evaluate(p, x)

julia> p(0.1)
0.09983341664682815

julia> sin(0.1) #just comparing last answer to the actual result of evaluating sin at 0.1
0.09983341664682815

And similar stuff for HomogeneousPolynomials and TaylorNs... If you think this is a worthwhile feature to add to TaylorSeries, I'd be more than happy to submit a PR! 😄

[lbenet]

It sounds as it could be interesting. Any comments or ideas, @dpsanders ?

[dpsanders]

Seems like a good idea! (Is it common to evaluate Taylor series like this?)

[PerezHz]

Thanks for the feedback! Well I guess it wouldn't hurt to be able to do p(0.1) as a shorter (and perhaps more legible) alternative to evaluate(p, 0.1), where p is a Taylor1 (and equivalent things for TaylorNs). Moreover, mathematically I think it does make sense to think about p(0.1) as the value of a polynomial function p at 0.1 (is that the question you're asking @dpsanders or did I misunderstand you?)

This notation is helpful, for example, when working with Legendre polynomials. Using this trick, one would be able to do things such as:

julia> using TaylorSeries

julia> t = Taylor1(16) #the independent variable
 1.0 t + 𝒪(t¹⁷)

julia> P_2 = (5t^3-3t)/2 #2nd-degree Legendre polynomial
 - 1.5 t + 2.5 t³ + 𝒪(t¹⁷)

julia> (p::Taylor1)(x) = evaluate(p, x) #add function-like behavior for Taylor1s

julia> P_2
 - 1.5 t + 2.5 t³ + 𝒪(t¹⁷)

julia> P_2(-1.0) #what is the value of P_2 at -1.0?
-1.0

julia> P_2(cos(t)) #what is the Taylor expansion of P_2(cos(t)), up to 16th order?
 1.0 - 3.0 t² + 2.125 t⁴ - 0.6333333333333333 t⁶ + 0.10171130952380952 t⁸ - 0.010170304232804232 t¹⁰ + 0.0006934235710277377 t¹² - 3.429014217704694e-5 t¹⁴ + 1.2858801882549402e-6 t¹⁶ + 𝒪(t¹⁷)

Thus, we get a nice, legible interface for evaluating polynomials, thanks to the design of evaluate

[blas-ko]

I like the idea too! I think it becomes much more clear! It's also natural to extend the definition into TaylorN polynomials

using TaylorSeries

julia> xN = TaylorN(2) + TaylorN(1)
1.0 x₁ + 1.0 x₂ + 𝒪(‖x‖⁷)

julia> (p::TaylorN)(x) = evaluate(p,x)

julia> xN([1,2])
3.0

[lbenet]

#118 is merged, so I am closing this.