Add method for evaluating a TaylorN on a Vector{TaylorN}

[PerezHz]

While working on PR #118 (see also discussion in #116), I was thinking that perhaps it'd be nice to have defined evaluate{T<:NumberNotSeries}(a::TaylorN{T}, vals::Array{TaylorN{T},1}), or something similar, in order to be able to compose Taylor expansions of N-variable functions. There is already a similar method for Taylor1, which allows to evaluate a Taylor1 on a Taylor1.

What I'm trying to say (using syntax from #118) is that whereas it is possible to compose two Taylor1s,

julia> using TaylorSeries

julia> t = Taylor1(25)
 1.0 t + 𝒪(t²⁶)

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 + 2.811457254345521e-15 t¹⁷ - 8.220635246624331e-18 t¹⁹ + 1.9572941063391263e-20 t²¹ - 3.868170170630684e-23 t²³ + 6.446950284384474e-26 t²⁵ + 𝒪(t²⁶)

julia> q = cos(t)
 1.0 - 0.5 t² + 0.041666666666666664 t⁴ - 0.001388888888888889 t⁶ + 2.48015873015873e-5 t⁸ - 2.7557319223985894e-7 t¹⁰ + 2.08767569878681e-9 t¹² - 1.1470745597729726e-11 t¹⁴ + 4.779477332387386e-14 t¹⁶ - 1.5619206968586228e-16 t¹⁸ + 4.1103176233121653e-19 t²⁰ - 8.896791392450574e-22 t²² + 1.6117375710961184e-24 t²⁴ + 𝒪(t²⁶)

julia> p(q) #evaluate a Taylor1 on a Taylor1; compare with: sc = x->sin(cos(x)); sc(t)
 0.8414709848078965 - 0.2701511529340699 t² - 0.08267127702314792 t⁴ + 0.028036523686489453 t⁶ - 0.0019241418790800983 t⁸ - 0.0004838563431246892 t¹⁰ + 0.00013040017931079692 t¹² - 1.210231179202225e-5 t¹⁴ - 3.316644672471871e-7 t¹⁶ + 2.429814411744226e-7 t¹⁸ - 3.288747013801372e-8 t²⁰ + 1.8233411897292427e-9 t²² + 1.2200826998513497e-10 t²⁴ + 𝒪(t²⁶)

julia> [p, q]([q, p]) #evaluate an array of Taylor1s on an array of Taylor1s
2-element Array{TaylorSeries.Taylor1{Float64},1}:
  0.8414709848078965 - 0.2701511529340699 t² - 0.08267127702314792 t⁴ + 0.028036523686489453 t⁶ - 0.0019241418790800983 t⁸ - 0.0004838563431246892 t¹⁰ + 0.00013040017931079692 t¹² - 1.210231179202225e-5 t¹⁴ - 3.316644672471871e-7 t¹⁶ + 2.429814411744226e-7 t¹⁸ - 3.288747013801372e-8 t²⁰ + 1.8233411897292427e-9 t²² + 1.2200826998513497e-10 t²⁴ + 𝒪(t²⁶)
                                    1.0 - 0.5 t² + 0.20833333333333331 t⁴ - 0.05138888888888889 t⁶ + 0.011334325396825395 t⁸ - 0.0022511574074074074 t¹⁰ + 0.00039391308922558923 t¹² - 6.30631883732082e-5 t¹⁴ + 9.459548466503244e-6 t¹⁶ - 1.3341203587285381e-6 t¹⁸ + 1.777225182337089e-7 t²⁰ - 2.2544681245974032e-8 t²² + 2.739783349785652e-9 t²⁴ + 𝒪(t²⁶)

analogous things cannot currently be done for TaylorNs:

julia> using TaylorSeries

julia> dx = set_variables("x", numvars=4, order=10);

julia> v = [1.0,2,3,4];

julia> dx[1](v) #TaylorN evaluated at a Vector{Float64}
1.0

julia> dx(v) #Vector{TaylorN} evaluated at a Vector{Float64}
4-element Array{Float64,1}:
 1.0
 2.0
 3.0
 4.0

julia> dx[1](dx)#TaylorN evaluated at a Vector{TaylorN}
ERROR: MethodError: no method matching evaluate(::TaylorSeries.TaylorN{Float64}, ::Array{TaylorSeries.TaylorN{Float64},1})
Closest candidates are:
  evaluate(::TaylorSeries.TaylorN{T<:Number}) where T<:Number at /Users/Jorge/forks/TaylorSeries.jl/src/evaluate.jl:211
  evaluate(::TaylorSeries.TaylorN{T<:Number}, ::Array{S<:Union{Complex, Real},1}) where {T<:Number, S<:Union{Complex, Real}} at /Users/Jorge/forks/TaylorSeries.jl/src/evaluate.jl:164
  evaluate(::TaylorSeries.TaylorN{T<:Number}, ::Array{TaylorSeries.Taylor1{S<:Union{Complex, Real}},1}) where {T<:Number, S<:Union{Complex, Real}} at /Users/Jorge/forks/TaylorSeries.jl/src/evaluate.jl:188
  ...
Stacktrace:
 [1] (::TaylorSeries.TaylorN{Float64})(::Array{TaylorSeries.TaylorN{Float64},1}) at /Users/Jorge/forks/TaylorSeries.jl/src/evaluate.jl:229

julia> dx(dx) #Vector{TaylorN} evaluated at a Vector{TaylorN}
ERROR: MethodError: no method matching evaluate(::Array{TaylorSeries.TaylorN{Float64},1}, ::Array{TaylorSeries.TaylorN{Float64},1})
Closest candidates are:
  evaluate(::Array{TaylorSeries.TaylorN{T<:Union{Complex, Real}},1}, ::Array{S<:Union{Complex, Real},1}) where {T<:Union{Complex, Real}, S<:Union{Complex, Real}} at /Users/Jorge/forks/TaylorSeries.jl/src/evaluate.jl:220
  evaluate(::Array{TaylorSeries.TaylorN{T<:Number},1}) where T<:Number at /Users/Jorge/forks/TaylorSeries.jl/src/evaluate.jl:226
  evaluate(::Array{TaylorSeries.TaylorN{T<:Number},1}, ::Array{T<:Number,1}) where T<:Number at /Users/Jorge/forks/TaylorSeries.jl/src/evaluate.jl:214
Stacktrace:
 [1] (::Array{TaylorSeries.TaylorN{Float64},1})(::Array{TaylorSeries.TaylorN{Float64},1}) at /Users/Jorge/forks/TaylorSeries.jl/src/evaluate.jl:234

I have some ideas on this that actually I was already testing while working on #118, so if there's interest I can submit a PR!

[lbenet]

I though also about this some time ago, and while I think it would be nice, it is quite tricky because the way things are currently. But, please, go ahead and submit a PR.

[blas-ko]

Also thought of this! This would also be useful if we address the many variable update! functionality discussed in #80 and #121.

[lbenet]

Solved by #121. Closing.