Implement an expand function?

[blas-ko]

I was thinking of implementing a function so it expands an independent variable of a given order around a function f? This could be something like

function expand(f::Function,order::Int64=15)
    a = Taylor1(order)
    return f(a)
end

I don't know if it's very efficient because the function doesn't know a priori the output's type, but I think it's practical for TaylorSeries users.

[PerezHz]

Sounds interesting!

julia> import Base.expand

julia> function expand(f::Function,order::Int64=15)
           a = Taylor1(order)
           return f(a)
       end
expand (generic function with 3 methods)

julia> expand(sin) #it works!
 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(t¹⁶)

julia> expand(λ->sin(cos(λ))) #it works for compositions of functions, too!
 0.8414709848078965 - 0.2701511529340699 t² - 0.08267127702314792 t⁴ + 0.028036523686489446 t⁶ - 0.0019241418790800977 t⁸ - 0.0004838563431246893 t¹⁰ + 0.00013040017931079692 t¹² - 1.2102311792022246e-5 t¹⁴ + 𝒪(t¹⁶)

julia> x = Taylor1(15)
 1.0 t + 𝒪(t¹⁶)

julia> sin(cos(x))
 0.8414709848078965 - 0.2701511529340699 t² - 0.08267127702314792 t⁴ + 0.028036523686489446 t⁶ - 0.0019241418790800977 t⁸ - 0.0004838563431246893 t¹⁰ + 0.00013040017931079692 t¹² - 1.2102311792022246e-5 t¹⁴ + 𝒪(t¹⁶)

julia> sin(cos(x)) == expand(λ->sin(cos(λ)))
true

Perhaps it would be helpful to be able to choose the initial point of expansion? Otherwise, it will always expand around zero. I mean something like:

julia> using TaylorSeries

julia> import Base.expand

julia> function expand{T<:Number}(f::Function, x0::T, order::Int64=15) #a Taylor expansion around x0
           a = Taylor1([x0, one(x0)], order)
           return f(a)
       end
expand (generic function with 3 methods)

julia> expand(λ->sin(cos(λ)), -1000.0)
 0.5332003773089166 + 0.6995309807177265 t - 0.4201657380038325 t² - 0.07232916974242763 t³ + 0.15121493557921284 t⁴ - 0.02436406884321303 t⁵ - 0.04058717887907187 t⁶ + 0.011469509530976701 t⁷ + 0.006593656037068762 t⁸ - 0.0032816768978896783 t⁹ - 0.000619734660543075 t¹⁰ + 0.0007119205288158886 t¹¹ + 2.058485176905429e-6 t¹² - 0.00011987116840574068 t¹³ + 1.4786492212018731e-5 t¹⁴ + 1.6049672402596366e-5 t¹⁵ + 𝒪(t¹⁶)

julia> sin(cos(-1000.0)) #The 0-th order Taylor coeff must be equal to the function evaluated at the point of expansion
0.5332003773089166

julia> x = Taylor1([-1000.0, 1.0], 15)
 - 1000.0 + 1.0 t + 𝒪(t¹⁶)

julia> sin(cos(x))
 0.5332003773089166 + 0.6995309807177265 t - 0.4201657380038325 t² - 0.07232916974242763 t³ + 0.15121493557921284 t⁴ - 0.02436406884321303 t⁵ - 0.04058717887907187 t⁶ + 0.011469509530976701 t⁷ + 0.006593656037068762 t⁸ - 0.0032816768978896783 t⁹ - 0.000619734660543075 t¹⁰ + 0.0007119205288158886 t¹¹ + 2.058485176905429e-6 t¹² - 0.00011987116840574068 t¹³ + 1.4786492212018731e-5 t¹⁴ + 1.6049672402596366e-5 t¹⁵ + 𝒪(t¹⁶)

julia> sin(cos(x)) == expand(λ->sin(cos(λ)), -1000.0) #test whether what we're doing is consistent
true

I really like the idea! (I did this on julia 0.5.0)

[lbenet]

What about defining it as a macro?

While I like the idea, I'm not sure whether expand or @expand are the best names; my point is that expand exists in Base, and actually has a quite different functionality.

@dpsanders What do you think?

[blas-ko]

Perhaps it would be helpful to be able to choose the initial point of expansion?

Sounds great, @PerezHz 👍
Maybe, so it doesn't clash with Base's expand, we can define it as expand_taylor or @expand_taylor. By the way, @lbenet, what would be the advantage of using a macro here instead of a normal function?

[lbenet]

Sorry for replying this late... The advantage would be that the replacement would be done at parse time.

[lbenet]

Implemented in #121. Closing!