Allocation-free 5-arg mul! for custom maps

[jagot]

This is basically a follow-up to #56;

If I have a custom type for which I have a defined a allocation-free 5-arg mul!

struct MyIntegralOperator{T} <: LinearMap{T}
...
end

mul!(y, A::MyIntegralOperator, x, α=true, β=false) = ...

which I want to form linear combinations of, how do I get allocation-free application? Right now, I think I need to duplicate the following functions, since it is not possible for types outside of LinearMaps.jl to belong to FreeMap:

https://github.com/Jutho/LinearMaps.jl/blob/95cc89cffff4b980af41aea32483af739a2d168c/src/linearcombination.jl#L83-L93

and

https://github.com/Jutho/LinearMaps.jl/blob/95cc89cffff4b980af41aea32483af739a2d168c/src/linearcombination.jl#L113-L138

A possible approach could be to tag the LinearCombination object with a trait, i.e. a type parameter (e.g. MulStyle) that indicates which method to dispatch to. Then the user would only need to overload mulstyle(::MyIntegralOperator) = FiveArgs. This pattern has been used to great effect in e.g. https://github.com/JuliaArrays/LazyArrays.jl/blob/db8b20746c26873d635df18bcaac06382def11bb/src/linalg/muladd.jl

[dkarrasch]

That sounds like a good idea. So, basically, at construction of a LinearCombination object, check whether all summands have the FiveArgs mulstyle, and in that case assign that mulstyle to the LinearCombinations object, right?

[jagot]

Yes, although in practice, I would normally construct a LinearCombination by successively adding new terms, so then only the MulStyle of the new term need be compared with that of the LinearCombination.

[dkarrasch]

@jagot I implemented your suggestion and I'd be happy if you could review it.

I realized that this is a very neat way of perhaps avoiding this nasty VERSION distinction (almost) altogether. The point is that depending on whether old or new mul! is available, matrices will get the FiveArg or ThreeArg trait. But otherwise, the dispatch and the underlying multiplication code for LinearCombinations are just the same.