Use 5-arg mul! for matrices

[dkarrasch]

Closes #56.

• introduce MatrixMap{T} <: WrappedMap{T}
• direct 5-arg mul! for MatrixMaps to LinearAlgebra's 5-arg mul!
• make MatrixMap invariant under adjoint/transpose
• use this, if possible, in linear combinations

[dkarrasch]

I tested this on a BlockMap, in which subsequent matvec-mul results are added to previous results. The only allocations are due to views, in contrast to earlier Julia/LinearMaps versions.

[coveralls]

Coverage decreased (-1.7%) to 94.684% when pulling b22a8b8 on going-5argmul into 3d2e53f on master.

[codecov]

Codecov Report

Merging #68 into master will decrease coverage by 3.72%.
The diff coverage is 93.33%.

@@            Coverage Diff            @@
##           master     #68      +/-   ##
=========================================
- Coverage   98.62%   94.9%   -3.73%     
=========================================
  Files           8       8              
  Lines         435     451      +16     
=========================================
- Hits          429     428       -1     
- Misses          6      23      +17

Impacted Files	Coverage Δ
src/wrappedmap.jl	100% <100%> (ø)	⬆️
src/linearcombination.jl	95.34% <90.9%> (-4.66%)	⬇️
src/transpose.jl	44.44% <0%> (-55.56%)	⬇️

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[codecov]

Codecov Report

Merging #68 into master will decrease coverage by 0.94%.
The diff coverage is 97.29%.

@@            Coverage Diff             @@
##           master      #68      +/-   ##
==========================================
- Coverage   98.62%   97.67%   -0.95%     
==========================================
  Files           8        8              
  Lines         435      473      +38     
==========================================
+ Hits          429      462      +33     
- Misses          6       11       +5

Impacted Files	Coverage Δ
src/LinearMaps.jl	100% <ø> (ø)	⬆️
src/wrappedmap.jl	100% <100%> (ø)	⬆️
src/linearcombination.jl	96.92% <96.96%> (-3.08%)	⬇️
src/transpose.jl	88.88% <0%> (-11.12%)	⬇️

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[dkarrasch]

Alright, this is tested really hard for zero allocations, for the case when the map is a wrapped map of some AbstractMatrix (called MatrixMap, not a type, just an abbreviation), a wrapped map of a uniformscaling object (called UniformScalingMap, which is going to be exported after #61 is merged), BTW these are joined as FreeMaps, and linear combinations of FreeMaps. In BlockMaps, any block can be the aforementioned FreeMap or LinearCombination of FreeMaps, and the only allocations are a few bytes for the views. Check out the newly added tests to get a flavor of what is allocation-free now.

@chriscoey Does this cover most of your applications? If not, I'd be interested to learn about the uncovered ones.

[chriscoey]

@dkarrasch thank you, you're brilliant! I'll let you know in the next couple months as we run benchmarks of our as-yet-unreleased optimization solver that calls methods from IterativeSolvers

[chriscoey]

Is this one ready for merge?

[dkarrasch]

I think it is, but I would love to have @Jutho's eyes on it, just in case.

[Jutho]

My apologies for the delay, I'll try to review before the end of the week.

[Jutho]

Looks great and ready to merge (or almost if you consider my small comments).

One final question, because I have lost track a bit: Are we at this point still using one(T) and zero(T) as default values of alpha and beta in the other 5-arg mul methods, or did we switch to true and false everywhere?

[Jutho]

One trick I learned recently is to add, in the body of a loop, a statement like

if !(@isdefined z)
     z = similar(y)
end

which would then only allocate z once the first map which is not a FreeMap is encountered, and thus not allocate at all if all the maps are FreeMap

[dkarrasch]

Thanks, @Jutho! Things are a bit spread over the place, but the idea is to have true and false as defaults for alpha and beta everywhere. As for the other mul! methods, this is done in #66 (I hope, but will make sure).

[Jutho]

Yes indeed; I should have remembered. Anyway, I think this is good to merge, aside from maybe the == method for MatrixMap. And please remind me of any open PRs which are ready for review (and feel free to keep reminding me if there is no reaction), because I have not followed closely.