Modified CG to use axpy!/axpby! directly instead of broadcasting

[vlc1]

When using IterativeSolvers.jl on custom vectors and matrices, I find it easier to add methods to generic BLAS routines (scal!, mul!, axpy!...) than customizing the broadcasting machinery on said types.

This is a very small modification that simply requires changing lines like

it.r .-= alpha .* it.c

to

axpy!(-alpha, it.c, it.r)

and so on as illustrated in the cg.jl file in this PR.

If this is a modification that is acceptable, I'll work my way through the other solvers (bicgstabl...).

[coveralls]

Pull Request Test Coverage Report for Build 4446585905

• 7 of 7 (100.0%) changed or added relevant lines in 1 file are covered.
• No unchanged relevant lines lost coverage.
• Overall coverage decreased (-0.0%) to 97.331%

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Totals
Change from base Build 2991758705:	-0.0%
Covered Lines:	1933
Relevant Lines:	1986

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💛 - Coveralls

[haampie]

I believe we've gone back and forth on this. Long ago there was overhead to broadcasting, so the BLAS routines were called. Then Julia improved and folks wanted broadcasting because it's more generic. I think I'm happy with the current broadcasting stuff, also because for example CG would need an axpby somewhere, which I think is not standard BLAS.

[dkarrasch]

These functions are now owned by LinearAlgebra (and not LinearAlgebra.BLAS) and admit generic fallbacks, likely using broadcasting but I don't remember. Another option could be to introduce _axp[b]y!, that use broadcasting generically but may be overloaded with a custom axpy!-like implementation.

[vlc1]

These functions are now owned by LinearAlgebra (and not LinearAlgebra.BLAS) and admit generic fallbacks, likely using broadcasting but I don't remember. Another option could be to introduce _axp[b]y!, that use broadcasting generically but may be overloaded with a custom axpy!-like implementation.

Fair enough, I might have to dig into how to customize broadcasting for my own datatypes...

These functions are now owned by LinearAlgebra (and not LinearAlgebra.BLAS) and admit generic fallbacks, likely using broadcasting but I don't remember. Another option could be to introduce _axp[b]y!, that use broadcasting generically but may be overloaded with a custom axpy!-like implementation.

That's pretty much what Krylov.jl does with some macros (@kaxpy...).