Change default solver to QR

[termi-official]

Supersedes #121

[termi-official]

@ChrisRackauckas is this good enough? The BackslashSolver from Claude in 121 has some type inference issues.

[ChrisRackauckas]

make this a column-pivoted QR, so it handles singularities?

[termi-official]

I am pretty sure SPQR already handles these faithfully:

julia> A = sprand(4,4,0.1)
4×4 SparseMatrixCSC{Float64, Int64} with 3 stored entries:
  ⋅         ⋅         ⋅          ⋅ 
 0.248639  0.469341   ⋅          ⋅ 
  ⋅         ⋅         ⋅          ⋅ 
  ⋅         ⋅        0.0588856   ⋅ 

julia> f = qr(A)
SparseArrays.SPQR.QRSparse{Float64, Int64}
Q factor:
4×4 SparseArrays.SPQR.QRSparseQ{Float64, Int64}
R factor:
4×4 SparseMatrixCSC{Float64, Int64} with 3 stored entries:
 0.248639   ⋅         0.469341   ⋅ 
  ⋅        0.0588856   ⋅         ⋅ 
  ⋅         ⋅          ⋅         ⋅ 
  ⋅         ⋅          ⋅         ⋅ 
Row permutation:
4-element Vector{Int64}:
 2
 4
 3
 1
Column permutation:
4-element Vector{Int64}:
 1
 3
 2
 4

Or do you have something else in mind? Note that if we start converting the coarse matrix from sparse to dense (as done in the pinv solver), then we again run into memory issues.

[ChrisRackauckas]

I dug into this topic quite a bit, and yes SPQR does work. There's now also https://github.com/SciML/SparseColumnPivotedQR.jl now, but SPQR is likely better for the matrices that people are using an AMG for.

[termi-official]

Thanks for taking time here! I tried to benchmark everything to look out for regressions, but ran into some trouble with the new OrdinaryDiffEq release (SciML/SciMLBenchmarks.jl#1601).

FYI if the coarse solver is accessible via LinearSolve, then it can be used via coarse_solver = AMG.LinearSolveWrapper(myalg) already since AMG v1.0 or so.