Add a wrapper for Krylov.jl and the block-GMRES implementation in ExaPF.jl

[amontoison]

François, I interfaced block-GMRES today and I observed that the linear systems with one right-hand side are inconsistent numerically.
This issue is triggered here.
I observed this phenomenon with the problem case300m.

[frapac]

I just ran the tests locally. The results are interesting:

• the tests are passing for case118.m
• case300.m is breaking because of GMRES, as you observed. I noticed we are not using any preconditioner, would that solve the issue? Or what do you mean exactly by "numerically inconsistent"?
• case2869pegase.m returns NaN in the 5th call to block-gmres.
• case1354pegase.m breaks with the following message:

CK-GMRES: system of size 2447 with 256 right-hand sides
 pass      k     ‖Rₖ‖  timer
    0      0  1.6e+01  0.00s
    1      1  8.2e+00  0.06s
    1      2  5.1e+00  0.11s
    1      3  3.6e+00  0.18s
    1      4  2.7e+00  0.27s
    1      5  2.0e+00  0.37s
    1      6  1.4e+00  0.49s
    1      7  1.1e+00  0.62s
    1      8  8.7e-01  0.76s
    1      9  5.9e-01  0.92s
    1     10  2.1e-01  1.09s
    1     11  7.6e-13  1.28s
ERROR: LAPACKException(5)
Stacktrace:
  [1] chklapackerror
    @ ~/.julia/juliaup/julia-1.9.3+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/lapack.jl:40 [inlined]
  [2] trtrs!(uplo::Char, trans::Char, diag::Char, A::Matrix{Float64}, B::Matrix{Float64})
    @ LinearAlgebra.LAPACK ~/.julia/juliaup/julia-1.9.3+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/lapack.jl:3424
  [3] ldiv!
    @ ~/.julia/juliaup/julia-1.9.3+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/triangular.jl:727 [inlined]
  [4] block_gmres!(solver::ExaPF.LinearSolvers.BlockGmresSolver{Float64, Float64, Vector{Float64}, Matrix{Float64}}, A::SparseArrays.SparseMatrixCSC{Float64, Int64}, B::Matrix{Float64}; M::UniformScaling{Bool}, N::UniformScaling{Bool}, ldiv::Bool, restart::Bool, reorthogonalization::Bool, atol::Float64, rtol::Float64, itmax::Int64, timemax::Float64, verbose::Int64, history::Bool)
    @ ExaPF.LinearSolvers ~/.julia/packages/ExaPF/63axT/src/LinearSolvers/block_gmres.jl:311
  [5] block_gmres!
    @ ~/.julia/packages/ExaPF/63axT/src/LinearSolvers/block_gmres.jl:107 [inlined]
  [6] ldiv!
    @ ~/dev/Argos.jl/src/krylov.jl:32 [inlined]
  [7] mul!
    @ ~/dev/Argos.jl/src/reduction.jl:25 [inlined]
  [8] adjoint_adjoint_reduction!(red::Argos.BatchReduction{Matrix{Float64}, Argos.ImplicitSensitivity{Argos.KrylovWrapper{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, ExaPF.LinearSolvers.BlockGmresSolver{Float64, Float64, Vector{Float64}, Matrix{Float64}}, Krylov.GmresSolver{Float64, Float64, Vector{Float64}}}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, hessvec::SubArray{Float64, 2, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, UnitRange{Int64}}, true}, H::Argos.HJDJ{Vector{Int64}, Vector{Float64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, w::Matrix{Float64})
    @ Argos ~/dev/Argos.jl/src/reduction.jl:136
  [9] reduce!(red::Argos.BatchReduction{Matrix{Float64}, Argos.ImplicitSensitivity{Argos.KrylovWrapper{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, ExaPF.LinearSolvers.BlockGmresSolver{Float64, Float64, Vector{Float64}, Matrix{Float64}}, Krylov.GmresSolver{Float64, Float64, Vector{Float64}}}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, dest::Matrix{Float64}, W::Argos.HJDJ{Vector{Int64}, Vector{Float64}, SparseArrays.SparseMatrixCSC{Float64, Int64}})
    @ Argos ~/dev/Argos.jl/src/reduction.jl:210
 [10] macro expansion
    @ ~/dev/Argos.jl/src/KKT/reduced_newton.jl:345 [inlined]
 [11] macro expansion
    @ ./timing.jl:393 [inlined]
 [12] build_kkt!(kkt::Argos.BieglerKKTSystem{Float64, Vector{Int64}, Vector{Float64}, Matrix{Float64}, SparseArrays.SparseMatrixCSC{Float64, Int64}, Argos.KrylovWrapper{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, ExaPF.LinearSolvers.BlockGmresSolver{Float64, Float64, Vector{Float64}, Matrix{Float64}}, Krylov.GmresSolver{Float64, Float64, Vector{Float64}}}})
    @ Argos ~/dev/Argos.jl/src/KKT/reduced_newton.jl:344
 [13] test_biegler_kkt(casename::String)
    @ Main ~/dev/Argos.jl/test/kkt.jl:40
 [14] top-level scope
    @ REPL[14]:1