fix docs for CSR

[oscardssmith]

With

A_gpu = CuSparseMatrixCSR(sprand(T, n, n, 0.05) + I)
b_cpu = CuVector(rand(T, n))
julia> x, stats = bicgstab(A_gpu, b_gpu, M=opM)

we before got

(Float32[1.0198809, 4.0956926, 1.28241, -1.7407427, -2.9416926, -0.32771796, 0.4703165, 5.368891, -3.1246433, -1.7737565  …  0.97625387, 3.7136111, -1.0120717, 2.8906355, -1.1602651, 0.46383825, 2.538051, 1.448916, 2.4476223, -3.592498], SimpleStats
 niter: 54
 solved: false
 inconsistent: false
 indefinite: false
 residuals: []
 Aresiduals: []
 κ₂(A): []
 timer: 48.24ms
 status: breakdown αₖ == 0
)

but with this, get

(Float32[-0.4150291, 2.639933, -1.3929893, -1.1049638, -1.2597414, 0.0031488854, -1.918298, 5.812737, -1.3443595, -1.2016623  …  1.6825913, 3.0281372, -1.4967421, 3.0308955, -1.1063535, 0.2894782, 1.1083188, -0.1900546, 2.3867438, 1.4211323], SimpleStats
 niter: 66
 solved: true
 inconsistent: false
 indefinite: false
 residuals: []
 Aresiduals: []
 κ₂(A): []
 timer: 57.85ms
 status: solution good enough given atol and rtol
)

[amontoison]

@oscardssmith The breakdown is independent of the good / bad preconditioner. What we should check is if we converge in one iteration with an ILU preconditioner of a dense matrix stored in sparse CSR format.

[codecov]

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[oscardssmith]

What we should check is if we converge in one iteration with an ILU preconditioner of a dense matrix stored in sparse CSR format.

This is true for the old, but not the new one.

The breakdown is independent of the good / bad preconditioner

What is it dependent on? Randomly failing to solve seems like a fairly large problem.

[amontoison]

What we should check is if we converge in one iteration with an ILU preconditioner of a dense matrix stored in sparse CSR format.

This is true for the old, but not the new one.

Ok, so it means that the old preconditioner is correct.

The breakdown is independent of the good / bad preconditioner

What is it dependent on? Randomly failing to solve seems like a fairly large problem.

Yes, it is why we should never use bicgstab :)
It could randomly breakdown.
Everybody wanted this method implemented in Krylov.jl but it can breakdown at any moment.
It is cheap but not robust.

[oscardssmith]

In that case, should the docs switch to gmres or some other better algorithm?

[amontoison]

I already suggest better algorithms in the docstring of bicgstab.
bilq and dqgmres are cheaper than gmres and more robust than bicgstab.
https://jso.dev/Krylov.jl/dev/solvers/unsymmetric/#BiCGSTAB