Update arnoldi.jl to include a reorthogonalisation option

[cneverett]

Added an option to reorthogonalise (reorth) each basis vector during its construction in arnoldi!. This helps numerical roundoff from leading to poor basis orthogonalisation, even with a modified Gram-Schmidt procedure, especially when the Krylov subspace dimension is large. This can lead to better stability and smaller errors for large, highly stiff, non-normal systems.

This is my first time contributing to SciML, so I am not familiar with the format of pull requests. This pull request only adds a small feature and I have updated the function documentation to include it. I am not sure if it needs any specific tests.

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

Add any other context about the problem here.

[ChrisRackauckas]

Do you have a test case where this was shown to be a tangible improvement?

[cneverett]

Unfortunately I don't have a simple test case that shows the improvement. I have been developing a modified version of arnoldi! to work with very large systems that I have been modelling (stiff, non-normal Jacobians of size around 10^4x10^4 needing Krylov dimensions of around 256 to captures dynamics accurately) and I doing so I have found that adding in this simple re-orthogonalisation process does help reduce the errest of functions like phiv! used for time stepping and reduce noise in the output results.

I can try to come up with a simple test-case to demonstrate these findings but it will take time. Or I could use the systems I am currently modelling and show some outputs with and without this re-orthogonalisation?

[ChrisRackauckas]

Tests fail.

This should get at least one test exercising it.

[cneverett]

I have now made reorth a kwarg in arnoldi_step! (set to false by default) so it shouldn't break downstream tests by changing the number of arguments.

I will come up with at least one test demonstrating the difference, which will take a little time.