Adding MGS H-GMRES(m) resolves #325

[pescap]

[pescap]

#325

[pescap]

Dear all, do you have any suggestion concerning this PR? Thank you!

[lukeolson]

@pescap Thanks for the PR! The preference is to keep the PR minimal -- so few formatting changes and consistent linting (I enabled CI including linting for this PR just now).

I can take a go at thinning this and also adding H to some of the parameterized tests...

[codecov-commenter]

Codecov Report

Merging #326 (2c9c117) into main (af81d67) will decrease coverage by 0.00%. The diff coverage is 76.92%.

@@            Coverage Diff             @@
##             main     #326      +/-   ##
==========================================
- Coverage   64.97%   64.96%   -0.01%     
==========================================
  Files          44       44              
  Lines        5524     5529       +5     
  Branches     1264     1265       +1     
==========================================
+ Hits         3589     3592       +3     
- Misses       1435     1436       +1     
- Partials      500      501       +1     

Impacted Files	Coverage Δ
pyamg/krylov/_gmres.py	50.00% <0.00%> (-12.50%)	:arrow_down:
pyamg/krylov/_gmres_mgs.py	74.16% <90.90%> (+0.66%)	:arrow_up:

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[lukeolson]

@pescap what do you think about renaming H? In terms of code readability I'd like to keep H for the upper Hessenbergs in order to match the typical notation (from Saad, van der Vorst, and others). Looking at the papers you link in #325 it doesn't appear that there's any consistent notation for the weight. But D might make sense? Or W (for weight)?

[pescap]

@pescap what do you think about renaming H? In terms of code readability I'd like to keep H for the upper Hessenbergs in order to match the typical notation (from Saad, van der Vorst, and others). Looking at the papers you link in #325 it doesn't appear that there's any consistent notation for the weight. But D might make sense? Or W (for weight)?

Sounds good! I recommend using D.

[pescap]

Hi @lukeolson, let me know if you have further questions. As soon as the PR is merged, I plan to create an issue to discuss and compare l2- and H^1-based GMRES for a real AMG case! Best wishes, Paul