Allow singular matrices in saturate

[mgkurtz]

Possible fix for #1101.

[fieker]

probably correct - but I'd expect it to be much slower (for suitable evil examples)

• hnf_with_transform is much slower than hnf without
• inv(T) is inverting a much larger matrix than inverting H - furthermore, H is square, upper triangular, so inverting is just writing it down. inv(T) is hard...

[mgkurtz]

I had some fun, adapting the original concept to the singular case. Indeed, the hnf_with_transform takes longer than the whole saturate. Curiously, for random input, inverting the unimodular transform U was faster than inverting the hnf matrix H. Probably, because (U⁻¹)ᵀ is the pretty looking result matrix, while the hnf and its inverse may have a few large entries.

[mgkurtz]

Hmph, I should listen better. You told me, not to compute the inverses, but to just solve the linear system directly. Well, this reduces the code to less than it was before, works out of the box for singular matrices, and speeds it up.

[mgkurtz]

Just look at commit ea0b343 – unless you are interested in all the comments in the previous commits.

[codecov]

Codecov Report

Patch coverage: 83.33% and project coverage change: +0.01 🎉

Comparison is base (e22dabd) 69.30% compared to head (ea0b343) 69.32%.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1102      +/-   ##
==========================================
+ Coverage   69.30%   69.32%   +0.01%     
==========================================
  Files         349      349              
  Lines      112890   112895       +5     
==========================================
+ Hits        78237    78259      +22     
+ Misses      34653    34636      -17     

Impacted Files	Coverage Δ
src/Misc/Matrix.jl	70.27% <83.33%> (-0.13%)	⬇️

... and 30 files with indirect coverage changes

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