Proper handling for BandedMatrices

[avik-pal]

This is being developed parallel to https://github.com/SciML/BoundaryValueDiffEq.jl/pull/124. I will mark it as ready once properly tested!

[codecov[bot]]

Codecov Report

Merging #393 (7d766d8) into main (0843b53) will increase coverage by 0.10%. The diff coverage is 5.55%.

@@            Coverage Diff             @@
##             main     #393      +/-   ##
==========================================
+ Coverage   68.81%   68.92%   +0.10%     
==========================================
  Files          24       26       +2     
  Lines        1889     1905      +16     
==========================================
+ Hits         1300     1313      +13     
- Misses        589      592       +3     

Files	Coverage Δ
src/common.jl	90.74% <ø> (+1.85%)	:arrow_up:
src/default.jl	66.86% <50.00%> (+1.20%)	:arrow_up:
ext/LinearSolveRecursiveArrayToolsExt.jl	0.00% <0.00%> (ø)
ext/LinearSolveBandedMatricesExt.jl	0.00% <0.00%> (ø)

... and 3 files with indirect coverage changes

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[ChrisRackauckas]

What's going on with v1.8? Should we just version gate ldiv!?

[avik-pal]

Seems like a weird one-off failure? The previous CIs for 1.8 did pass

[avik-pal]

This is good to go on my end!