Add banded paths to generic solve

[jishnub]

Currently, the generic solve only checks for diagonal and triangular matrices when dispatching to the faster methods. We also have solve methods for Bidiagonal and SymTridiagonal matrices, so we may add these branches as well.

With these,

julia> n = 5000; dv, ev = rand(n), rand(n-1);

julia> A = diagm(0=>dv, 1=>ev, -1=>ev); # symmetric tridiagonal by construction

julia> @btime $A \ $dv;
  656.843 ms (9 allocations: 190.81 MiB) # master
  11.549 ms (22 allocations: 235.09 KiB) # PR

After this change, almost all the time is spent on checking if the matrix is tridiagonal.

[stevengj]

(This all seems a bit weird to me; I feel like if you want a structured solve algorithm, you should use a structured-matrix type. But I guess since we're already checking for triangular matrices there is some argument for this…)

[jishnub]

I see this as a generic fallback, where the structure may not be obvious from the type, but may be obtained by looking at the values. Suppose we have a fast solver for a Diagonal, but we call the method with a Transpose of a Diagonal, we may still use the banded structure even when we aren't directly dispatching to the method for the Diagonal. One way around this is to define \ for a Transpose{<:Any, <:Diagonal}, which would also work. The alternate, as in this PR, is to check for the banded structure from the values in the fallback method, and dispatch to the fast paths. This makes sense when the alternate approach --- factorizing the matrix --- is significantly more expensive than checking for a banded structure. In addition, this method would now work with arbitrary banded matrices that have 3 or fewer populated bands, even when the package might not be implementing a fast solver for these.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 93.81%. Comparing base (c64c328) to head (f348c95).

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1372   +/-   ##
=======================================
  Coverage   93.81%   93.81%           
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  Files          34       34           
  Lines       15717    15723    +6     
=======================================
+ Hits        14745    14751    +6     
  Misses        972      972           

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[dkarrasch]

Wouldn't you get this "for free" with istrium1 && istril1?

[jishnub]

Yes, but there were some triangular lu-related test failures for unusual types, so I've removed the triangular branch for now. We may add this back once those issues are resolved.