Simple but inelegant fix for #16548

[Sacha0]

This pull request contains a simple but inelegant fix for JuliaLang/LinearAlgebra.jl#333.

By inelegant I mean this fix: (1) replicates code nearly verbatim between factorize and \ in base/sparse/linalg.jl, and replicates some logic between those methods and \ in base/linalg/generic.jl; and (2) includes a brittle test, almost certain to break if someone eliminates the code replication between the two preceding \ methods. (A note on the test's brittleness and expected eventual removal appears adjacent to the test.)

Hence, though this fix may serve for the immediate future, eventually a better approach should replace it. A couple thoughts on better approaches:

A little metaprogramming could superficially eliminate the code replication between factorize and \ in base/sparse/linalg.jl, but those methods are short enough that the likely net impact of such metaprogramming would be obfuscation.

Taking a step back, \ classifies the LHS, factors the classified LHS, and then solves a system with the factored LHS. factorize classifies the LHS and then factors the classified LHS. Of course somehow decoupling classification, factorization, and solution, then composing those elements to form \ (both dense and sparse) and factorize from the same code, would be ideal. But decoupling seems to make type inference an issue, so no dice?

@andreasnoack @tkelman, other thoughts? Should this be discussed immediately, or should a simple fix like this one stand for now and additional discussion be deferred to another day/thread?

Best!

[StefanKarpinski]

Thanks! Better an inelegant fix than nothing. As long as the behavior is fixed, we can come up with a more elegant fix later and backport it to 0.5.1.

[vtjnash]

But decoupling seems to make type inference an issue, so no dice?

yep, looks like this would require more aggressive Union splitting / method considerations to realize the output type is constant. @JeffBezanson maybe we could have abstract_call_gf_by_type consider more methods as long as all are returning the same leaftype?

[Sacha0]

Thanks for the review and merge!