yak shaving, full deprecation style

[Sacha0]

tl;dr
This pull request eliminates a few uses of full from base/linalg/special.jl, via a minor yak shave. Ref. JuliaLang/LinearAlgebra.jl#229, JuliaLang/LinearAlgebra.jl#231, #18850, and linked threads.

details
The eliminated uses of full were part of ad hoc checks for upper bidiagonal, lower bidiagonal, and tridiagonal band structure in conversions between structured and/or annotated matrix types.

The first commit generalizes existing and introduces new (not exported) generic functions that perform the necessary upper bidiagonal, lower bidiagonal, and tridiagonal band structure checks. Specifically, the first commit extends istri[u|l](A) to istri[u|l](A, k) matching tri[u|l](A, k), implements isbanded(A, kl, ku) as a short child of istri[u|l](A, k), (re)implements isdiag(A), isbidiag[u|l](A), and istridiag(A) as short children of isbanded(A, ul, uk), and tests and documents all those functions.

The second commit eliminates the uses of full in question via the functions provided by the first commit.

The third commit then cleans up the complete set of surrounding structured and/or annotated matrix interconversion methods, at a pleasing LOC reduction.

Best!

[alyst]

maybe @inbounds?

[Sacha0]

I am not certain this is safe for arbitrary AbstractArrays, for example OffsetArrays, and so hesitate to add an @inbounds?

[andreasnoack]

Eventually, these functions should use either be restricted to StridedMatrix or the index calculations be changed to work with OffsetArrays.

[Sacha0]

Good call :). Do you have thoughts on the best way to generalize these loops? Thanks!

[andreasnoack]

The easiest might be to reuse the existing loops and then use the i and j to index into the actual indices for the array, e.g. ii = indices(A, 1)[i], and then index with ii and jj.

[fredrikekre]

Looks great, I like the isbanded function. I am a bit skeptical to isbidiag[u|l] and istridiag though. I am not sure they are very useful (seem to be used only a couple of times in this PR, which should indicate something?), it is simple enough to do isbanded(A, 0, 1) / isbanded(A, -1, 0) / isbanded(A, -1, 1)? With introduction of isbanded we could perhaps instead @deprecate isdiag(A) isbanded(A, 0, 0) rather than introducing more specific functions like that? Thoughts? 😄

[StefanKarpinski]

I think isdiag is traditional and common enough that we should keep it. 100% agree that isbidiag, istridiag etc. should just be expressed using isbanded.

[Sacha0]

Cheers, agreed :).

[Sacha0]

Nixed isbidiag[u|l] and istridiag. Thoughts? Thanks!

[Sacha0]

Thanks all! :)