Eigen for AbstractFill and eigvals for Tridiagonal{<:Number,<:AbstractFillVector}

[jishnub]

The various tridiagonal matrices that wrap AbstractFillVectors are Toeplitz matrices, and their eigenvalues may be computed analytically in O(n) time, and eigenvectors in O(n^2). Aside from this, since a square AbstractFillMatrix is of rank-1, it has only one non-zero eigenvalue that may be computed trivially in O(1) time, and the eigenvectors may also be evaluated in O(n^2).

julia> T = Tridiagonal(Fill(2,400), Fill(5,401), Fill(2,400));

julia> @btime eigvals($T);
  12.503 μs (1 allocation: 3.31 KiB)  # This PR
  8.211 ms (11 allocations: 1.37 MiB) # master

julia> @btime eigen($T);
  875.773 μs (4 allocations: 2.46 MiB) # This PR
  26.678 ms (16 allocations: 3.83 MiB) # master

julia> F = Fill(2, 40, 40);

julia> @btime eigvals($F); # This PR
  4.211 ns (0 allocations: 0 bytes)

julia> @btime eigen($F);
  1.444 μs (1 allocation: 12.62 KiB)

julia> @btime eigvals($(Matrix(F))); # as on aster
  106.181 μs (10 allocations: 27.94 KiB)

[codecov]

Codecov Report

Merging #256 (d3ad175) into master (91f9f6f) will increase coverage by 0.03%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #256      +/-   ##
==========================================
+ Coverage   99.74%   99.77%   +0.03%     
==========================================
  Files           5        5              
  Lines         771      882     +111     
==========================================
+ Hits          769      880     +111     
  Misses          2        2              

Impacted Files	Coverage Δ
src/FillArrays.jl	99.42% <ø> (ø)
src/fillalgebra.jl	100.00% <100.00%> (ø)

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[jishnub]

I'm not fully sure about d5ff1e4, where I evaluate sin(pi*q*j/(n+1)) iteratively. This is faster than calling sinpi on each iteration, but this might reduce the accuracy for large matrices as floating-point errors pile up. Should we revert this, and go back to calling sinpi in each iteration?

[dlfivefifty]

I don't really think this PR belongs here, it's too complicated for this package.... ToeplitzMatrices.jl would be a better home even if it means type-piracy...

[jishnub]

Perhaps this can be simplified somewhat by removing the eigenvalue sorting. The Toeplitz component may be moved to ToeplitzMatrices, but the eigen for FillArrays may remain here?

[dlfivefifty]

Yes I'm happy for the eigen for Fill to be here... just Toeplitz matrices becomes a bit too algorithmic and then its harder to maintain...

[jishnub]

I've moved the tridiagonal methods to ToeplitzMatrices.jl. I'm closing this for now, as eigen for Fill arrays is perhaps not a very useful operation (and it's trivial in any case)