Add support for calculating eigenvalues and eigenvectors

[hyrodium]

Before this PR

julia> eigvals(RotXYZ(1,2,3))
ERROR: Only hermitian matrices are diagonalizable by *StaticArrays*. Non-Hermitian matrices should be converted to `Array` first.
Stacktrace:
 [1] error(s::String)
   @ Base ./error.jl:33
 [2] #eigvals#619
   @ ~/.julia/packages/StaticArrays/vxjOO/src/eigen.jl:12 [inlined]
 [3] eigvals(a::RotXYZ{Float64})
   @ StaticArrays ~/.julia/packages/StaticArrays/vxjOO/src/eigen.jl:7
 [4] top-level scope
   @ REPL[2]:1

julia> eigvecs(RotXYZ(1,2,3))
3×3 StaticArrays.SMatrix{3, 3, ComplexF64, 9} with indices SOneTo(3)×SOneTo(3):
  0.130632+0.39127im    0.130632-0.39127im   0.812211+0.0im
  0.689496-0.0im        0.689496+0.0im       -0.22179+0.0im
 0.0867797-0.588989im  0.0867797+0.588989im  0.539559+0.0im

After this PR

julia> eigvals(RotXYZ(1,2,3))
3-element StaticArrays.SVector{3, ComplexF64} with indices SOneTo(3):
 -0.7278678429175085 + 0.6857174368838911im
 -0.7278678429175085 - 0.6857174368838911im
                 1.0 + 0.0im

julia> eigvecs(RotXYZ(1,2,3))
3×3 StaticArrays.SMatrix{3, 3, ComplexF64, 9} with indices SOneTo(3)×SOneTo(3):
  -0.130632+0.39127im     0.39127-0.130632im   -0.812211+0.0im
  -0.689496+0.0im             0.0-0.689496im     0.22179+0.0im
 -0.0867797-0.588989im  -0.588989-0.0867797im  -0.539559+0.0im

[codecov]

Codecov Report

Merging #170 (143f9aa) into master (e9b567f) will increase coverage by 0.20%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #170      +/-   ##
==========================================
+ Coverage   85.95%   86.15%   +0.20%     
==========================================
  Files          12       13       +1     
  Lines        1289     1315      +26     
==========================================
+ Hits         1108     1133      +25     
- Misses        181      182       +1     

Impacted Files	Coverage Δ
src/eigen.jl	100.00% <100.00%> (ø)
src/unitquaternion.jl	95.62% <0.00%> (-0.55%)	⬇️

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[hyrodium]

Ah, I was forgetting to add methods for Rotation{2}.
I'll deal with it tommorow.

[andyferris]

This is cool.

Out of curiousity, what do you think you might do with it?

Do we support log yet?

[andyferris]

How much work is repeated here?

[hyrodium]

Sorry, I didn't quite understand the question.

[andyferris]

I meant, some of the calculation going into the eigenvalues is probably being recalculated in eigvecs. Not the way it is currently written but it usual that it falls out that way.

I doubt it has a massive performance impact here.

[hyrodium]

Ah, I've updated code and avoid converting to AngleAxis twice.

Before performance update

julia> R = rand(RotMatrix{3})
3×3 RotMatrix3{Float64} with indices SOneTo(3)×SOneTo(3):
  0.196251   -0.948589   0.248325
  0.979358    0.202123  -0.00188397
 -0.0484051   0.243568   0.968675

julia> @benchmark eigen(R)
BenchmarkTools.Trial: 10000 samples with 434 evaluations.
 Range (min … max):  234.707 ns …  2.399 μs  ┊ GC (min … max): 0.00% … 89.78%
 Time  (median):     238.424 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   246.933 ns ± 73.440 ns  ┊ GC (mean ± σ):  1.26% ±  3.77%

  ▃▆█▆▅▃▁▁▁▁             ▁▁▁        ▁                          ▁
  ████████████▇█▇▇▆▆▇▆▆▅█████▆▆▅▇▇▇█████▇▆▆▅▆▆▇▇▇▇▅▅▅▄▅▄▄▃▄▄▃▄ █
  235 ns        Histogram: log(frequency) by time       315 ns <

 Memory estimate: 208 bytes, allocs estimate: 1.

After performance update

julia> R = rand(RotMatrix{3})
3×3 RotMatrix3{Float64} with indices SOneTo(3)×SOneTo(3):
 -0.81819   -0.492907   -0.295987
 -0.287214  -0.0955672   0.953087
 -0.49807    0.864818   -0.0633776

julia> @benchmark eigen(R)
BenchmarkTools.Trial: 10000 samples with 700 evaluations.
 Range (min … max):  179.454 ns … 929.644 ns  ┊ GC (min … max): 0.00% … 80.02%
 Time  (median):     183.104 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   187.316 ns ±  40.447 ns  ┊ GC (mean ± σ):  1.17% ±  4.40%

   ▃▇██▇▆▄▃▄▄▃▂▁                                                ▂
  ███████████████▆▆▅▆▄▁▃▁▃▁▁▁▁▃▁▁▁▃▁▁▁▁▁▁▃▁▁▃▃▄▄▅▃▄▆▇▇▇▇▇███▇█▇ █
  179 ns        Histogram: log(frequency) by time        232 ns <

 Memory estimate: 208 bytes, allocs estimate: 1.

[hyrodium]

Out of curiousity, what do you think you might do with it?

I don't have applications of eigenvalues of rotations for now. Just for fun:smile:

Do we support log yet?

I thought the return value type should be Lie algebra and this should be done after solving #30. But, I realized that we just need to return SMatrix{3,3} for now.

[hyrodium]

I've added the methods for 2d rotations, and I think this is ready to merge.