Julia: Matrix functions for UniformScaling

[jagot]

I was surprised by this:

julia> sqrt(5.0I)
ERROR: MethodError: no method matching sqrt(::UniformScaling{Float64})
Closest candidates are:
  sqrt(::Float16) at math.jl:1085
  sqrt(::Complex{Float16}) at math.jl:1086
  sqrt(::Missing) at math.jl:1138
  ...
Stacktrace:
 [1] top-level scope at REPL[25]:1

julia> sqrt(Matrix(5I, 5, 5))
5×5 Array{Float64,2}:
 2.23607  0.0      0.0      0.0      0.0
 0.0      2.23607  0.0      0.0      0.0
 0.0      0.0      2.23607  0.0      0.0
 0.0      0.0      0.0      2.23607  0.0
 0.0      0.0      0.0      0.0      2.23607

Since we already have a few matrix functions defined, such as
https://github.com/JuliaLang/julia/blob/e36fe95a639c6956aa39367bef8e727b174e52b2/stdlib/LinearAlgebra/src/dense.jl#L702-L759
it would seem reasonable to expect at least these elementary functions to be defined for UniformScaling as well:

• [x] ^(::UniformScaling, ::Number) (already added in https://github.com/JuliaLang/julia/pull/34008)
• [x] exp, log
• [x] expm1, logp1
• [x] sin, cos, tan, csc, sec, cot + inverses and hyperbolic versions
• [x] sqrt, cbrt
• [x] pinv, tr
• [x] real, imag, reim

All of the above have been added in #28872, the question is now which additional functions are of interest, i.e. which are used for matrices? Candidates are

• [ ] abs - one could argue that the matrix generalization is det
  (which yields a scalar), but you can compute abs(A) as
  Q*abs.(λ)*Q', which is something different.
• [ ] abs2
• [ ] log10, log2
• [ ] exp2, exp10
• [ ] sinc, cosc

I am unsure if the functions below are sensible for matrices, possibly
with the exception of sinpi and cospi.

• [ ] sinpi, cospi
• [ ] sind, cosd, tand, secd, cscd, cotd
• [ ] asind, acosd, atand, asecd, acscd, acotd
• [ ] deg2rad, rad2deg
• [ ] mod2pi

From #28872 remains also the idea of matrix decompositions.

[jagot]

Initial implementation: https://github.com/jagot/UniformScalingMatrixFunctions.jl

[dkarrasch]

There was some initial effort made in #28872, but then got stale.

[jagot]

Good to know! Maybe that PR should be revived instead?

[dkarrasch]

Yes. Actually, that PR had all that you are asking for here. So we could resolve merge conflicts there, merge, and open a new issue to track the ambitious unresolved broadcasting aspects.

[JeffreySarnoff]

sqrt
exp, log
sin, cos, tan + asin, acos, atan
sinh, cosh, tanh + asinh, acosh, atanh

other similar from Base.Math: cbrt , expm1, log1p

[jagot]

I've rebased #28872 and pushed to my fork. Just need to build and test and then make a new PR?

[dkarrasch]

I've just resolved the merge conflicts there and one qualification issue, tests are running. If they pass, I'd suggest to merge #28872, and then continue from there.

[jagot]

Hmm, double work double the fun?

[jagot]

@antoine-levitt suggested this list of functions:
https://github.com/stevengj/ChangePrecision.jl/blob/402b3eec7710552b655555e8335faca2737235e4/src/ChangePrecision.jl#L33-L46
supplemented by
https://github.com/JuliaDiff/DiffRules.jl/blob/master/src/rules.jl

[dkarrasch]

@antoine-levitt suggested this list of functions:
...
supplemented by
...

Sounds good. Maybe update the OP to reflect the missing parts now that #28872 is merged? One open issue there were matrix decompositions.

[jagot]

Updated!