sqrt(::Matrix{Float64}) is not type stable

[LilithHafner]

julia> using StableRNGs

julia> rng = StableRNG(0)
StableRNGs.LehmerRNG(state=0x00000000000000000000000000000001)

julia> sqrt(rand(rng,3,3))
3×3 Matrix{ComplexF64}:
    0.410548+0.369007im      0.49482-0.209855im   0.394519-0.231836im
 -0.00928873+0.0595217im    0.935186-0.0338502im   0.11389-0.0373957im
    0.661866-0.458654im   -0.0589303+0.260838im   0.524852+0.288158im

julia> sqrt(rand(rng,3,3))
3×3 Matrix{Float64}:
 0.301226   0.392135   -0.452243
 0.183647   0.25732     1.14379
 0.185353  -0.0785732   0.578105

For scalars, we throw when sqrting a negative unless the negative is of a complex type in which case we return a complex result. IMO that is a much better solution. Switching to that would be breaking in the case of square rooting real matrices with negative real eigenvalues.

[jishnub]

Previous discussion in #21

[giordano]

Shouldn't this be in the LinearAlgebra repo?

[stevengj]

A non-breaking change could be to pass an additional argument, e.g. sqrt(A, Real) if you want a real result (throwing an exception if not real) and sqrt(A, Complex) if you want a complex result (even if the result is real).

Note that eigen (and eigvals and eigvecs) have the same issue — for a real matrix, they return real matrices/vectors if possible and complex matrices/vectors if not. So any change to sqrt should probably be mirrored there.