Integer matrix exponentiation in schurpow (#51992)

This improves type-inference by avoiding recursion, as the `A^p` method
calls `schurpow` if `p` is a float. After this, the result of `schurpow`
is inferred as a small `Union`:

```julia
julia> @inferred Union{Matrix{ComplexF64}, Matrix{Float64}} LinearAlgebra.schurpow([1.0 2; 3 4], 2.0)
2×2 Matrix{Float64}:
  7.0  10.0
 15.0  22.0
```

One concern here might be that for large `p`, the `A^Int(p)` computation
might be expensive by repeated multiplication, as opposed to
diagonalization. However, this may only be the case for really large
`p`, which may not be commonly encountered.

I've added a test, but I'm unsure if `schurpow` is deemed to be an
internal function, and this test is unwise. Unfortunately, the return
type of `A^p` isn't concretely inferred yet as there are too many
possible types that are returned, so I couldn't test for that.

Author: jishnub

src/dense.jl

@@ -490,7 +490,7 @@ end
 function schurpow(A::AbstractMatrix, p)
     if istriu(A)
         # Integer part
-        retmat = A ^ floor(p)
+        retmat = A ^ floor(Integer, p)
         # Real part
         if p - floor(p) == 0.5
             # special case: A^0.5 === sqrt(A)
@@ -501,7 +501,7 @@ function schurpow(A::AbstractMatrix, p)
     else
         S,Q,d = Schur{Complex}(schur(A))
         # Integer part
-        R = S ^ floor(p)
+        R = S ^ floor(Integer, p)
         # Real part
         if p - floor(p) == 0.5
             # special case: A^0.5 === sqrt(A)

test/dense.jl

@@ -1028,8 +1028,8 @@ end
     @test lyap(1.0+2.0im, 3.0+4.0im) == -1.5 - 2.0im
 end
 
-@testset "Matrix to real power" for elty in (Float64, ComplexF64)
-# Tests proposed at Higham, Deadman: Testing Matrix Function Algorithms Using Identities, March 2014
+@testset "$elty Matrix to real power" for elty in (Float64, ComplexF64)
+    # Tests proposed at Higham, Deadman: Testing Matrix Function Algorithms Using Identities, March 2014
     #Aa : only positive real eigenvalues
     Aa = convert(Matrix{elty}, [5 4 2 1; 0 1 -1 -1; -1 -1 3 0; 1 1 -1 2])
 
@@ -1065,6 +1065,9 @@ end
         @test (A^(2/3))*(A^(1/3)) ≈ A
         @test (A^im)^(-im) ≈ A
     end
+
+    Tschurpow = Union{Matrix{real(elty)}, Matrix{complex(elty)}}
+    @test (@inferred Tschurpow LinearAlgebra.schurpow(Aa, 2.0)) ≈ Aa^2
 end
 
 @testset "diagonal integer matrix to real power" begin