sqrt, cbrt and log for dense diagonal matrices (#1156)

This PR improves performance by only applying the functions to the
diagonal elements:
```julia
julia> A = diagm(0=>ones(100));

julia> @Btime log($A);
  364.163 μs (22 allocations: 401.62 KiB) # master
  13.528 μs (7 allocations: 80.02 KiB) # this PR
```
Similar improvements for `sqrt` and `cbrt` as well.

Author: jishnub

src/dense.jl

@@ -890,7 +890,9 @@ julia> log(A)
 """
 function log(A::AbstractMatrix)
     # If possible, use diagonalization
-    if ishermitian(A)
+    if isdiag(A)
+        return applydiagonal(log, A)
+    elseif ishermitian(A)
         logHermA = log(Hermitian(A))
         return ishermitian(logHermA) ? copytri!(parent(logHermA), 'U', true) : parent(logHermA)
     elseif istriu(A)
@@ -969,7 +971,9 @@ sqrt(::AbstractMatrix)
 
 function sqrt(A::AbstractMatrix{T}) where {T<:Union{Real,Complex}}
     if checksquare(A) == 0
-        return copy(A)
+        return copy(float(A))
+    elseif isdiag(A)
+        return applydiagonal(sqrt, A)
     elseif ishermitian(A)
         sqrtHermA = sqrt(Hermitian(A))
         return ishermitian(sqrtHermA) ? copytri!(parent(sqrtHermA), 'U', true) : parent(sqrtHermA)
@@ -1035,7 +1039,9 @@ true
 """
 function cbrt(A::AbstractMatrix{<:Real})
     if checksquare(A) == 0
-        return copy(A)
+        return copy(float(A))
+    elseif isdiag(A)
+        return applydiagonal(cbrt, A)
     elseif issymmetric(A)
         return cbrt(Symmetric(A, :U))
     else

test/dense.jl

@@ -817,6 +817,7 @@ end
 
     A13 = convert(Matrix{elty}, [2 0; 0 2])
     @test typeof(log(A13)) == Array{elty, 2}
+    @test exp(log(A13)) ≈ log(exp(A13)) ≈ A13
 
     T = elty == Float64 ? Symmetric : Hermitian
     @test typeof(log(T(A13))) == T{elty, Array{elty, 2}}
@@ -968,6 +969,10 @@ end
         @test typeof(sqrt(A8)) == Matrix{elty}
     end
 end
+@testset "sqrt for diagonal" begin
+    A = diagm(0 => [1, 2, 3])
+    @test sqrt(A)^2 ≈ A
+end
 
 @testset "issue #40141" begin
     x = [-1 -eps() 0 0; eps() -1 0 0; 0 0 -1 -eps(); 0 0 eps() -1]
@@ -1280,6 +1285,7 @@ end
     T = cbrt(Symmetric(S,:U))
     @test T*T*T ≈ S
     @test eltype(S) == eltype(T)
+    @test cbrt(Array(Symmetric(S,:U))) == T
     # Real valued symmetric
     S =  (A -> (A+A')/2)(randn(N,N))
     T = cbrt(Symmetric(S,:L))
@@ -1300,6 +1306,16 @@ end
     T = cbrt(A)
     @test T*T*T ≈ A
     @test eltype(A) == eltype(T)
+    @testset "diagonal" begin
+        A = diagm(0 => [1, 2, 3])
+        @test cbrt(A)^3 ≈ A
+    end
+    @testset "empty" begin
+        A = Matrix{Float64}(undef, 0, 0)
+        @test cbrt(A) == A
+        A = Matrix{Int}(undef, 0, 0)
+        @test cbrt(A) isa Matrix{Float64}
+    end
 end
 
 @testset "tr" begin