Don't access parent of triangular matrix in powm (#52583)

Since the values stored in the parent corresponding to the structural
zeros of a tridiagonal matrix aren't well-defined, using it in `ldiv!`
is a footgun that may lead to heisenbugs (one seen in
https://buildkite.com/julialang/julia-master/builds/31285#018c7cc7-6c77-41ac-a01b-1c7d14cb1b15).
This PR changes it to using the tridiagonal matrix directly in `ldiv!`,
which should lead to predictable results, and be bug-free. The failing
tests for #52571 pass locally with this change.

(cherry picked from commit ef549ae)

Author: jishnub

stdlib/LinearAlgebra/src/triangular.jl

@@ -1551,7 +1551,7 @@ end
 #   Higham and Lin, "An improved Schur-Padé algorithm for fractional powers of
 #     a matrix and their Fréchet derivatives", SIAM. J. Matrix Anal. & Appl.,
 #     34(3), (2013) 1341–1360.
-function powm!(A0::UpperTriangular{<:BlasFloat}, p::Real)
+function powm!(A0::UpperTriangular, p::Real)
     if abs(p) >= 1
         throw(ArgumentError("p must be a real number in (-1,1), got $p"))
     end
@@ -1583,22 +1583,22 @@ function powm!(A0::UpperTriangular{<:BlasFloat}, p::Real)
         end
         copyto!(Stmp, S)
         mul!(S, A, c)
-        ldiv!(Stmp, S.data)
+        ldiv!(Stmp, S)
 
         c = (p - j) / (j4 - 2)
         for i = 1:n
             @inbounds S[i, i] = S[i, i] + 1
         end
         copyto!(Stmp, S)
         mul!(S, A, c)
-        ldiv!(Stmp, S.data)
+        ldiv!(Stmp, S)
     end
     for i = 1:n
         S[i, i] = S[i, i] + 1
     end
     copyto!(Stmp, S)
     mul!(S, A, -p)
-    ldiv!(Stmp, S.data)
+    ldiv!(Stmp, S)
     for i = 1:n
         @inbounds S[i, i] = S[i, i] + 1
     end

stdlib/LinearAlgebra/test/dense.jl

@@ -1067,6 +1067,11 @@ end
     end
 end
 
+@testset "BigFloat triangular real power" begin
+    A = Float64[3 1; 0 3]
+    @test A^(3/4) ≈ big.(A)^(3/4)
+end
+
 @testset "diagonal integer matrix to real power" begin
     A = Matrix(Diagonal([1, 2, 3]))
     @test A^2.3 ≈ float(A)^2.3