Fix log_quasitriu for internal scaling s=0

NEWS.md

@@ -148,6 +148,7 @@ Standard library changes
   Custom array types may specialize this function to return an appropriate result ([#55252]).
 * The matrix multiplication `A * B` calls `matprod_dest(A, B, T::Type)` to generate the destination.
   This function is now public ([#55537]).
+* Now `exp(log(A))` for matrix `A` correctly returns `A`, up to numerical precision ([#56311]).
 
 #### Logging
 

stdlib/LinearAlgebra/src/triangular.jl

@@ -2218,6 +2218,11 @@ end
 # SIAM J. Sci. Comput., 34(4), (2012) C153–C169. doi: 10.1137/110852553
 # Algorithm 5.1
 Base.@propagate_inbounds function _sqrt_pow_diag_block_2x2!(A, A0, s)
+    if iszero(s)
+        A[1,1] -= 1
+        A[2,2] -= 1
+        return
+    end
     _sqrt_real_2x2!(A, A0)
     if isone(s)
         A[1,1] -= 1

stdlib/LinearAlgebra/test/triangular.jl

@@ -1386,4 +1386,14 @@ end
     end
 end
 
+@testset "log_quasitriu with internal scaling s=0 (issue #54833)" begin
+    M = [0.9949357359852791 -0.015567763143324862 -0.09091193493947397 -0.03994428739762443 0.07338356301650806;
+    0.011813655598647289 0.9968988574699793 -0.06204555000202496 0.04694097614450692 0.09028834462782365;
+    0.092737943594701 0.059546719185135925 0.9935850721633324 0.025348893985651405 -0.018530261590167685;
+    0.0369187299165628 -0.04903571106913449 -0.025962938675946543 0.9977767446862031 0.12901494726320517;
+    0.0 0.0 0.0 0.0 1.0]
+
+    @test exp(log(M)) ≈ M
+end
+
 end # module TestTriangular