bug fixes in matrix log (JuliaLang#32327)

* bug fixes in matrix log

* patches to matrix log (JuliaLang#33245)

* patches to matrix log

Avoid integer overflow if `s > 63`.
Correct logic for `s == 0`.
Only use fancy divided difference formulae if eigenvalues are close - avoids dangerous roundoff error if they are in opposite sectors.

* add tests

stdlib/LinearAlgebra/src/triangular.jl

@@ -2294,32 +2294,14 @@ function log(A0::UpperTriangular{T}) where T<:BlasFloat
  end
 
  # Compute accurate superdiagonal of T
- p = 1 / 2^s
- for k = 1:n-1
- Ak = A0[k,k]
- Akp1 = A0[k+1,k+1]
- Akp = Ak^p
- Akp1p = Akp1^p
- A[k,k] = Akp
- A[k+1,k+1] = Akp1p
- if Ak == Akp1
- A[k,k+1] = p * A0[k,k+1] * Ak^(p-1)
- elseif 2 * abs(Ak) < abs(Akp1) || 2 * abs(Akp1) < abs(Ak)
- A[k,k+1] = A0[k,k+1] * (Akp1p - Akp) / (Akp1 - Ak)
- else
- logAk = log(Ak)
- logAkp1 = log(Akp1)
- w = atanh((Akp1 - Ak)/(Akp1 + Ak)) + im*pi*ceil((imag(logAkp1-logAk)-pi)/(2*pi))
- dd = 2 * exp(p*(logAk+logAkp1)/2) * sinh(p*w) / (Akp1 - Ak)
- A[k,k+1] = A0[k,k+1] * dd
- end
- end
+ blockpower!(A, A0, 0.5^s)
 
  # Compute accurate diagonal of T
  for i = 1:n
  a = A0[i,i]
  if s == 0
- r = a - 1
+ A[i,i] = a - 1
+ continue
  end
  s0 = s
  if angle(a) >= pi / 2
@@ -2356,7 +2338,7 @@ function log(A0::UpperTriangular{T}) where T<:BlasFloat
  end
 
  # Scale back
- lmul!(2^s, Y)
+ lmul!(2.0^s, Y)
 
  # Compute accurate diagonal and superdiagonal of log(T)
  for k = 1:n-1
@@ -2368,11 +2350,16 @@ function log(A0::UpperTriangular{T}) where T<:BlasFloat
  Y[k+1,k+1] = logAkp1
  if Ak == Akp1
  Y[k,k+1] = A0[k,k+1] / Ak
- elseif 2 * abs(Ak) < abs(Akp1) || 2 * abs(Akp1) < abs(Ak)
+ elseif 2 * abs(Ak) < abs(Akp1) || 2 * abs(Akp1) < abs(Ak) || iszero(Akp1 + Ak)
  Y[k,k+1] = A0[k,k+1] * (logAkp1 - logAk) / (Akp1 - Ak)
  else
- w = atanh((Akp1 - Ak)/(Akp1 + Ak) + im*pi*(ceil((imag(logAkp1-logAk) - pi)/(2*pi))))
- Y[k,k+1] = 2 * A0[k,k+1] * w / (Akp1 - Ak)
+ z = (Akp1 - Ak)/(Akp1 + Ak)
+ if abs(z) > 1
+ Y[k,k+1] = A0[k,k+1] * (logAkp1 - logAk) / (Akp1 - Ak)
+ else
+ w = atanh(z) + im * pi * (unw(logAkp1-logAk) - unw(log1p(z)-log1p(-z)))
+ Y[k,k+1] = 2 * A0[k,k+1] * w / (Akp1 - Ak)
+ end
  end
  end
 
@@ -2519,14 +2506,19 @@ function blockpower!(A::UpperTriangular, A0::UpperTriangular, p)
 
  if Ak == Akp1
  A[k,k+1] = p * A0[k,k+1] * Ak^(p-1)
- elseif 2 * abs(Ak) < abs(Akp1) || 2 * abs(Akp1) < abs(Ak)
+ elseif 2 * abs(Ak) < abs(Akp1) || 2 * abs(Akp1) < abs(Ak) || iszero(Akp1 + Ak)
  A[k,k+1] = A0[k,k+1] * (Akp1p - Akp) / (Akp1 - Ak)
  else
  logAk = log(Ak)
  logAkp1 = log(Akp1)
- w = atanh((Akp1 - Ak)/(Akp1 + Ak)) + im * pi * unw(logAkp1-logAk)
- dd = 2 * exp(p*(logAk+logAkp1)/2) * sinh(p*w) / (Akp1 - Ak);
- A[k,k+1] = A0[k,k+1] * dd
+ z = (Akp1 - Ak)/(Akp1 + Ak)
+ if abs(z) > 1
+ A[k,k+1] = A0[k,k+1] * (Akp1p - Akp) / (Akp1 - Ak)
+ else
+ w = atanh(z) + im * pi * (unw(logAkp1-logAk) - unw(log1p(z)-log1p(-z)))
+ dd = 2 * exp(p*(logAk+logAkp1)/2) * sinh(p*w) / (Akp1 - Ak);
+ A[k,k+1] = A0[k,k+1] * dd
+ end
  end
  end
 end

stdlib/LinearAlgebra/test/dense.jl

@@ -903,4 +903,19 @@ end
  @test adjoint(factorize(adjoint(a))) == factorize(a)
 end
 
+@testset "Matrix log issue #32313" begin
+ for A in ([30 20; -50 -30], [10.0im 0; 0 -10.0im], randn(6,6))
+ @test exp(log(A)) ≈ A
+ end
+end
+
+@testset "Matrix log PR #33245" begin
+ # edge case for divided difference
+ A1 = triu(ones(3,3),1) + diagm([1.0, -2eps()-1im, -eps()+0.75im])
+ @test exp(log(A1)) ≈ A1
+ # case where no sqrt is needed (s=0)
+ A2 = [1.01 0.01 0.01; 0 1.01 0.01; 0 0 1.01]
+ @test exp(log(A2)) ≈ A2
+end
+
 end # module TestDense