Make the algorithm for real powers of a matrix robust

base/linalg/diagonal.jl

@@ -251,6 +251,14 @@ expm(D::Diagonal) = Diagonal(exp.(D.diag))
 logm(D::Diagonal) = Diagonal(log.(D.diag))
 sqrtm(D::Diagonal) = Diagonal(sqrt.(D.diag))
 
+# Matrix functions
+^(D::Diagonal, p::Real) = Diagonal((D.diag).^p)
+for (funm, func) in ([:expm,:exp], [:sqrtm,:sqrt], [:logm,:log])
+ @eval begin
+ ($funm)(D::Diagonal) = Diagonal(($func)(D.diag))
+ end
+end
+
 #Linear solver
 function A_ldiv_B!(D::Diagonal, B::StridedVecOrMat)
  m, n = size(B, 1), size(B, 2)

base/linalg/triangular.jl

@@ -1672,7 +1672,7 @@ powm(A::LowerTriangular, p::Real) = powm(A.', p::Real).'
 # Based on the code available at http://eprints.ma.man.ac.uk/1851/02/logm.zip,
 # Copyright (c) 2011, Awad H. Al-Mohy and Nicholas J. Higham
 # Julia version relicensed with permission from original authors
-function logm{T<:Union{Float32,Float64,Complex{Float32},Complex{Float64}}}(A0::UpperTriangular{T})
+function logm{T<:Union{Float64,Complex{Float64}}}(A0::UpperTriangular{T})
  theta = [1.586970738772063e-005,
  2.313807884242979e-003,
  1.938179313533253e-002,
@@ -1718,7 +1718,7 @@ function logm{T<:Union{Float32,Float64,Complex{Float32},Complex{Float64}}}(A0::U
  scale!(2^s,Y.data)
 
  # Compute accurate diagonal and superdiagonal of log(A)
- for k = 1:n-1
+ @inbounds for k = 1:n-1
  Ak = complex(A0[k,k])
  Akp1 = complex(A0[k+1,k+1])
  logAk = log(Ak)
@@ -1749,7 +1749,7 @@ logm(A::LowerTriangular) = logm(A.').'
 # Numer. Algorithms, 59, (2012), 393–402.
 function sqrt_diag!(A0::UpperTriangular, A::UpperTriangular, s)
  n = checksquare(A0)
- for i = 1:n
+ @inbounds for i = 1:n
  a = complex(A0[i,i])
  if s == 0
  A[i,i] = a - 1
@@ -1806,60 +1806,53 @@ function invsquaring(A0::UpperTriangular, theta)
  d2 = sqrt(norm(AmI^2, 1))
  d3 = cbrt(norm(AmI^3, 1))
  alpha2 = max(d2, d3)
- foundm = false
  if alpha2 <= theta[2]
  m = alpha2<=theta[1]?1:2
- foundm = true
- end
-
- while ~foundm
- more = false
- if s > s0
- d3 = cbrt(norm(AmI^3, 1))
- end
- d4 = norm(AmI^4, 1)^(1/4)
- alpha3 = max(d3, d4)
- if alpha3 <= theta[tmax]
- for j = 3:tmax
- if alpha3 <= theta[j]
+ else
+ while true
+ more = false
+ if s > s0
+ d3 = cbrt(norm(AmI^3, 1))
+ end
+ d4 = norm(AmI^4, 1)^(1/4)
+ alpha3 = max(d3, d4)
+ if alpha3 <= theta[tmax]
+ for j = 3:tmax
+ if alpha3 <= theta[j]
+ break
+ end
+ end
+ if j <= 6
+ m = j
  break
+ elseif alpha3 / 2 <= theta[5] && p < 2
+ more = true
+ p = p + 1
  end
  end
- if j <= 6
- m = j
- foundm = true
- break
- elseif alpha3 / 2 <= theta[5] && p < 2
- more = true
- p = p + 1
- end
- end
-
- if ~more
- d5 = norm(AmI^5, 1)^(1/5)
- alpha4 = max(d4, d5)
- eta = min(alpha3, alpha4)
- if eta <= theta[tmax]
- j = 0
- for j = 6:tmax
- if eta <= theta[j]
- m = j
- break
+ if ~more
+ d5 = norm(AmI^5, 1)^(1/5)
+ alpha4 = max(d4, d5)
+ eta = min(alpha3, alpha4)
+ if eta <= theta[tmax]
+ j = 0
+ for j = 6:tmax
+ if eta <= theta[j]
+ m = j
+ break
+ end
  end
+ break
  end
- foundm = true
+ end
+ if s == maxsqrt
+ m = tmax
  break
  end
+ A = sqrtm(A)
+ AmI = A - I
+ s = s + 1
  end
-
- if s == maxsqrt
- m = tmax
- foundm = true
- break
- end
- A = sqrtm(A)
- AmI = A - I
- s = s + 1
  end
 
  # Compute accurate superdiagonal of T