Merge pull request #21184 from iamnapo/anj/powm

Fixed the algorithm for powers of a matrix.

Author: tkelman

base/linalg/dense.jl

@@ -29,7 +29,7 @@ function scale!(X::Array{T}, s::Real) where T<:BlasComplex
     X
 end
 
-#Test whether a matrix is positive-definite
+# Test whether a matrix is positive-definite
 isposdef!(A::StridedMatrix{<:BlasFloat}, UL::Symbol) = LAPACK.potrf!(char_uplo(UL), A)[2] == 0
 
 """
@@ -323,18 +323,67 @@ kron(a::AbstractVector, b::AbstractVector)=vec(kron(reshape(a,length(a),1),resha
 kron(a::AbstractMatrix, b::AbstractVector)=kron(a,reshape(b,length(b),1))
 kron(a::AbstractVector, b::AbstractMatrix)=kron(reshape(a,length(a),1),b)
 
-^(A::AbstractMatrix, p::Integer) = p < 0 ? inv(A^-p) : Base.power_by_squaring(A,p)
-
-function ^(A::AbstractMatrix, p::Number)
+# Matrix power
+^{T}(A::AbstractMatrix{T}, p::Integer) = p < 0 ? Base.power_by_squaring(inv(A), -p) : Base.power_by_squaring(A, p)
+function ^{T}(A::AbstractMatrix{T}, p::Real)
+    # For integer powers, use repeated squaring
     if isinteger(p)
-        return A^Integer(real(p))
+        TT = Base.promote_op(^, eltype(A), typeof(p))
+        return (TT == eltype(A) ? A : copy!(similar(A, TT), A))^Integer(p)
+    end
+
+    # If possible, use diagonalization
+    if T <: Real && issymmetric(A)
+        return (Symmetric(A)^p)
+    end
+    if ishermitian(A)
+        return (Hermitian(A)^p)
+    end
+
+    n = checksquare(A)
+
+    # Quicker return if A is diagonal
+    if isdiag(A)
+        retmat = copy(A)
+        for i in 1:n
+            retmat[i, i] = retmat[i, i] ^ p
+        end
+        return retmat
+    end
+
+    # Otherwise, use Schur decomposition
+    if istriu(A)
+        # Integer part
+        retmat = A ^ floor(p)
+        # Real part
+        if p - floor(p) == 0.5
+            # special case: A^0.5 === sqrtm(A)
+            retmat = retmat * sqrtm(A)
+        else
+            retmat = retmat * powm!(UpperTriangular(float.(A)), real(p - floor(p)))
+        end
+    else
+        S,Q,d = schur(complex(A))
+        # Integer part
+        R = S ^ floor(p)
+        # Real part
+        if p - floor(p) == 0.5
+            # special case: A^0.5 === sqrtm(A)
+            R = R * sqrtm(S)
+        else
+            R = R * powm!(UpperTriangular(float.(S)), real(p - floor(p)))
+        end
+        retmat = Q * R * Q'
+    end
+
+    # if A has nonpositive real eigenvalues, retmat is a nonprincipal matrix power.
+    if isreal(retmat)
+        return real(retmat)
+    else
+        return retmat
     end
-    checksquare(A)
-    v, X = eig(A)
-    any(v.<0) && (v = complex(v))
-    Xinv = ishermitian(A) ? X' : inv(X)
-    (X * Diagonal(v.^p)) * Xinv
 end
+^(A::AbstractMatrix, p::Number) = expm(p*logm(A))
 
 # Matrix exponential
 
@@ -466,7 +515,7 @@ function rcswap!(i::Integer, j::Integer, X::StridedMatrix{<:Number})
 end
 
 """
-    logm(A::StridedMatrix)
+    logm(A{T}::StridedMatrix{T})
 
 If `A` has no negative real eigenvalue, compute the principal matrix logarithm of `A`, i.e.
 the unique matrix ``X`` such that ``e^X = A`` and ``-\\pi < Im(\\lambda) < \\pi`` for all
@@ -497,8 +546,11 @@ julia> logm(A)
  0.0  1.0
 ```
 """
-function logm(A::StridedMatrix)
+function logm{T}(A::StridedMatrix{T})
     # If possible, use diagonalization
+    if issymmetric(A) && T <: Real
+        return full(logm(Symmetric(A)))
+    end
     if ishermitian(A)
         return full(logm(Hermitian(A)))
     end

base/linalg/diagonal.jl

@@ -288,6 +288,7 @@ end
 # identity matrices via eye(Diagonal{type},n)
 eye(::Type{Diagonal{T}}, n::Int) where {T} = Diagonal(ones(T,n))
 
+# Matrix functions
 expm(D::Diagonal) = Diagonal(exp.(D.diag))
 expm(D::Diagonal{<:AbstractMatrix}) = Diagonal(expm.(D.diag))
 logm(D::Diagonal) = Diagonal(log.(D.diag))

base/linalg/linalg.jl

@@ -262,7 +262,6 @@ include("hessenberg.jl")
 include("lq.jl")
 include("eigen.jl")
 include("svd.jl")
-include("schur.jl")
 include("symmetric.jl")
 include("cholesky.jl")
 include("lu.jl")
@@ -274,6 +273,8 @@ include("givens.jl")
 include("special.jl")
 include("bitarray.jl")
 include("ldlt.jl")
+include("schur.jl")
+
 
 include("arpack.jl")
 include("arnoldi.jl")

base/linalg/schur.jl

@@ -107,6 +107,12 @@ function schur(A::StridedMatrix)
     SchurF = schurfact(A)
     SchurF[:T], SchurF[:Z], SchurF[:values]
 end
+schur(A::Symmetric) = schur(full(A))
+schur(A::Hermitian) = schur(full(A))
+schur(A::UpperTriangular) = schur(full(A))
+schur(A::LowerTriangular) = schur(full(A))
+schur(A::Tridiagonal) = schur(full(A))
+
 
 """
     ordschur!(F::Schur, select::Union{Vector{Bool},BitVector}) -> F::Schur

base/linalg/symmetric.jl

@@ -1,6 +1,6 @@
 # This file is a part of Julia. License is MIT: https://julialang.org/license
 
-#Symmetric and Hermitian matrices
+# Symmetric and Hermitian matrices
 struct Symmetric{T,S<:AbstractMatrix} <: AbstractMatrix{T}
     data::S
     uplo::Char
@@ -181,7 +181,7 @@ trace(A::Hermitian) = real(trace(A.data))
 Base.conj(A::HermOrSym) = typeof(A)(conj(A.data), A.uplo)
 Base.conj!(A::HermOrSym) = typeof(A)(conj!(A.data), A.uplo)
 
-#tril/triu
+# tril/triu
 function tril(A::Hermitian, k::Integer=0)
     if A.uplo == 'U' && k <= 0
         return tril!(A.data',k)
@@ -235,7 +235,7 @@ end
 ## Matvec
 A_mul_B!{T<:BlasFloat}(y::StridedVector{T}, A::Symmetric{T,<:StridedMatrix}, x::StridedVector{T}) = BLAS.symv!(A.uplo, one(T), A.data, x, zero(T), y)
 A_mul_B!{T<:BlasComplex}(y::StridedVector{T}, A::Hermitian{T,<:StridedMatrix}, x::StridedVector{T}) = BLAS.hemv!(A.uplo, one(T), A.data, x, zero(T), y)
-##Matmat
+## Matmat
 A_mul_B!{T<:BlasFloat}(C::StridedMatrix{T}, A::Symmetric{T,<:StridedMatrix}, B::StridedMatrix{T}) = BLAS.symm!('L', A.uplo, one(T), A.data, B, zero(T), C)
 A_mul_B!{T<:BlasFloat}(C::StridedMatrix{T}, A::StridedMatrix{T}, B::Symmetric{T,<:StridedMatrix}) = BLAS.symm!('R', B.uplo, one(T), B.data, A, zero(T), C)
 A_mul_B!{T<:BlasComplex}(C::StridedMatrix{T}, A::Hermitian{T,<:StridedMatrix}, B::StridedMatrix{T}) = BLAS.hemm!('L', A.uplo, one(T), A.data, B, zero(T), C)
@@ -403,7 +403,54 @@ function svdvals!{T<:Real,S}(A::Union{Hermitian{T,S}, Symmetric{T,S}, Hermitian{
     return sort!(vals, rev = true)
 end
 
-#Matrix-valued functions
+# Matrix functions
+function ^{T<:Real}(A::Symmetric{T}, p::Integer)
+    if p < 0
+        return Symmetric(Base.power_by_squaring(inv(A), -p))
+    else
+        return Symmetric(Base.power_by_squaring(A, p))
+    end
+end
+function ^{T<:Real}(A::Symmetric{T}, p::Real)
+    F = eigfact(A)
+    if all(λ -> λ ≥ 0, F.values)
+        retmat = (F.vectors * Diagonal((F.values).^p)) * F.vectors'
+    else
+        retmat = (F.vectors * Diagonal((complex(F.values)).^p)) * F.vectors'
+    end
+    return Symmetric(retmat)
+end
+function ^(A::Hermitian, p::Integer)
+    n = checksquare(A)
+    if p < 0
+        retmat = Base.power_by_squaring(inv(A), -p)
+    else
+        retmat = Base.power_by_squaring(A, p)
+    end
+    for i = 1:n
+        retmat[i,i] = real(retmat[i,i])
+    end
+    return Hermitian(retmat)
+end
+function ^{T}(A::Hermitian{T}, p::Real)
+    n = checksquare(A)
+    F = eigfact(A)
+    if all(λ -> λ ≥ 0, F.values)
+        retmat = (F.vectors * Diagonal((F.values).^p)) * F.vectors'
+        if T <: Real
+            return Hermitian(retmat)
+        else
+            for i = 1:n
+                retmat[i,i] = real(retmat[i,i])
+            end
+            return Hermitian(retmat)
+        end
+    else
+        retmat = (F.vectors * Diagonal((complex(F.values).^p))) * F.vectors'
+        return retmat
+    end
+end
+
 function expm(A::Symmetric)
     F = eigfact(A)
     return Symmetric((F.vectors * Diagonal(exp.(F.values))) * F.vectors')
@@ -423,10 +470,8 @@ function expm(A::Hermitian{T}) where T
 end
 
 for (funm, func) in ([:logm,:log], [:sqrtm,:sqrt])
-
     @eval begin
-
-        function ($funm)(A::Symmetric)
+        function ($funm){T<:Real}(A::Symmetric{T})
             F = eigfact(A)
             if isposdef(F)
                 retmat = (F.vectors * Diagonal(($func).(F.values))) * F.vectors'
@@ -454,7 +499,5 @@ for (funm, func) in ([:logm,:log], [:sqrtm,:sqrt])
                 return retmat
             end
         end
-
     end
-
 end