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meoperators.jl
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meoperators.jl
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# TODO: These types aren't used anywhere for multiple dispatch, so why have them?
abstract type LyapunovMatrixEquationsMaps{T} <: LinearMaps.LinearMap{T} end
abstract type SylvesterMatrixEquationsMaps{T} <: LinearMaps.LinearMap{T} end
const MatrixEquationsMaps{T} = Union{LyapunovMatrixEquationsMaps{T},SylvesterMatrixEquationsMaps{T}}
abstract type ContinuousOrDiscrete end
struct Continuous <: ContinuousOrDiscrete end
struct Discrete <: ContinuousOrDiscrete end
abstract type TtypeOrHtype end
struct Ttype <: TtypeOrHtype end
struct Htype <: TtypeOrHtype end
struct trmatop{T} <: LinearMaps.LinearMap{T}
size::Dims{2}
function trmatop{T}(dims::Dims{2}) where {T}
all(≥(0), dims) || throw(ArgumentError("dims must be non-negative"))
return new{T}(dims)
end
function trmatop(m::Int, n::Int)
(m ≥ (0) & n ≥ (0)) || throw(ArgumentError("dimensions must be non-negative"))
return new{Bool}((m, n))
end
end
trmatop(dims::Dims{2}) = trmatop{Bool}(dims)
trmatop(n::Int) = trmatop(n, n)
Base.size(A::trmatop) = (N = prod(A.size); return (N,N))
LinearAlgebra.issymmetric(A::trmatop) = true
LinearAlgebra.ishermitian(A::trmatop) = true
function LinearMaps._unsafe_mul!(y::AbstractVector, A::trmatop, x::AbstractVector)
(m, n) = A.size
k = 1
for j = 1:n
for i = 1:m
y[(i-1)*n+j] = x[k]
k += 1
end
end
return y
end
"""
M = trmatop(n, m)
M = trmatop(A)
Define the transposition operator `M: X -> X'` for all `n x m` matrices or for all matrices of size of `A`.
The corresponding commutation matrix (see [here](https://en.wikipedia.org/wiki/Commutation_matrix))
can be generated as `Matrix(M)`.
"""
function trmatop(A)
trmatop{eltype(A)}(size(A))
end
"""
M = eliminationop(n)
M = eliminationop(A)
Define the elimination operator of all `n×n` matrices to select their upper triangular parts or of all square matrices of size of `A`.
See [here](https://en.wikipedia.org/wiki/Duplication_and_elimination_matrices) for the
definition of an elimination matrix for the selection of lower triangular parts.
"""
struct eliminationop{T} <: LinearMaps.LinearMap{T}
dim::Int
function eliminationop{T}(dim::Int) where {T}
dim >= 0 || throw(ArgumentError("dimension must be non-negative"))
return new{T}(dim)
end
function eliminationop(n::Int)
(n ≥ (0)) || throw(ArgumentError("dimension must be non-negative"))
return new{Bool}(n)
end
end
Base.size(A::eliminationop) = (n = A.dim; return (div(n*(n+1),2), n*n))
LinearAlgebra.adjoint(A::eliminationop{T}) where {T} = LinearAlgebra.transpose(A)
function LinearMaps._unsafe_mul!(y::AbstractVector, A::eliminationop, x::AbstractVector)
# X = reshape(x, A.dim, A.dim)
# y[:] = triu2vec(X)
n = A.dim
k = 1
for j = 1:n
for i = 1:j
y[k] = x[(i-1)*n + j]
k += 1
end
end
return y
end
function LinearMaps._unsafe_mul!(x::AbstractVector, AT::LinearMaps.TransposeMap{T,<:eliminationop{T}}, y::AbstractVector) where {T}
# X = vec2triu(y)
n = AT.lmap.dim
ZERO = zero(eltype(y))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
x[(i-1)*n + j] = y[k]
k += 1
end
for i = j+1:n
x[(i-1)*n + j] = ZERO
end
end
end
return x
end
eliminationop(A) = eliminationop{eltype(A)}(LinearAlgebra.checksquare(A))
"""
M = duplicationop(n)
M = duplicationop(A)
Define the duplication operator of all `n×n` matrices to reconstruct a hermitian matrix from its upper triangular elements
or of all square matrices of size of `A`.
See [here](https://en.wikipedia.org/wiki/Duplication_and_elimination_matrices) for the
definition of a duplication matrix from the lower triangular parts.
"""
struct duplicationop{T} <: LinearMaps.LinearMap{T}
dim::Int
function duplicationop{T}(dim::Int) where {T}
dim >= 0 || throw(ArgumentError("dimension must be non-negative"))
return new{T}(dim)
end
function duplicationop(n::Int)
(n ≥ (0)) || throw(ArgumentError("dimension must be non-negative"))
return new{Bool}(n)
end
end
Base.size(A::duplicationop) = (n = A.dim; return (n*n, div(n*(n+1),2)))
LinearAlgebra.adjoint(A::duplicationop{T}) where {T} = LinearAlgebra.transpose(A)
function LinearMaps._unsafe_mul!(y::AbstractVector, A::duplicationop, x::AbstractVector)
# y[:] = vec2triu(x, her = true)
n = A.dim
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[(i-1)*n + j] = x[k]
k += 1
end
for i = 1:j-1
y[(j-1)*n + i] = conj(y[(i-1)*n + j])
end
end
end
return y
end
function LinearMaps._unsafe_mul!(x::AbstractVector, AT::LinearMaps.TransposeMap{T,<:duplicationop{T}}, y::AbstractVector) where {T}
n = AT.lmap.dim
# Y = reshape(y, n, n)
# x[:] = triu2vec(Y+transpose(Y)-Diagonal(Y))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
#x[k] = i == j ? Y[j,j] : Y[i,j] + Y[j,i]
x[k] = i == j ? y[(j-1)*n + j] : y[(i-1)*n + j] + y[(j-1)*n + i]
k += 1
end
end
end
return x
end
duplicationop(A) = duplicationop{eltype(A)}(LinearAlgebra.checksquare(A))
struct LyapLikeMap{T,TA <: AbstractMatrix,CD <: ContinuousOrDiscrete,TH<:TtypeOrHtype} <: LyapunovMatrixEquationsMaps{T}
A::TA
adj::Bool
isig::Int
function LyapLikeMap{T,TA,CD,TH}(A::TA; isig = 1, adj=false) where {T,TA<:AbstractMatrix{T},CD,TH}
abs(isig) == 1 || throw(ArgumentError("only 1 or -1 values allowed for isig; got isig = $isig"))
return new{T,TA,CD,TH}(A, adj, isig)
end
end
LyapLikeMap(A::TA, ::CD = Continuous(), ::TH = Ttype(); adj::Bool = false, isig::Int = 1) where {T,TA<:AbstractMatrix{T},CD<:ContinuousOrDiscrete,TH<:TtypeOrHtype} =
LyapLikeMap{T,TA,CD,TH}(A; adj, isig)
function Base.size(L::LyapLikeMap)
m, n = size(L.A)
return (m*m, m*n)
end
"""
L = lyaplikeop(A; isig = 1, adj = false, htype = false)
For a matrix `A`, define for `adj = false` the continuous T-Lyapunov operator `L:X -> A*X+isig*transpose(X)*transpose(A)` if `htype = false`
or the continuous H-Lyapunov operator `L:X -> A*X+isig*X'*A'` if `htype = true`, or
define for `adj = true` the continuous T-Lyapunov operator `L:X -> A*transpose(X)+X*transpose(A)` if `htype = false`,
or the continuous H-Lyapunov operator `L:X -> A*X'+isig*X*A'` if `htype = true`.
"""
function lyaplikeop(A::AbstractMatrix; disc=false, adj=false, isig = 1, htype = false)
LyapLikeMap(A, ifelse(disc, Discrete(), Continuous()), ifelse(htype, Htype(), Ttype()); adj, isig)
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::LyapLikeMap{T,TA,Continuous,Ttype}, x::AbstractVector) where {T,TA}
m, n = size(L.A)
T1 = promote_type(T, eltype(x))
X = L.adj ? reshape(convert(AbstractVector{T1}, x), m, n) : reshape(convert(AbstractVector{T1}, x), n, m)
Y = L.adj ? L.A*transpose(X) : L.A*X
y[:] .= L.isig == 1 ? (Y+transpose(Y))[:] : (Y-transpose(Y))[:]
return y
end
function LinearMaps._unsafe_mul!(x::AbstractVector, LT::LinearMaps.TransposeMap{T,<:LyapLikeMap{T,<:Any,Continuous,Ttype}}, y::AbstractVector) where {T}
m = size(LT.lmap.A,1)
T1 = promote_type(T, eltype(y))
Y = reshape(convert(AbstractVector{T1}, y), m, m)
temp = LT.lmap.isig ==1 ? Y+transpose(Y) : (LT.lmap.adj ? transpose(Y)-Y : Y-transpose(Y))
x[:] = LT.lmap.adj ? (temp*LT.lmap.A)[:] : (transpose(LT.lmap.A)*temp)[:]
return x
end
function LinearMaps._unsafe_mul!(x::AbstractVector, LT::LinearMaps.AdjointMap{T,<:LyapLikeMap{T,<:Any,Continuous,Ttype}}, y::AbstractVector) where {T}
m = size(LT.lmap.A,1)
T1 = promote_type(T, eltype(y))
Y = reshape(convert(AbstractVector{T1}, y), m, m)
temp = LT.lmap.isig ==1 ? Y+transpose(Y) : (LT.lmap.adj ? transpose(Y)-Y : Y-transpose(Y))
x[:] = LT.lmap.adj ? (temp*conj(LT.lmap.A))[:] : (LT.lmap.A'*temp)[:]
return x
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::LyapLikeMap{T,TA,Continuous,Htype}, x::AbstractVector) where {T,TA}
m, n = size(L.A)
T1 = promote_type(T, eltype(x))
X = L.adj ? reshape(convert(AbstractVector{T1}, x), m, n) : reshape(convert(AbstractVector{T1}, x), n, m)
Y = L.adj ? L.A*X' : L.A*X
y[:] .= L.isig == 1 ? (Y+Y')[:] : (Y-Y')[:]
return y
end
function LinearMaps._unsafe_mul!(x::AbstractVector, LT::LinearMaps.TransposeMap{T,<:LyapLikeMap{T,<:Any,Continuous,Htype}}, y::AbstractVector) where {T}
m = size(LT.lmap.A,1)
T1 = promote_type(T, eltype(y))
Y = reshape(convert(AbstractVector{T1}, y), m, m)
temp = LT.lmap.isig ==1 ? Y+transpose(Y) : (LT.lmap.adj ? transpose(Y)-Y : Y-transpose(Y))
x[:] = LT.lmap.adj ? (temp*LT.lmap.A)[:] : (transpose(LT.lmap.A)*temp)[:]
return x
end
function LinearMaps._unsafe_mul!(x::AbstractVector, LT::LinearMaps.AdjointMap{T,<:LyapLikeMap{T,<:Any,Continuous,Htype}}, y::AbstractVector) where {T}
m = size(LT.lmap.A,1)
T1 = promote_type(T, eltype(y))
Y = reshape(convert(AbstractVector{T1}, y), m, m)
temp = LT.lmap.isig ==1 ? Y+Y' : (LT.lmap.adj ? Y'-Y : Y-Y')
x[:] = LT.lmap.adj ? (temp*LT.lmap.A)[:] : (LT.lmap.A'*temp)[:]
return x
end
struct UTLyapLikeMap{T,TU <: AbstractMatrix,CD <: ContinuousOrDiscrete,TH<:TtypeOrHtype} <: LyapunovMatrixEquationsMaps{T}
U::TU
adj::Bool
function UTLyapLikeMap{T,TU,CD,TH}(U::TU, adj=false) where {T,TU<:AbstractMatrix{T},CD,TH}
LinearAlgebra.checksquare(U)
istriu(U) || error("U must be upper triangular")
#(!adj && istriu(U.parent)) || (adj && istriu(U)) || error("U must be upper triangular")
return new{T,TU,CD,TH}(U, adj)
end
end
UTLyapLikeMap(U::TU, ::CD = Continuous(), ::TH = Ttype(); adj::Bool = false) where {T,TU<:AbstractMatrix{T},CD<:ContinuousOrDiscrete,TH<:TtypeOrHtype} =
UTLyapLikeMap{T,TU,CD,TH}(U, adj)
function Base.size(L::UTLyapLikeMap)
n = size(L.U, 1)
N = Int(n * (n + 1) / 2)
return (N, N)
end
"""
L = tulyaplikeop(U; adj = false)
Define, for an upper triangular matrix `U`, the continuous T-Lyapunov operator `L:X -> transpose(U)*X+transpose(X)*U`, if `adj = false`,
or `L:X -> U*transpose(X)+X*transpose(U)` if `adj = true`.
"""
function tulyaplikeop(U::AbstractMatrix; disc=false, adj = false)
UTLyapLikeMap(U, ifelse(disc, Discrete(), Continuous()), Ttype(); adj )
end
"""
L = hulyaplikeop(U; adj = false)
Define, for an upper triangular matrix `U`, the continuous H-Lyapunov operator `L:X -> U'*X+X'*U`, if `adj = false`,
`L:X -> U*X'+X*U'` if `adj = true`.
"""
function hulyaplikeop(U::AbstractMatrix; disc=false, adj = false)
UTLyapLikeMap(U, ifelse(disc, Discrete(), Continuous()), Htype(); adj)
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::UTLyapLikeMap{T,TU,Continuous,Ttype}, x::AbstractVector) where {T,TU}
T1 = promote_type(T, eltype(x))
X = vec2triu(convert(AbstractVector{T1}, x), her=false)
Y = L.adj ? L.U*transpose(X) : transpose(L.U)*X
y[:] .= triu2vec(Y+transpose(Y))
return y
end
#LinearAlgebra.adjoint(L::UTLyapLikeMap{T,TU,Continuous,Ttype}) where {T,TU}= LinearAlgebra.transpose(L)
#LinearAlgebra.transpose(L::UTLyapLikeMap{T,TU,Continuous,Ttype}) where {T,TU}= LinearAlgebra.adjoint(L)
function LinearMaps._unsafe_mul!(x::AbstractVector, LT::LinearMaps.TransposeMap{T,<:UTLyapLikeMap{T,<:Any,Continuous,Ttype}}, y::AbstractVector) where {T}
n = size(LT.lmap.U,2)
T1 = promote_type(T, eltype(y))
Y = vec2triu(convert(AbstractVector{T1}, y), her=false)
x[:] = LT.lmap.adj ? triu2vec((Y+transpose(Y))*LT.lmap.U) : triu2vec(LT.lmap.U*(Y+transpose(Y)))
#x[:] = LT.lmap.adj ? triu2vec(Y*LT.lmap.U+Y'*conj(LT.lmap.U)) : triu2vec(LT.lmap.U*Y+conj(LT.lmap.U)*Y')
return x
end
function LinearMaps._unsafe_mul!(x::AbstractVector, LT::LinearMaps.AdjointMap{T,<:UTLyapLikeMap{T,<:Any,Continuous,Ttype}}, y::AbstractVector) where {T}
n = size(LT.lmap.U,2)
T1 = promote_type(T, eltype(y))
Y = vec2triu(convert(AbstractVector{T1}, y), her=false)
#x[:] = LT.lmap.adj ? triu2vec((Y+Y')*conj(LT.lmap.U)) : triu2vec(conj(LT.lmap.U)*(Y+Y'))
x[:] = LT.lmap.adj ? triu2vec((Y+transpose(Y))*conj(LT.lmap.U)) : triu2vec(conj(LT.lmap.U)*(Y+transpose(Y)))
return x
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::UTLyapLikeMap{T,TU,Continuous,Htype}, x::AbstractVector) where {T,TU}
T1 = promote_type(T, eltype(x))
X = vec2triu(convert(AbstractVector{T1}, x), her=false)
Y = L.adj ? L.U*X' : L.U'*X
y[:] .= triu2vec(Y+Y')
return y
end
# function LinearMaps._unsafe_mul!(x::AbstractVector, LT::LinearMaps.TransposeMap{T,<:UTLyapLikeMap{T,TU,Continuous,Htype}}, y::AbstractVector) where {T,TU}
# n = size(LT.lmap.U,2)
# T1 = promote_type(T, eltype(y))
# Y = vec2triu(convert(AbstractVector{T1}, y), her=false)
# x[:] = LT.lmap.adj ? triu2vec(Y*LT.lmap.U+Y'*conj(LT.lmap.U)) : triu2vec(LT.lmap.U*Y+conj(LT.lmap.U)*Y')
# return x
# end
function LinearMaps._unsafe_mul!(x::AbstractVector, LT::LinearMaps.AdjointMap{T,<:UTLyapLikeMap{T,TU,Continuous,Htype}}, y::AbstractVector) where {T,TU}
n = size(LT.lmap.U,2)
T1 = promote_type(T, eltype(y))
Y = vec2triu(convert(AbstractVector{T1}, y), her=false)
# the following ensures Matrix(L') == Matrix(L)'
#x[:] = LT.lmap.adj ? triu2vec(Y*LT.lmap.U+Y'*conj(LT.lmap.U)) : triu2vec(LT.lmap.U*Y+conj(LT.lmap.U)*Y')
x[:] = LT.lmap.adj ? triu2vec((Y+Y')*LT.lmap.U) : triu2vec(LT.lmap.U*(Y+Y'))
return x
end
# function LinearMaps._unsafe_mul!(y::AbstractVector, LT::LinearMaps.AdjointMap{T,<:UTHLyapunovMap{T}}, x::AbstractVector) where {T}
# n = size(LT.lmap.U,2)
# T1 = promote_type(T, eltype(x))
# X = vec2triu(convert(AbstractVector{T1}, x), her=false)
# y[:] = LT.lmap.adj ? triu2vec((X+X')*LT.lmap.U) : triu2vec(LT.lmap.U'*(X+X'))
# return y
# end
# struct UTLyapLikeMap{T,TU <: AbstractMatrix,CD <: ContinuousOrDiscrete} <: LyapunovMatrixEquationsMaps{T}
# U::TU
# adj::Bool
# function UTLyapLikeMap{T,TU,CD}(U::TU, adj=false) where {T,TU<:AbstractMatrix{T},CD}
# LinearAlgebra.checksquare(U)
# (!adj && istriu(U.parent)) || (adj && istriu(U)) || error("U must be upper triangular")
# return new{T,TU,CD}(U, adj)
# end
# end
# UTLyapLikeMap(U::TU, ::CD = Continuous(), adj::Bool = false) where {T,TU<:AbstractMatrix{T},CD<:ContinuousOrDiscrete} =
# UTLyapLikeMap{T,TU,CD}(U, adj)
# #LinearAlgebra.transpose(L::UTLyapLikeMap{<:Any,<:Any,CD}) where {CD} = LinearAlgebra.adjoint(L)
# function Base.size(L::UTLyapLikeMap)
# n = size(L.U, 1)
# N = Int(n * (n + 1) / 2)
# return (N, N)
# end
# function LinearMaps._unsafe_mul!(y::AbstractVector, L::UTLyapLikeMap{T,<:Any,Continuous,Htype}, x::AbstractVector) where {T,TU}
# T1 = promote_type(T, eltype(x))
# X = vec2triu(convert(AbstractVector{T1}, x), her=false)
# Y = L.adj ? L.U*X' : L.U*X
# y[:] .= triu2vec(Y+Y')
# return y
# end
# function LinearMaps._unsafe_mul!(y::AbstractVector, LT::LinearMaps.AdjointMap{T,<:UTLyapLikeMap{T}}, x::AbstractVector) where {T}
# n = size(LT.lmap.U,2)
# T1 = promote_type(T, eltype(x))
# X = vec2triu(convert(AbstractVector{T1}, x), her=false)
# y[:] = LT.lmap.adj ? triu2vec((X+X')*LT.lmap.U) : triu2vec(LT.lmap.U'*(X+X'))
# return y
# end
struct LyapunovMap{T,TA <: AbstractMatrix,CD <: ContinuousOrDiscrete} <: LyapunovMatrixEquationsMaps{T}
A::TA
her::Bool
function LyapunovMap{T,TA,CD}(A::TA, her=false) where {T,TA<:AbstractMatrix{T},CD}
LinearAlgebra.checksquare(A)
return new{T,TA,CD}(A, her)
end
end
LyapunovMap(A::TA, ::CD = Continuous(), her::Bool = false) where {T,TA<:AbstractMatrix{T},CD<:ContinuousOrDiscrete} =
LyapunovMap{T,TA,CD}(A, her)
# LinearAlgebra.adjoint(L::LyapunovMap{<:Any,<:Any,CD}) where {CD} = LyapunovMap(L.A', CD(), L.her)
# LinearAlgebra.transpose(L::LyapunovMap{<:Any,<:Any,CD}) where {CD} = LyapunovMap(L.A', CD(), L.her)
# LinearAlgebra.transpose(L::LyapunovMap{<:Any,<:Any,CD}) where {CD} = LyapunovMap(L.A', CD(), L.her)
"""
L = lyapop(A; disc = false, her = false)
Define, for an `n x n` matrix `A`, the continuous Lyapunov operator `L:X -> AX+XA'`
if `disc = false` or the discrete Lyapunov operator `L:X -> AXA'-X` if `disc = true`.
If `her = false` the Lyapunov operator `L:X -> Y` maps general square matrices `X`
into general square matrices `Y`, and the associated matrix `M = Matrix(L)` is
``n^2 \\times n^2``.
If `her = true` the Lyapunov operator `L:X -> Y` maps symmetric/Hermitian matrices `X`
into symmetric/Hermitian matrices `Y`, and the associated matrix `M = Matrix(L)` is
``n(n+1)/2 \\times n(n+1)/2``.
For the definitions of the Lyapunov operators see:
M. Konstantinov, V. Mehrmann, P. Petkov. On properties of Sylvester and Lyapunov
operators. Linear Algebra and its Applications 312:35–71, 2000.
"""
lyapop(A::AbstractMatrix; disc=false, her=false) =
LyapunovMap(A, ifelse(disc, Discrete(), Continuous()), her)
lyapop(A::Schur; disc=false, her=false) =
LyapunovMap(A.T, ifelse(disc, Discrete(), Continuous()), her)
lyapop(A::Number; disc=false, her=false) =
LyapunovMap(fill(A, 1, 1), ifelse(disc, Discrete(), Continuous()), her)
function Base.size(L::LyapunovMap)
n = size(L.A, 1)
N = L.her ? Int(n * (n + 1) / 2) : n * n
return (N, N)
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::LyapunovMap{T,TA,Discrete}, x::AbstractVector) where {T,TA}
require_one_based_indexing(y, x)
n = size(L.A, 1)
T1 = promote_type(T, eltype(x))
if L.her
X = vec2triu(convert(AbstractVector{T1}, x), her=true)
# (y .= triu2vec(utqu(X,L.A') - X))
muldsym!(y, L.A, X)
else
X = reshape(convert(AbstractVector{T1}, x), n, n)
# (y .= (L.A*X*L.A' - X)[:])
mul!(y, -1, x)
Y = reshape(y, n, n)
mul!(Y, L.A, X * L.A', true, true)
end
return y
end
for ttype in (LinearMaps.TransposeMap, LinearMaps.AdjointMap)
@eval function LinearMaps._unsafe_mul!(x::AbstractVector, L::$ttype{T,LyapunovMap{T,TA,Discrete}}, y::AbstractVector) where {T,TA}
require_one_based_indexing(y, x)
n = size(L.lmap.A, 1)
T1 = promote_type(T, eltype(y))
if L.lmap.her
Y = vec2triu(convert(AbstractVector{T1}, y), her=false)
# x[:] = triu2vec(L.lmap.A'*Y*L.lmap.A - Y)
muldsym!(x, L.lmap.A', Y, dual = true)
else
# (x .= (L.A'*Y*L.A - Y)[:])
mul!(x, -1, y)
Y = reshape(convert(AbstractVector{T1}, y), n, n)
X = reshape(x, n, n)
mul!(X, L.lmap.A'*Y, L.lmap.A, true, true)
end
return x
end
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::LyapunovMap{T,TA,Continuous}, x::AbstractVector) where {T,TA}
require_one_based_indexing(y, x)
n = size(L.A, 1)
T1 = promote_type(T, eltype(x))
if L.her
X = vec2triu(convert(AbstractVector{T1}, x), her=true)
#y[:] = triu2vec(L.A*X + X*L.A')
mulcsym!(y, L.A, X)
else
X = reshape(convert(AbstractVector{T1}, x), n, n)
# (y[:] = (L.A*X + X*L.A')[:])
Y = reshape(y, n, n)
mul!(Y, X, L.A')
mul!(Y, L.A, X, true, true)
end
return y
end
for ttype in (LinearMaps.TransposeMap, LinearMaps.AdjointMap)
@eval function LinearMaps._unsafe_mul!(x::AbstractVector, L::$ttype{T,LyapunovMap{T,TA,Continuous}}, y::AbstractVector) where {T,TA}
require_one_based_indexing(y, x)
n = size(L.lmap.A, 1)
T1 = promote_type(T, eltype(y))
if L.lmap.her
Y = vec2triu(convert(AbstractVector{T1}, y), her=false)
#x[:] = triu2vec(L.lmap.A'*Y + Y*L.lmap.A)
mulcsym!(x, L.lmap.A', Y, dual = true)
else
Y = reshape(convert(AbstractVector{T1}, y), n, n)
# (x[:] = (L.A'*Y + Y*L.A)[:])
X = reshape(x, n, n)
mul!(X, Y, L.lmap.A)
mul!(X, L.lmap.A', Y, true, true)
end
return x
end
end
function mulcsym!(y::AbstractVector, A::AbstractMatrix, X::AbstractMatrix; dual = false)
require_one_based_indexing(y, A, X)
# A*X + X*A'
n = size(A, 1)
if dual
#Y = A*X+X*A'
Y = similar(X, n, n)
mul!(Y, X, A')
mul!(Y, A, X, true, true)
# y[:] = triu2vec(Y+transpose(Y)-Diagonal(Y))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[k] = i == j ? Y[j,j] : Y[i,j] + Y[j,i]
k += 1
end
end
end
return y
end
ZERO = zero(eltype(y))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
temp = ZERO
for l = 1:n
temp += A[i,l] * X[l,j] + X[i,l] * conj(A[j,l])
end
y[k] = temp
k += 1
end
end
end
return y
end
function muldsym!(y::AbstractVector, A::AbstractMatrix, X::AbstractMatrix; dual = false)
require_one_based_indexing(y, X)
# A*X*A' - X
n = size(A, 1)
if dual
#Y = A*X*A' - X
Y = similar(X, n, n)
mul!(Y, -1, X)
mul!(Y, A*X, A', true, true)
# y[:] = triu2vec(Y+transpose(Y)-Diagonal(Y))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[k] = i == j ? Y[j,j] : Y[i,j] + Y[j,i]
k += 1
end
end
end
return y
end
# t = triu(X)-diag(X)/2
t = UpperTriangular(X) - Diagonal(X[diagind(X)] ./ 2)
Y = similar(t, n, n)
# Y = A*t*A'
mul!(Y, A * t, A')
# Y + Y' - X
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[k] = Y[i,j] + conj(Y[j,i]) - X[i,j]
k += 1
end
end
end
return y
end
struct GeneralizedLyapunovMap{T,TA <: AbstractMatrix{T},TE <: AbstractMatrix{T},CD<:ContinuousOrDiscrete} <: LyapunovMatrixEquationsMaps{T}
A::TA
E::TE
her::Bool
function GeneralizedLyapunovMap{T,TA,TE,CD}(A::TA, E::TE, her=false) where {T,TA<:AbstractMatrix{T},TE<:AbstractMatrix{T},CD}
n = LinearAlgebra.checksquare(A)
n == LinearAlgebra.checksquare(E) ||
throw(DimensionMismatch("E must be a square matrix of dimension $n"))
return new{T,TA,TE,CD}(A, E, her)
end
end
function GeneralizedLyapunovMap(A::AbstractMatrix, E::AbstractMatrix, ::CD = Continuous(), her::Bool = false) where {CD<:ContinuousOrDiscrete}
n = LinearAlgebra.checksquare(A)
n == LinearAlgebra.checksquare(E) ||
throw(DimensionMismatch("E must be a square matrix of dimension $n"))
T = promote_type(eltype(A), eltype(E))
A = convert(AbstractMatrix{T}, A)
E = convert(AbstractMatrix{T}, E)
return GeneralizedLyapunovMap{T,typeof(A),typeof(E),CD}(A, E, her)
end
# LinearAlgebra.adjoint(L::GeneralizedLyapunovMap{<:Any,<:Any,<:Any,CD}) where {CD} =
# GeneralizedLyapunovMap(L.A', L.E', CD(), L.her)
# LinearAlgebra.transpose(L::GeneralizedLyapunovMap{<:Any,<:Any,<:Any,CD}) where {CD} =
# GeneralizedLyapunovMap(L.A', L.E', CD(), L.her)
"""
L = lyapop(A, E; disc = false, her = false)
Define, for a pair `(A,E)` of `n x n` matrices, the continuous Lyapunov operator `L:X -> AXE'+EXA'`
if `disc = false` or the discrete Lyapunov operator `L:X -> AXA'-EXE'` if `disc = true`.
If `her = false` the Lyapunov operator `L:X -> Y` maps general square matrices `X`
into general square matrices `Y`, and the associated matrix `M = Matrix(L)` is
``n^2 \\times n^2``.
If `her = true` the Lyapunov operator `L:X -> Y` maps symmetric/Hermitian matrices `X`
into symmetric/Hermitian matrices `Y`, and the associated `M = Matrix(L)` is a
``n(n+1)/2 \\times n(n+1)/2``.
For the definitions of the Lyapunov operators see:
M. Konstantinov, V. Mehrmann, P. Petkov. On properties of Sylvester and Lyapunov
operators. Linear Algebra and its Applications 312:35–71, 2000.
"""
lyapop(A::AbstractMatrix, E::AbstractMatrix; disc=false, her=false) =
GeneralizedLyapunovMap(A, E, ifelse(disc, Discrete(), Continuous()), her)
lyapop(F::GeneralizedSchur; disc=false, her=false) =
GeneralizedLyapunovMap(F.S, F.T, ifelse(disc, Discrete(), Continuous()), her)
function lyapop(A::Number, E::Number; disc=false, her=false)
A, E = promote(A, E)
GeneralizedLyapunovMap(fill(A, 1, 1), fill(E, 1, 1), ifelse(disc, Discrete(), Continuous()), her)
end
function Base.size(L::GeneralizedLyapunovMap)
n = size(L.A, 1)
N = L.her ? Int(n * (n + 1) / 2) : n * n
return (N, N)
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::GeneralizedLyapunovMap{T,<:Any,<:Any,Discrete}, x::AbstractVector) where {T}
require_one_based_indexing(y, x)
n = size(L.A, 1)
T1 = promote_type(T, eltype(x))
if L.her
X = vec2triu(convert(AbstractVector{T1}, x), her=true)
# (y .= triu2vec(utqu(X,L.A') - utqu(X,L.E')))
muldsym!(y, L.A, L.E, X)
else
X = reshape(convert(AbstractVector{T1}, x), n, n)
Y = reshape(y, n, n)
temp = similar(Y, (n, n))
# (y .= (L.A*X*L.A' - L.E*X*L.E')[:])
# Y .= X
mul!(temp, X, L.A')
mul!(Y, L.A, temp)
mul!(temp, X, L.E')
mul!(Y, L.E, temp, -1, 1)
end
return y
end
for ttype in (LinearMaps.TransposeMap, LinearMaps.AdjointMap)
@eval function LinearMaps._unsafe_mul!(x::AbstractVector, L::$ttype{T,GeneralizedLyapunovMap{T,TA,TE,Discrete}}, y::AbstractVector) where {T,TA,TE}
require_one_based_indexing(y, x)
n = size(L.lmap.A, 1)
T1 = promote_type(T, eltype(y))
if L.lmap.her
Y = vec2triu(convert(AbstractVector{T1}, y), her=false)
# x[:] = triu2vec(L.lmap.A'*Y*L.lmap.A - Y)
muldsym!(x, L.lmap.A', L.lmap.E', Y, dual = true)
else
# (x .= (L.A'*Y*L.A - L.E'*X*L.E)[:])
X = reshape(x, n, n)
Y = reshape(convert(AbstractVector{T1}, y), n, n)
temp = similar(Y, (n, n))
mul!(temp, Y, L.lmap.A)
mul!(X, L.lmap.A', temp)
mul!(temp, Y, L.lmap.E)
mul!(X, L.lmap.E', temp, -1, 1)
end
return x
end
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::GeneralizedLyapunovMap{T,<:Any,<:Any,Continuous}, x::AbstractVector) where {T}
require_one_based_indexing(y, x)
n = size(L.A, 1)
T1 = promote_type(T, eltype(x))
if L.her
X = vec2triu(convert(AbstractVector{T1}, x), her=true)
mulcsym!(y, L.A, L.E, X)
else
X = reshape(convert(AbstractVector{T1}, x), n, n)
Y = reshape(y, n, n)
temp = similar(Y, (n, n))
# (y[:] = (L.A*X*L.E' + L.E*X*L.A')[:])
mul!(temp, L.E, X)
mul!(Y, temp, L.A')
mul!(temp, X, L.E')
mul!(Y, L.A, temp, 1, 1)
end
return y
end
for ttype in (LinearMaps.TransposeMap, LinearMaps.AdjointMap)
@eval function LinearMaps._unsafe_mul!(x::AbstractVector, L::$ttype{T,GeneralizedLyapunovMap{T,TA,TE,Continuous}}, y::AbstractVector) where {T,TA,TE}
require_one_based_indexing(y, x)
n = size(L.lmap.A, 1)
T1 = promote_type(T, eltype(y))
if L.lmap.her
Y = vec2triu(convert(AbstractVector{T1}, y), her=false)
#x[:] = triu2vec(L.lmap.A'*Y + Y*L.lmap.A)
mulcsym!(x, L.lmap.A', L.lmap.E', Y, dual = true)
else
X = reshape(x, n, n)
Y = reshape(convert(AbstractVector{T1}, y), n, n)
temp = similar(Y, (n, n))
# (x[:] = (L.A'*Y*L.E + L.E'*Y*L.A)[:])
mul!(temp, L.lmap.E', Y)
mul!(X, temp, L.lmap.A)
mul!(temp, Y, L.lmap.E)
mul!(X, L.lmap.A', temp, 1, 1)
end
return x
end
end
function mulcsym!(y::AbstractVector, A::AbstractMatrix, E::AbstractMatrix, X::AbstractMatrix; dual = false)
require_one_based_indexing(y, A)
# AXE' + EXA'
n = size(A, 1)
Y = similar(X, n, n)
if dual
#Y = AXE' + EXA'
Y = similar(X, n, n)
temp = similar(Y, (n, n))
mul!(temp, E, X)
mul!(Y, temp, A')
mul!(temp, X, E')
mul!(Y, A, temp, 1, 1)
# y[:] = triu2vec(Y+transpose(Y)-Diagonal(Y))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[k] = i == j ? Y[j,j] : Y[i,j] + Y[j,i]
k += 1
end
end
end
else
ZERO = zero(eltype(y))
# Y = XE'
mul!(Y, X, E')
# AY + Y'A'
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
temp = ZERO
for l = 1:n
temp += A[i,l] * Y[l,j] + conj(Y[l,i] * A[j,l])
end
y[k] = temp
k += 1
end
end
end
end
return y
end
function muldsym!(y::AbstractVector, A::AbstractMatrix, E::AbstractMatrix, X::AbstractMatrix; dual = false)
require_one_based_indexing(y)
# AXA' - EXE'
n = size(A, 1)
Y = similar(X, n, n)
if dual
#Y = AXA' - EXE'
temp = similar(Y, (n, n))
mul!(temp, A, X)
mul!(Y, temp, A')
mul!(temp, X, E')
mul!(Y, E, temp, -1, 1)
# y[:] = triu2vec(Y+transpose(Y)-Diagonal(Y))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[k] = i == j ? Y[j,j] : Y[i,j] + Y[j,i]
k += 1
end
end
end
else
# t = triu(X)-diag(X)/2
t = UpperTriangular(X) - Diagonal(X[diagind(X)] ./ 2)
# Y = A*t*A' - E*t*E'
mul!(Y, A * t, A')
mul!(Y, E * t, E', -1, 1)
# Y + Y'
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[k] = Y[i,j] + conj(Y[j,i])
k += 1
end
end
end
end
return y
end
struct InverseLyapunovMap{T,TA <: AbstractMatrix,adj,CD} <: LyapunovMatrixEquationsMaps{T}
A::TA
her::Bool
sf::Bool
function InverseLyapunovMap{T,TA,adj,CD}(A::TA, her::Bool, sf::Bool) where {T <: BlasFloat, TA <: AbstractMatrix{T}, adj, CD}
LinearAlgebra.checksquare(A)
return new{T,TA,adj,CD}(A, her, sf)
end
end
function InverseLyapunovMap(A::AbstractMatrix{T}, ::CD = Continuous(), her::Bool = false) where {T <: BlasFloat,CD <: ContinuousOrDiscrete}
schur_flag = isschur(A)
return InverseLyapunovMap{T,typeof(A),false,CD}(A, her, schur_flag)
end
function InverseLyapunovMap(A::Adjoint{T,<:AbstractMatrix{T}}, ::CD = Continuous(), her::Bool = false) where {T <: BlasFloat,CD <: ContinuousOrDiscrete}
Ap = A.parent
schur_flag = isschur(Ap)
return InverseLyapunovMap{T,typeof(Ap),true,CD}(Ap, her, schur_flag)
end
LinearAlgebra.adjoint(L::InverseLyapunovMap{<:Any,<:Any,adj,CD}) where {adj,CD} =
eltype(L) <: Complex ? InverseLyapunovMap(adj ? L.A : L.A', CD(), L.her) : LinearAlgebra.transpose(L)
# LinearAlgebra.adjoint(L::InverseLyapunovMap{<:Any,<:Any,adj,CD}) where {adj,CD} =
# InverseLyapunovMap(adj ? L.A : L.A', CD(), L.her)
# LinearAlgebra.transpose(L::InverseLyapunovMap{<:Any,<:Any,adj,CD}) where {adj,CD} =
# InverseLyapunovMap(adj ? L.A : L.A', CD(), L.her)
LinearAlgebra.inv(L::LyapunovMap{T,TA,CD}) where {T,TA,CD} =
InverseLyapunovMap(L.A, CD(), L.her)
LinearAlgebra.inv(L::InverseLyapunovMap{<:Any,<:Any,adj,CD}) where {adj,CD} =
LyapunovMap(adj ? L.A' : L.A, CD(), L.her)
"""
LINV = invlyapop(A; disc = false, her = false)
Define `LINV`, the inverse of the continuous Lyapunov operator `L:X -> AX+XA'` for `disc = false`
or the inverse of the discrete Lyapunov operator `L:X -> AXA'-X` for `disc = true`, where
`A` is an `n x n` matrix.
If `her = false` the inverse Lyapunov operator `LINV:Y -> X` maps general square matrices `Y`
into general square matrices `X`, and the associated matrix `M = Matrix(LINV)` is
``n^2 \\times n^2``.
If `her = true` the inverse Lyapunov operator `LINV:Y -> X` maps symmetric/Hermitian matrices `Y`
into symmetric/Hermitian matrices `X`, and the associated matrix `M = Matrix(LINV)` is
``n(n+1)/2 \\times n(n+1)/2``.
For the definitions of the Lyapunov operators see:
M. Konstantinov, V. Mehrmann, P. Petkov. On properties of Sylvester and Lyapunov
operators. Linear Algebra and its Applications 312:35–71, 2000.
"""
invlyapop(A::AbstractMatrix{<:BlasFloat}; disc=false, her=false) =
InverseLyapunovMap(A, ifelse(disc, Discrete(), Continuous()), her)
function invlyapop(A::AbstractMatrix; disc=false, her=false)
A = convert(AbstractMatrix{promote_type(eltype(A), Float64)}, A)
InverseLyapunovMap(A, ifelse(disc, Discrete(), Continuous()), her)
end
invlyapop(A::Schur{<:BlasFloat}; disc=false, her=false) =
InverseLyapunovMap(A.T, ifelse(disc, Discrete(), Continuous()), her)
function Base.size(L::InverseLyapunovMap)
n = size(L.A, 1)
N = L.her ? Int(n * (n + 1) / 2) : n * n
return (N, N)
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::InverseLyapunovMap{T,<:Any,adj,Discrete}, x::AbstractVector) where {T <: BlasFloat,adj}
n = size(L.A, 1)
T1 = promote_type(T, eltype(x))
try
if L.sf && !(T <: Real && T1 <: Complex)
if L.her
Y = vec2triu(-convert(AbstractVector{T1}, x), her=true)
lyapds!(L.A, Y, adj=adj)
copyto!(y, triu2vec(Y))
else
Y = reshape(-convert(AbstractVector{T1}, x), n, n)
adj ? sylvds!(-L.A, L.A, Y, adjA=true) : sylvds!(-L.A, L.A, Y, adjB=true)
copyto!(y, Y)
end
else
if L.her
Y = vec2triu(-convert(AbstractVector{T1}, x), her=true)
y .= triu2vec(lyapd(adj ? L.A' : L.A, Y))
else
Y = reshape(-convert(AbstractVector{T1}, x), n, n)
copyto!(y, lyapd(adj ? L.A' : L.A, Y))
end
end
return y
catch err
findfirst("SingularException", string(err)) === nothing &&
findfirst("LAPACKException", string(err)) === nothing ? rethrow() : throw("ME:SingularException: Singular operator")
end
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::LinearMaps.TransposeMap{<:Any,<:InverseLyapunovMap{T,<:Any,adj,Discrete}}, x::AbstractVector) where {T <: BlasFloat,adj}
n = size(L.lmap.A, 1)
T1 = promote_type(T, eltype(x))
try
if L.lmap.sf && !(T <: Real && T1 <: Complex)
if L.lmap.her
Y = vec2triu(-convert(AbstractVector{T1}, x), her=false)
adj ? sylvds!(-L.lmap.A, L.lmap.A, Y, adjB=true) : sylvds!(-L.lmap.A, L.lmap.A, Y, adjA=true)
# lyapds!(L.A, Y, adj=adj)
# copyto!(y, triu2vec(Y))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[k] = i == j ? Y[j,j] : Y[i,j] + Y[j,i]
k += 1
end
end
end
else
Y = reshape(-convert(AbstractVector{T1}, x), n, n)
adj ? sylvds!(-L.lmap.A, L.lmap.A, Y, adjB=true) : sylvds!(-L.lmap.A, L.lmap.A, Y, adjA=true)
copyto!(y, Y)
end
else
if L.lmap.her
Y1 = vec2triu(-convert(AbstractVector{T1}, x), her=false)
#y .= triu2vec(lyapd(adj ? L.A' : L.A, Y))
adj ? (Y = sylvd(-L.lmap.A, L.lmap.A', Y1)) : (Y = sylvd(-L.lmap.A', L.lmap.A, Y1))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[k] = i == j ? Y[j,j] : Y[i,j] + Y[j,i]
k += 1
end
end
end
else
Y = reshape(-convert(AbstractVector{T1}, x), n, n)
copyto!(y, lyapd(adj ? L.lmap.A : L.lmap.A', Y))
end
end
return y
catch err
findfirst("SingularException", string(err)) === nothing &&
findfirst("LAPACKException", string(err)) === nothing ? rethrow() : throw("ME:SingularException: Singular operator")
end
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::InverseLyapunovMap{T,<:Any,adj,Continuous}, x::AbstractVector) where {T <: BlasFloat,adj}
n = size(L.A, 1)
T1 = promote_type(T, eltype(x))
try
if L.sf && !(T <: Real && T1 <: Complex)
if L.her
Y = vec2triu(-convert(AbstractVector{T1}, x), her=true)
lyapcs!(L.A, Y; adj=adj)
copyto!(y, triu2vec(Y))
else
Y = copy(reshape(convert(AbstractVector{T1}, x), n, n))
adj ? (sylvcs!(L.A, L.A, Y, adjA=true)) : (sylvcs!(L.A, L.A, Y, adjB=true))
copyto!(y, Y)
end
else
if L.her
Y = vec2triu(-convert(AbstractVector{T1}, x), her=true)
y .= triu2vec(lyapc(adj ? L.A' : L.A, Y))
else
Y = reshape(-convert(AbstractVector{T1}, x), n, n)
copyto!(y, lyapc(adj ? L.A' : L.A, Y))
end
end
return y
catch err
findfirst("SingularException", string(err)) === nothing &&
findfirst("LAPACKException", string(err)) === nothing ? rethrow() : throw("ME:SingularException: Singular operator")
end
end
function LinearMaps._unsafe_mul!(y::AbstractVector, L::LinearMaps.TransposeMap{<:Any,<:InverseLyapunovMap{T,<:Any,adj,Continuous}}, x::AbstractVector) where {T <: BlasFloat,adj}
n = size(L.lmap.A, 1)
T1 = promote_type(T, eltype(x))
try
if L.lmap.sf && !(T <: Real && T1 <: Complex)
if L.lmap.her
Y = vec2triu(convert(AbstractVector{T1}, x), her=false)
adj ? (sylvcs!(L.lmap.A, L.lmap.A, Y, adjB=true)) : (sylvcs!(L.lmap.A, L.lmap.A, Y, adjA=true))
@inbounds begin
k = 1
for j = 1:n
for i = 1:j
y[k] = i == j ? Y[j,j] : Y[i,j] + Y[j,i]
k += 1
end
end
end
else
Y = copy(reshape(convert(AbstractVector{T1}, x), n, n))
adj ? (sylvcs!(L.lmap.A, L.lmap.A, Y, adjB=true)) : (sylvcs!(L.lmap.A, L.lmap.A, Y, adjA=true))
copyto!(y, Y)