CarlemanLinearization.build_matrixMethod
build_matrix(F₁, F₂, N; compress=false)

Compute the Carleman linearization matrix associated to the quadratic system $x' = F₁x + F₂(x⊗x)$, truncated at order $N$.

Input

  • F₁ – sparse matrix of size $n × n$
  • F₂ – sparse matrix of size $n × n^2$
  • N – integer representing the truncation order, should be at least two
  • compress – if true produces the matrix corresponding to the commutative monomials only

Output

Sparse matrix A.

Algorithm

See references [1] and [2] of CARLIN.md.

CarlemanLinearization.build_matrix_compressedMethod
build_matrix_compressed(F₁, F₂, N)

Compute the compressed Carleman linearization matrix associated to the quadratic system $x' = F₁x + F₂(x⊗x)$, truncated at order $N$.

Input & Output are the same as for build_matrix.

CarlemanLinearization.generate_monomialsMethod
generate_monomials(n, N)

Return a list of n-tuples of nonegative integers with the sum at most N, ordered by the total degree (no other guarantees on the ordering).

CarlemanLinearization.kron_idMethod
kron_id(n, k)

Compute the k-fold Kronecker product of the identity: I ⊗ I ⊗ ... ⊗ I, k times.

Input

  • n – integer representing the order (dimension of the identity)
  • k – integer representing the power

Output

A Diagonal matrix with element 1 in the diagonal and order $n^k$.

Examples

julia> kron_id(2, 2)
4×4 Diagonal{Float64,Array{Float64,1}}:
 1.0   ⋅    ⋅    ⋅
  ⋅   1.0   ⋅    ⋅
  ⋅    ⋅   1.0   ⋅
  ⋅    ⋅    ⋅   1.0
CarlemanLinearization.kron_powMethod
kron_pow(x::Vector{<:AbstractVariable}, pow::Int)

Compute the higher order concrete Kronecker power: x ⊗ x ⊗ ... ⊗ x, pow times for a vector of symbolic monomials.

Input

  • x – polynomial variable
  • pow – integer

Output

Vector of multivariate monomial corresponding to x^{⊗ pow}.

Examples

julia> using DynamicPolynomials

julia> @polyvar x[1:2]
(PolyVar{true}[x₁, x₂],)

julia> x
2-element Vector{PolyVar{true}}:
 x₁
 x₂

julia> kron_pow(x, 2)
4-element Vector{Monomial{true}}:
 x₁²
 x₁x₂
 x₁x₂
 x₂²
CarlemanLinearization.kron_sandwichMethod
kron_sandwich(F, n, k1, k2)

Compute A ⊗ F ⊗ C where A = I^{⊗ k1} and C = I^{⊗ k2} where I is the identity matrix of order n, the k1-fold and k2-fold Kronecker product of the identity and F in between.

Input

  • F – matrix
  • n – integer, dimension of the identity
  • k1 – nonnegative integer
  • k2 – nonnegative integer

Output

The kronecker product I^{⊗ k1} ⊗ F ⊗ I^{⊗ k2}, represented as a sparse matrix if F is sparse.

Examples

julia> F = sparse([0 1; -1 0.])
2×2 SparseMatrixCSC{Float64,Int64} with 2 stored entries:
  [2, 1]  =  -1.0
  [1, 2]  =  1.0

julia> Q = kron_sandwich(F, 2, 2, 2);

julia> size(Q)
(32, 32)

julia> I2 = I(2)
IdentityMultiple{Float64} of value 1.0 and order 2

julia> Q == reduce(kron, [I2, I2, F, I2, I2])
true
CarlemanLinearization.kron_sumMethod
kron_sum(F::AbstractMatrix, k::Int)

Compute the Kronecker sum of order k defined as:

\[ F ⊕ F ⊕ ... ⊕ F := F ⊗ I ⊗ ... ⊗ I + I ⊗ F ⊗ I ⊗ ... ⊗ I + ... + I ⊗ ... ⊗ I ⊗ F\]

where each term has k products and there are a total of k summands and I is the identity matrix of order n.

Examples

It holds that:

  • $k = 1: F$
  • $k = 2: F ⊗ I + I ⊗ F$
  • $k = 3: F ⊗ I ⊗ I + I ⊗ F ⊗ I + I ⊗ I ⊗ F$
julia> F = sparse([0 1; -1 0.])
2×2 SparseMatrixCSC{Float64,Int64} with 2 stored entries:
  [2, 1]  =  -1.0
  [1, 2]  =  1.0

julia> kron_sum(F, 1) == F
true

julia> I2 = I(2)
IdentityMultiple{Float64} of value 1.0 and order 2

julia> kron_sum(F, 2) == kron(F, I2) + kron(I2, F)
true

julia> kron_sum(F, 3) == sum(reduce(kron, X) for X in [[F, I2, I2], [I2, F, I2], [I2, I2, F]])
true
CarlemanLinearization.quadratic_matrix_formMethod
quadratic_matrix_form(f, vars)

Extract the linear and quadratic matrices of a system of polynomial equations.

Input

  • f – vector of polynomials
  • vars – vector of variables

Output

Matrices F1 and F2 of size n × n and n × n^2 respectively, such that F1 contains the linear coefficients of the polynomial vector field f and F2 the quadratic coefficients, which indexing that respects the Kronecker power vars ⊗ vars.

Examples

TO-DO