Series

Series

Series{C,M}

Class representing multivariate series. The series is a dictionary, which associates values of type C to monomials of type M.

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TensorDec.dualFunction.
dual(p::Polynomial, d:: Int64) -> Series{T}

Compute the series associated to the tensor p of degree d. T is the type of the coefficients of the polynomial p.

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dual(p::Polynomial) -> Series{T}

Compute the series associated to the polynomial p, replacing the variables xi by its dual dxi. T is the type of the coefficients of the polynomial p.

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dot 
TensorDec.momentFunction.
moment(w::Vector{T}, P::Matrix{T}) -> Vector{Int64} -> T

Compute the moment function $α -> ∑_{i} ω_{i} P_{i}^α$ associated to the sequence P of r points of dimension n, which is a matrix of size r*n and the weights w.

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moment(p::Polynomial, zeta::Vector{T}) -> Vector{Int64} -> T

Compute the moment function $α \rightarrow p(ζ^α)$.

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TensorDec.seriesFunction.
series(f,L) -> Series{T}

Compute the generating series $\sum_{x^{α} \in L} f(α) z^α$ for a function $f: \mathbb{N}^n \rightarrow T$ and a sequence L of monomials.

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series(w:: Vector{T}, P::Matrix{T}, L::Vector{M}) -> Series{T}

Compute the series of the moment sequence $∑_{i} ω_{i} P_{i}^α$ for $α \in L$.

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series(w:: Vector{T}, P::Matrix{T}, X, d::Int64) -> Series{T}

Compute the series of the moment sequence $∑_i ω_{i} P_{i}^α$ for $|α| \leq d$.

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series(p::Polynomial, zeta, X, d::Int64) -> Series{T}

Compute the series of moments $p(ζ^α)$ for $|α| \leq d$.

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series(H::Matrix{C}, L1::Vector{M}, L2::Vector{M}) -> Series{C,M}

Compute the series associated to the Hankel matrix H, with rows (resp. columns) indexed by the array of monomials L1 (resp. L2).

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TensorDec.hankelFunction.
hankel(σ::Series{T}, L1, L2) -> Array{T,2}

Hankel matrix of $σ$ with the rows indexed by the list of polynomials L1 and the columns by L2. The entries are the dot product for $σ$ of the corresponding elements in L1 and L2.

Example

julia> L =[1, x1, x2, x1^2, x1*x2, x2^2]

julia> H = hankel(s,L,L)
6x6 Array{Float64,2}:
  4.0   5.0   7.0    5.0  11.0  13.0
  5.0   5.0  11.0   -1.0  17.0  23.0
  7.0  11.0  13.0   17.0  23.0  25.0
  5.0  -1.0  17.0  -31.0  23.0  41.0
 11.0  17.0  23.0   23.0  41.0  47.0
 13.0  23.0  25.0   41.0  47.0  49.0
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Compute the Hankel matrix (a.k.a. Catalecticant matrix) in degree d of the symmetric tensor F.

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maxdegree