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MultivariateSeries

Package for the decomposition of tensors and polynomial-exponential series.

Introduction

The package MultivariateSeries.jl provides tools for the manipulation of sequences (\sigma_{\alpha})_{\alpha} \in \mathbb{K}^{\mathbb{N}^{n}} indexed by multivariate indices \alpha \in {\mathbb{N}^{n}} which are represented as series:

$$\sigma(\mathbf{z}) = \sum_{\alpha \in {\mathbb{N}^{n}}} \sigma_{\alpha} \mathbf{y}^{\alpha}$$

The sequence \sigma or the series \sigma(y) represents a linear functional on the polynomials:

$$\sigma: p= \sum_{\alpha \in \mathbb{N}^n} p_{\alpha} \mathbf{x}^{\alpha} \mapsto \sum_{\alpha \in \mathbb{N}^n} p_{\alpha} \sigma_{\alpha}$$

The series are implemented as association tables (or dictionnaries) between (a finite set of) monomials and coefficients. They are printed using dual variables dxi=$y_i$:

using MultivariateSeries
X = @ring x1 x2

julia> p = (1+x1)^3 + 0.5*(1+x1+x2)^2
x1³ + 3.5x1² + x1x2 + 0.5x2² + 4.0x1 + x2 + 1.5

julia> sigma = dual(p)
dx1^3 + 1.5 + 3.5dx1^2 + 4.0dx1 + dx2 + dx1*dx2 + 0.5dx2^2

Since the series is represented by a table, the order in which the dual monomials are printed is the order used in the table. It is not necessarily sorted by a monomial ordering.

julia> sigma.terms
Dict{Monomial{true},Float64} with 7 entries:
   x1³  => 1.0
   1    => 1.5
   x1²  => 3.5
   x1   => 4.0
   x2   => 1.0
   x1x2 => 1.0
   x2²  => 0.5

Series act as linear functionals on polynomials via the dot product:

julia> dot(sigma,x1^2)
3.5
julia> dot(sigma,x2^4)
0.0

Polynomial-exponential decomposition

The package provide tools for solving the following decomposition problem:

Given (the first terms of) sequence \sigma \in \mathbb{K}^{\mathbb{N}^{n}} or the series \sigma(\mathbf{y}) \in \mathbb{K}[[\mathbf{y}]], we want to decompose it as polynomial-exponential series

$$\sigma(\mathbf{y}) = \sum_{i=1}^r \omega_i(\mathbf{y}) e^{\xi_{i,1} y_1+ \cdots + \xi_{i,n} y_n}$$

with polynomials \omega_{i}(\mathbf{y}) and points \xi_{i}= (\xi_{i,1}, \ldots, \xi_{i,n})\in \mathbb{K}^{n}. \omega_i are called the weights and \xi_i the frequencies of the decomposition.

These types of decompositions appear in many problems (see [Examples](@ref sec_examples)).

The package MultivariateSeries provides functions to manipulate (truncated) series, to construct truncated Hankel matrices, and to compute such a decomposition from these Hankel matrices.

[Examples](@id sec_examples)

Pages = map(file -> joinpath("expl", file), filter(x ->endswith(x, "md"), readdir("expl")))

Functions and types

Pages = map(file -> joinpath("code", file), filter(x ->endswith(x, "md"), readdir("code")))

[Installation](@id sec_installation)

The package is available at https://github.com/AlgebraicGeometricModeling/MultivariateSeries.jl.git

To install it from Julia:

] add https://github.com/AlgebraicGeometricModeling/MultivariateSeries.jl.git

It can then be used as follows:

using MultivariateSeries

See the [Examples](@ref sec_examples) for more details.

Dependencies

The package MultivariateSeries depends on the following packages:

  • DynamicPolynomials package on multivariate polynomials represented as lists of monomials.
  • MultivariatePolynomials generic interface package for multivariate polynomials.

These packages will be installed with MultivariateSeries (see [installation](@ref sec_installation)).