MultivariateBases

abstract type AbstractPolynomialBasis end

Polynomial basis of a subspace of the polynomials [Section~3.1.5, BPT12].

[BPT12] Blekherman, G.; Parrilo, P. A. & Thomas, R. R. Semidefinite Optimization and Convex Algebraic Geometry. Society for Industrial and Applied Mathematics, 2012.

maxdegree_basis(B::Type{<:AbstractPolynomialBasis}, variables, maxdegree::Int)

Return the basis of type B generating all polynomials of degree up to maxdegree with variables variables.

basis_covering_monomials(B::Type{<:AbstractPolynomialBasis}, monos::AbstractVector{<:AbstractMonomial})

Return the minimal basis of type B that can generate all polynomials of the monomial basis generated by monos.

Examples

For example, to generate all the polynomials with nonzero coefficients for the monomials x^4 and x^2, we need three polynomials as otherwise, we generate polynomials with nonzero constant term.

julia> using DynamicPolynomials

julia> @polyvar x
(x,)

julia> basis_covering_monomials(ChebyshevBasis, [x^2, x^4])
ChebyshevBasisFirstKind([1.0, -1.0 + 2.0x², 1.0 - 8.0x² + 8.0x⁴])

struct FixedPolynomialBasis{PT<:MP.AbstractPolynomialLike, PV<:AbstractVector{PT}} <: AbstractPolynomialBasis
    polynomials::PV
end

Polynomial basis with the polynomials of the vector polynomials. For instance, FixedPolynomialBasis([1, x, 2x^2-1, 4x^3-3x]) is the Chebyshev polynomial basis for cubic polynomials in the variable x.

struct OrthonormalCoefficientsBasis{PT<:MP.AbstractPolynomialLike, PV<:AbstractVector{PT}} <: AbstractPolynomialBasis
    polynomials::PV
end

Polynomial basis with the polynomials of the vector polynomials that are orthonormal with respect to the inner produce derived from the inner product of their coefficients. For instance, FixedPolynomialBasis([1, x, 2x^2-1, 4x^3-3x]) is the Chebyshev polynomial basis for cubic polynomials in the variable x.

struct MonomialBasis{MT<:MP.AbstractMonomial, MV<:AbstractVector{MT}} <: AbstractPolynomialBasis
    monomials::MV
end

Monomial basis with the monomials of the vector monomials. For instance, MonomialBasis([1, x, y, x^2, x*y, y^2]) is the monomial basis for the subspace of quadratic polynomials in the variables x, y.

This basis is orthogonal under a scalar product defined with the complex Gaussian measure as density. Once normalized so as to be orthonormal with this scalar product, one get ths ScaledMonomialBasis.

struct ScaledMonomialBasis{MT<:MP.AbstractMonomial, MV<:AbstractVector{MT}} <: AbstractPolynomialBasis
    monomials::MV
end

Scaled monomial basis (see [Section 3.1.5, BPT12]) with the monomials of the vector monomials. Given a monomial $x^\alpha = x_1^{\alpha_1} \cdots x_n^{\alpha_n}$ of degree $d = \sum_{i=1}^n \alpha_i$, the corresponding polynomial of the basis is

\[{d \choose \alpha}^{\frac{1}{2}} x^{\alpha} \quad \text{ where } \quad {d \choose \alpha} = \frac{d!}{\alpha_1! \alpha_2! \cdots \alpha_n!}.\]

For instance, create a polynomial with the basis $[xy^2, xy]$ creates the polynomial $\sqrt{3} a xy^2 + \sqrt{2} b xy$ where a and b are new JuMP decision variables. Constraining the polynomial $axy^2 + bxy$ to be zero with the scaled monomial basis constrains a/√3 and b/√2 to be zero.

This basis is orthonormal under the scalar product:

\[\langle f, g \rangle = \int_{\mathcal{C}^n} f(z) \overline{g(z)} d\nu_n\]

where $\nu_n$ is the Gaussian measure on $\mathcal{C}^n$ with the density $\pi^{-n} \exp(-\lVert z \rVert^2)$. See [Section 4; B07] for more details.

[BPT12] Blekherman, G.; Parrilo, P. A. & Thomas, R. R. Semidefinite Optimization and Convex Algebraic Geometry. Society for Industrial and Applied Mathematics (2012).

[B07] Barvinok, Alexander. Integration and optimization of multivariate polynomials by restriction onto a random subspace. Foundations of Computational Mathematics 7.2 (2007): 229-244.

abstract type AbstractMultipleOrthogonalBasis{P} <: AbstractPolynomialVectorBasis{P, Vector{P}} end

Polynomial basis such that $\langle p_i(x), p_j(x) \rangle = 0$ if $i \neq j$ where

\[\langle p(x), q(x) \rangle = \int p(x)q(x) w(x) dx\]

where the weight is a product of weight functions $w(x) = w_1(x_1)w_2(x_2) \cdots w_n(x_n)$ in each variable. The polynomial of the basis are product of univariate polynomials: $p(x) = p_1(x_1)p_2(x_2) \cdots p_n(x_n)$. where the univariate polynomials of variable x_i form an univariate orthogonal basis for the weight function w_i(x_i). Therefore, they satisfy the recurrence relation

\[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

where reccurence_first_coef gives a_k, reccurence_second_coef gives b_k, reccurence_third_coef gives c_k and reccurence_deno_coef gives d_k.

univariate_orthogonal_basis(B::Type{<:AbstractMultipleOrthogonalBasis},
                            variable::MP.AbstractVariable, degree::Integer)

Return the vector of univariate polynomials of the basis B up to degree with variable variable.

reccurence_first_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)

Return a_{degree} in recurrence equation

\[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

reccurence_second_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)

Return b_{degree} in recurrence equation

\[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

reccurence_third_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)

Return c_{degree} in recurrence equation

\[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

reccurence_deno_coef(B::Type{<:AbstractMultipleOrthogonalBasis}, degree::Integer)

Return d_{degree} in recurrence equation

\[d_k p_k(x_i) = (a_k x_i + b_k) p_{k-1}(x_i) + c_k p_{k-2}(x_i)\]

struct ProbabilistsHermiteBasis{P} <: AbstractHermiteBasis{P}
    polynomials::Vector{P}
end

Orthogonal polynomial with respect to the univariate weight function $w(x) = \exp(-x^2/2)$ over the interval $[-\infty, \infty]$.

struct PhysicistsHermiteBasis{P} <: AbstractHermiteBasis{P}
    polynomials::Vector{P}
end

Orthogonal polynomial with respect to the univariate weight function $w(x) = \exp(-x^2)$ over the interval $[-\infty, \infty]$.

struct LaguerreBasis{P} <: AbstractMultipleOrthogonalBasis{P}
    polynomials::Vector{P}
end

Orthogonal polynomial with respect to the univariate weight function $w(x) = \exp(-x)$ over the interval $[0, \infty]$.

struct AbstractGegenbauerBasis{P} <: AbstractMultipleOrthogonalBasis{P}
    polynomials::Vector{P}
end

Orthogonal polynomial with respect to the univariate weight function $w(x) = (1 - x^2)^{\alpha - 1/2}$ over the interval $[-1, 1]$.

struct LegendreBasis{P} <: AbstractGegenbauerBasis{P}
    polynomials::Vector{P}
end

Orthogonal polynomial with respect to the univariate weight function $w(x) = 1$ over the interval $[-1, 1]$.

struct ChebyshevBasisFirstKind{P} <: AbstractChebyshevBasis{P}
    polynomials::Vector{P}
end

Orthogonal polynomial with respect to the univariate weight function $w(x) = \frac{1}{\sqrt{1 - x^2}}$ over the interval $[-1, 1]$.

struct ChebyshevBasisSecondKind{P} <: AbstractChebyshevBasis{P}
    polynomials::Vector{P}
end

Orthogonal polynomial with respect to the univariate weight function $w(x) = \sqrt{1 - x^2}$ over the interval $[-1, 1]$.