MultivariateBases.Monomial — Typestruct Monomial <: AbstractMonomialIndexed endMonomial basis with the monomials of the vector monomials. For instance, SubBasis{Monomial}([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 ScaledMonomial.
MultivariateBases.ScaledMonomial — Typestruct ScaledMonomial <: AbstractMonomial endScaled 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 +SubBasis{ChebyshevFirstKind}([1, x², x⁴])
MultivariateBases.Monomial — Typestruct Monomial <: AbstractMonomialIndexed endMonomial basis with the monomials of the vector monomials. For instance, SubBasis{Monomial}([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 ScaledMonomial.
MultivariateBases.ScaledMonomial — Typestruct ScaledMonomial <: AbstractMonomial endScaled 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.
MultivariateBases.AbstractMultipleOrthogonal — Typeabstract type AbstractMultipleOrthogonal <: AbstractMonomialIndexed endPolynomial 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.
MultivariateBases.univariate_orthogonal_basis — Functionunivariate_orthogonal_basis(
B::Type{<:AbstractMultipleOrthogonal},
variable::MP.AbstractVariable,
@@ -14,4 +14,4 @@
polynomials::Vector{P}
endOrthogonal polynomial with respect to the univariate weight function $w(x) = \exp(-x^2/2)$ over the interval $[-\infty, \infty]$.
MultivariateBases.PhysicistsHermite — Typestruct PhysicistsHermite{P} <: AbstractHermite{P}
polynomials::Vector{P}
-endOrthogonal polynomial with respect to the univariate weight function $w(x) = \exp(-x^2)$ over the interval $[-\infty, \infty]$.
MultivariateBases.Laguerre — Typestruct LaguerreBasis <: AbstractMultipleOrthogonal endOrthogonal polynomial with respect to the univariate weight function $w(x) = \exp(-x)$ over the interval $[0, \infty]$.
MultivariateBases.AbstractGegenbauer — Typestruct AbstractGegenbauer <: AbstractMultipleOrthogonal endOrthogonal polynomial with respect to the univariate weight function $w(x) = (1 - x^2)^{\alpha - 1/2}$ over the interval $[-1, 1]$.
MultivariateBases.Legendre — Typestruct Legendre <: AbstractGegenbauer endOrthogonal polynomial with respect to the univariate weight function $w(x) = 1$ over the interval $[-1, 1]$.
MultivariateBases.ChebyshevFirstKind — Typestruct ChebyshevFirstKind <: AbstractChebyshev endOrthogonal polynomial with respect to the univariate weight function $w(x) = \frac{1}{\sqrt{1 - x^2}}$ over the interval $[-1, 1]$.
MultivariateBases.ChebyshevSecondKind — Typestruct ChebyshevSecondKind <: AbstractChebyshevBasis endOrthogonal polynomial with respect to the univariate weight function $w(x) = \sqrt{1 - x^2}$ over the interval $[-1, 1]$.
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