archimedes.approximation.OrthogonalPolynomialBasis

class archimedes.approximation.OrthogonalPolynomialBasis(

measure: Measure,

n_basis: int,

density: bool = False,

)

A classical orthonormal polynomial basis

Orthonormal polynomials \(\{p_0, p_1, \ldots, p_{n-1}\}\), constructed to be orthonormal with respect to a given measure (reference domain and inner product weight).

The monic polynomials orthogonal w.r.t. any Measure satisfy the three-term recurrence

\[\pi_{k+1}(x) = (x - \alpha_k) \, \pi_k(x) - \beta_k \, \pi_{k-1}(x)\]

which holds for any classical orthogonal polynomial family. The squared norm for the associated weight function \(w(x)\) is \(\int \pi_k^2 \, w \, dx = \beta_0 \beta_1 \cdots \beta_k\).

The basis functions are additionally normalized with \(p_k = \pi_k / \sqrt{\beta_0 \cdots \beta_k}\) for conditioning at high polynomial degree.

Supported measures include:

• LegendreMeasure

• JacobiMeasure (also Chebyshev as a special case)

• PhysicistsHermiteMeasure

• ProbabilistsHermiteMeasure

• LaguerreMeasure

However, customized measures may be constructed by defining a domain and weight function, with recursions computed automatically using the discretized Stieltjes procedure. See the [documentation on quadrature](handbook/quadrature) for details.

An orthogonal polynomial basis can be constructed from any such custom measure, with the basis functions defined automatically via the three-term recurrence.

Methods

boundary_dofs([order])	Indices of the degrees of freedom at the left/right ends of the domain.
evaluate(x[, deriv, side])	Evaluate all n_basis basis functions at x.

Attributes

Parameters	The target-domain parameters this basis expects.
density	Whether this basis is orthonormal with respect to a probability measure (unit mass) rather than the raw weight of the associated Measure.
ndim	Number of independent variables the basis functions take.
measure	Defines the orthogonality weight and reference domain.
n_basis	Number of basis functions in this basis.

__init__(

measure: Measure,

n_basis: int,

density: bool = False,

) → None

boundary_dofs(order: int = 0) → tuple[int | None, int | None]

Indices of the degrees of freedom at the left/right ends of the domain.

Can be used for example to set boundary conditions or enforce continuity at the domain endpoints.

Parameters:

order (int, optional) – Derivative order to look up. Default 0 (the endpoint value).

Returns:

left, right – Index into this basis’s functions, or None where there is no degree of freedom of that order at that end.

Return type:

int or None

Notes

Only meaningful for nodal families (e.g. LagrangeBasis) with nodes at the endpoints. Modal families (e.g. OrthogonalPolynomialBasis) and nodal bases with interior-only nodes (Gauss-Legendre points, for instance) do not have boundary DOFs.

evaluate(

x,

deriv: int = 0,

*,

side: str = RIGHT,

**domain_kwargs,

)

Evaluate all n_basis basis functions at x.

Parameters:

• x (array_like) – Evaluation points, shape (npts,).

• deriv (int, optional) – Order of derivative to evaluate. Default 0.

• side ({"right", "left"}, optional) – Which one-sided limit to take where the basis is two-valued. Default "right". Irrelevant for smooth bases.

• **domain_kwargs – Target-domain parameters; see the subclass docstring.

Returns:

phi – Basis values (or deriv-th derivatives), shape (npts, n_basis).

Return type:

ndarray

property Parameters: type

The target-domain parameters this basis expects.

A Parameters subclass (e.g. UnitInterval.Parameters). Returns the type, not an instance.

Determines which reference domain the basis is defined on, for example:

• UnitInterval for \([-1, 1]\).

• HalfLine for \([0, \infty)\).

• RealLine for \((-\infty, \infty)\).

density: bool = False

Whether this basis is orthonormal with respect to a probability measure (unit mass) rather than the raw weight of the associated Measure.

Only meaningful for orthogonal polynomial families based on a Measure (in particular OrthogonalPolynomialBasis); other families should leave this False.

measure: Measure

Defines the orthogonality weight and reference domain.

n_basis: int

Number of basis functions in this basis.

ndim: int = 1

Number of independent variables the basis functions take.

Typically 1, since most bases are univariate. TensorBasis is the exception, with one variable per tensored factor.