archimedes.approximation.Basis

class archimedes.approximation.Basis

A finite-dimensional family of basis functions \(\{\phi_i\}_{i=1}^n\).

Basis only defines the basis functions and evaluates them, without any notion of a target domain or coefficient vector. FunctionSpace combines a Basis with a domain and quadrature rule to define an inner product space, and Function further combines a FunctionSpace with expansion coefficients.

Available implementations include:

• BSplineBasis: B-spline basis functions

• ConcatBasis: Concatenation of multiple bases

• ConstrainedBasis: Constrained linear combination of another basis

• CubicHermiteBasis: Cubic Hermite basis functions

• FourierBasis: Trigonometric basis functions on a periodic interval

• LagrangeBasis: Lagrange polynomial basis functions

• MonomialBasis: Monomial (power series) basis functions

• OrthogonalPolynomialBasis: Classical orthogonal polynomial bases

• PiecewiseBasis: Basis constructed by tiling multiple local bases

• TensorBasis: Multi-dimensional basis formed by a tensor product

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.
n_basis	Number of basis functions in this basis.

__init__()

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.

abstractmethod evaluate(

x: ndarray,

deriv: int = 0,

*,

side: str = RIGHT,

**domain_kwargs,

) → ndarray

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

abstract property Parameters: type[ReferenceDomain.Parameters]

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.

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.