archimedes.approximation.MonomialBasis¶

class archimedes.approximation.MonomialBasis(n_basis: int)¶

Monomial (power series) basis on an interval.

The basis functions are \(\{1, t, t^2, \ldots, t^{n-1}\}\), with reference domain \([-1, 1]\). Other intervals \(x \in [a, b]\) are affinely mapped onto the reference domain.

Parameters:: n_basis (int) – Number of basis functions. The basis spans polynomials of degree 0 through n_basis - 1.

Warning

Monomial bases are inherently ill-conditioned for high degrees; prefer orthogonal polynomial bases for high-degree polynomial approximation.

See also

OrthogonalPolynomialBasis: Orthogonal polynomial families (Legendre, Chebyshev, Jacobi, Hermite, Laguerre); preferred for numerical conditioning and spectral accuracy.
FunctionSpace.monomial: Convenience constructor for a FunctionSpace built on this basis.

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__(n_basis: int) → 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.

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.