Polydim::Interpolation::Lagrange Namespace Reference

Functions
Eigen::VectorXd	Lagrange_1D_coefficients (const Eigen::VectorXd &interpolation_points_x)
	Compute the barycentric weights of the 1D Lagrange basis.
Eigen::MatrixXd	Lagrange_1D_values (const Eigen::VectorXd &interpolation_points_x, const Eigen::VectorXd &lagrange_1D_coefficients, const Eigen::VectorXd &evaluation_points_x)
	Evaluate the 1D Lagrange basis functions at given points.
Eigen::MatrixXd	Lagrange_1D_derivative_values (const Eigen::VectorXd &interpolation_points_x, const Eigen::VectorXd &lagrange_1D_coefficients, const Eigen::VectorXd &evaluation_points_x)
	Evaluate the derivatives of the 1D Lagrange basis functions at given points.

Function Documentation

Lagrange_1D_coefficients()

Eigen::VectorXd Polydim::Interpolation::Lagrange::Lagrange_1D_coefficients

(

const Eigen::VectorXd &

interpolation_points_x

)

Compute the barycentric weights of the 1D Lagrange basis.

Returns the weights \(w_i = \dfrac{1}{\prod_{j \neq i} (x_i - x_j)}\) associated with the interpolation nodes \(\{x_i\}\). These are computed once and reused by Lagrange_1D_values() and Lagrange_1D_derivative_values() to evaluate the basis and its derivatives efficiently.

Parameters

interpolation_points_x

Interpolation nodes \(\{x_i\}\).

Returns

The vector of barycentric weights \(w_i\) (empty if no nodes are given).

Lagrange_1D_derivative_values()

Eigen::MatrixXd Polydim::Interpolation::Lagrange::Lagrange_1D_derivative_values	(	const Eigen::VectorXd &	interpolation_points_x,
		const Eigen::VectorXd &	lagrange_1D_coefficients,
		const Eigen::VectorXd &	evaluation_points_x
	)

Evaluate the derivatives of the 1D Lagrange basis functions at given points.

Computes \(\ell_i'(x) = w_i \sum_{j \neq i} \prod_{k \neq i,\,j} (x - x_k)\) (product rule applied to Lagrange_1D_values()) for every evaluation point and every basis function. With a single interpolation node the derivative is identically zero.

Parameters

interpolation_points_x	Interpolation nodes \(\{x_i\}\).
lagrange_1D_coefficients	Barycentric weights from Lagrange_1D_coefficients().
evaluation_points_x	Points at which to evaluate the derivatives.

Returns

A \(N_{\text{eval}} \times N_{\text{nodes}}\) matrix whose \((p, i)\) entry is \(\ell_i'\) evaluated at the \(p\)-th point.

Lagrange_1D_values()

Eigen::MatrixXd Polydim::Interpolation::Lagrange::Lagrange_1D_values	(	const Eigen::VectorXd &	interpolation_points_x,
		const Eigen::VectorXd &	lagrange_1D_coefficients,
		const Eigen::VectorXd &	evaluation_points_x
	)

Evaluate the 1D Lagrange basis functions at given points.

Computes \(\ell_i(x) = w_i \prod_{j \neq i} (x - x_j)\) for every evaluation point \(x\) and every basis function \(\ell_i\), where \(w_i\) are the precomputed barycentric weights. With a single interpolation node the basis reduces to the constant \(1\).

Parameters

interpolation_points_x	Interpolation nodes \(\{x_i\}\).
lagrange_1D_coefficients	Barycentric weights from Lagrange_1D_coefficients().
evaluation_points_x	Points at which to evaluate the basis.

Returns

A \(N_{\text{eval}} \times N_{\text{nodes}}\) matrix whose \((p, i)\) entry is \(\ell_i\) evaluated at the \(p\)-th point.