n_eval: Evaluate N basis

In dspline: Tools for Computations with Discrete Splines

Evaluate N basis

Description

Evaluates the discrete B-spline basis of a given order, with respect to given design points, evaluated at arbitrary query points.

Usage

n_eval(k, xd, x, normalized = TRUE, knot_idx = NULL, N = NULL)

Arguments

k	Order for the discrete B-spline basis. Must be >= 0.
xd	Design points. Must be sorted in increasing order, and have length at least k+1.
x	Query points. Must be sorted in increasing order.
normalized	Should the discrete B-spline basis vectors be normalized to attain a maximum value of 1 over the design points? The default is TRUE.
knot_idx	Vector of indices, a subset of (k+1):(n-1) where n = length(xd), that indicates which design points should be used as knot points for the discrete B-splines. Must be sorted in increasing order. The default is NULL, which is taken to mean (k+1):(n-1).
N	Matrix of discrete B-spline evaluations at the design points. The default is NULL, which means that this is precomputed before constructing the matrix of discrete B-spline evaluations at the query points. If N is non-NULL, then the argument normalized will be ignored (as this would have only been used to construct N at the design points).

Details

The discrete B-spline basis functions of order k, defined with respect to design points x_1 < \ldots < x_n, are denoted \eta^k_1, \ldots, \eta^k_n. For a discussion of their properties and further references, see the help file for n_mat(). The current function produces a matrix of evaluations of the discrete B-spline basis at an arbitrary sequence of query points. For each query point x, this matrix has a corresponding row with entries:

\eta^k_j(x), \; j = 1, \ldots, n.

Unlike the falling factorial basis, the discrete B-spline basis is not generally available in closed-form. Therefore, the current function (unlike h_eval()) will first check if it should precompute the evaluations of the discrete B-spline basis at the design points. If the argument N is non-NULL, then it will use this as the matrix of evaluations at the design points; if N is NULL, then it will call n_mat() to produce such a matrix, and will pass to this function the arguments normalized and knot_idx accordingly.

After obtaining the matrix of discrete B-spline evaluations at the design points, the fast interpolation scheme from dspline_interp() is used to produce evaluations at the query points.

Value

Sparse matrix of dimension length(x) by length(knot_idx) + k + 1.

See Also

n_mat() for constructing evaluations of the discrete B-spline basis at the design points.

Examples

xd = 1:10 / 10
x = 1:9 / 10 + 0.05
n_mat(2, xd, knot_idx = c(3, 5, 7))
n_eval(2, xd, x, knot_idx = c(3, 5, 7))

dspline documentation built on June 8, 2025, 9:40 p.m.