scipy.interpolate.BSpline.from_power_basis

classmethod BSpline.from_power_basis(pp, bc_type='not-a-knot')

Construct a polynomial in the B-spline basis from a piecewise polynomial in the power basis.

For now, accepts CubicSpline instances only.

Parameters

ppCubicSpline

A piecewise polynomial in the power basis, as created by CubicSpline

bc_typestring, optional

Boundary condition type as in CubicSpline: one of the not-a-knot, natural, clamped, or periodic. Necessary for construction an instance of BSpline class. Default is not-a-knot.

Returns

bBSpline object

A new instance representing the initial polynomial in the B-spline basis.

Notes

New in version 1.8.0.

Accepts only CubicSpline instances for now.

The algorithm follows from differentiation the Marsden’s identity [1]: each of coefficients of spline interpolation function in the B-spline basis is computed as follows:

\[c_j = \sum_{m=0}^{k} \frac{(k-m)!}{k!} c_{m,i} (-1)^{k-m} D^m p_{j,k}(x_i)\]

\(c_{m, i}\) - a coefficient of CubicSpline, \(D^m p_{j, k}(x_i)\) - an m-th defivative of a dual polynomial in \(x_i\).

k always equals 3 for now.

First n - 2 coefficients are computed in \(x_i = x_j\), e.g.

\[c_1 = \sum_{m=0}^{k} \frac{(k-1)!}{k!} c_{m,1} D^m p_{j,3}(x_1)\]

Last nod + 2 coefficients are computed in x[-2], nod - number of derivatives at the ends.

For example, consider \(x = [0, 1, 2, 3, 4]\), \(y = [1, 1, 1, 1, 1]\) and bc_type = natural

The coefficients of CubicSpline in the power basis:

\([[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [1, 1, 1, 1, 1]]\)

The knot vector: \(t = [0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4]\)

In this case

\[c_j = \frac{0!}{k!} c_{3, i} k! = c_{3, i} = 1,~j = 0, ..., 6\]

References

1

Tom Lyche and Knut Morken, Spline Methods, 2005, Section 3.1.2