`scipy.interpolate._bsplines:BSpline.from_power_basis`

source: /scipy/interpolate/_bsplines.py :792

Summary

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

Extended Summary

For now, accepts CubicSpline instances only.

Parameters

pp : CubicSpline: A piecewise polynomial in the power basis, as created by CubicSpline
bc_type : string, 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

b : `BSpline` object: A new instance representing the initial polynomial in the B-spline basis.

Notes

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} = m = 0 \sum 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} = m = 0 \sum 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

Aliases

scipy.interpolate.BSpline.from_power_basis