from_power_basis#
- classmethod BSpline.from_power_basis(pp, bc_type='not-a-knot')[source]#
- Construct a polynomial in the B-spline basis from a piecewise polynomial in the power basis. - For now, accepts - CubicSplineinstances 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- BSplineclass. Default is- not-a-knot.
 
- Returns:
- bBSplineobject
- A new instance representing the initial polynomial in the B-spline basis. 
 
- b
 - Notes - Added in version 1.8.0. - Accepts only - CubicSplineinstances 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 \(x_i\). - kalways equals 3 for now.- First - n - 2coefficients 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 + 2coefficients 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