bsplev_single#

rateslib.splines.bsplev_single(x, i, k, t, org_k)#

Calculate the m th order derivative from the right of an indexed b-spline at x.

Parameters:

  • x (float) – The x value at which to evaluate the b-spline.

  • i (int) – The index of the b-spline to evaluate.

  • k (int) – The order of the b-spline (note that k=4 is a cubic spline).

  • t (sequence of float) – The knot sequence of the pp spline.

  • org_k (int, optional) – The original k input. Used only internally when recursively calculating successive b-splines. Users will not typically use this parameters.

Notes

B-splines can be recursively defined as:

\[B_{i,k,\mathbf{t}}(x) = \frac{x-t_i}{t_{i+k-1}-t_i}B_{i,k-1,\mathbf{t}}(x) + \frac{t_{i+k}-x}{t_{i+k}-t_{i+1}}B_{i+1,k-1,\mathbf{t}}(x)\]

and such that the basic, stepwise, b-spline or order 1 are:

\[\begin{split}B_{i,1,\mathbf{t}}(x) = \left \{ \begin{matrix} 1, & t_i \leq x < t_{i+1} \\ 0, & \text{otherwise} \end{matrix} \right .\end{split}\]

For continuity on the right boundary the rightmost basic b-spline is also set equal to 1 there: \(B_{n,1,\mathbf{t}}(t_{n+k})=1\).