`scipy.interpolate._interpolate:BPoly._construct_from_derivatives`

source: /scipy/interpolate/_interpolate.py :1753

Signature

staticmethod def _construct_from_derivatives ( xa , xb , ya , yb )

Summary

Compute the coefficients of a polynomial in the Bernstein basis given the values and derivatives at the edges.

Extended Summary

Return the coefficients of a polynomial in the Bernstein basis defined on [xa, xb] and having the values and derivatives at the endpoints xa and xb as specified by ya and yb. The polynomial constructed is of the minimal possible degree, i.e., if the lengths of ya and yb are na and nb, the degree of the polynomial is na + nb - 1.

Parameters

xa : float: Left-hand end point of the interval
xb : float: Right-hand end point of the interval
ya : array_like: Derivatives at xa. ya[0] is the value of the function, and ya[i] for i > 0 is the value of the ith derivative.
yb : array_like: Derivatives at xb.

Returns

: array: coefficient array of a polynomial having specified derivatives

Notes

This uses several facts from life of Bernstein basis functions. First of all,

$b_{a, n}^{'} = n (b_{a - 1, n - 1} - b_{a, n - 1})$

If B(x) is a linear combination of the form

$B (x) = a = 0 \sum n c_{a} b_{a, n},$

then :math: B'(x) = n sum_{a=0}^{n-1} (c_{a+1} - c_{a}) b_{a, n-1}. Iterating the latter one, one finds for the q-th derivative

$B^{q} (x) = n! / (n - q)! a = 0 \sum n - q Q_{a} b_{a, n - q},$

with

$Q_{a} = j = 0 \sum q (-)^{j + q} co mb (q, j) c_{j + a}$

This way, only a=0 contributes to :math: B^{q}(x = xa), and c_q are found one by one by iterating q = 0, ..., na.

At x = xb it's the same with a = n - q.

Aliases

scipy.interpolate.BPoly._construct_from_derivatives