scipy.interpolate.BPoly¶

class scipy.interpolate.BPoly(c, x, extrapolate=None)[source]¶

Piecewise polynomial in terms of coefficients and breakpoints

The polynomial in the i-th interval x[i] <= xp < x[i+1] is written in the Bernstein polynomial basis:

S = sum(c[a, i] * b(a, k; x) for a in range(k+1))

where k is the degree of the polynomial, and:

b(a, k; x) = comb(k, a) * t**k * (1 - t)**(k - a)

with t = (x - x[i]) / (x[i+1] - x[i]).

Parameters:

Parameters:	c : ndarray, shape (k, m, ...) Polynomial coefficients, order k and m intervals x : ndarray, shape (m+1,) Polynomial breakpoints. These must be sorted in increasing order. extrapolate : bool, optional Whether to extrapolate to ouf-of-bounds points based on first and last intervals, or to return NaNs. Default: True.

c : ndarray, shape (k, m, ...)

Polynomial coefficients, order k and m intervals

x : ndarray, shape (m+1,)

Polynomial breakpoints. These must be sorted in increasing order.

extrapolate : bool, optional

Whether to extrapolate to ouf-of-bounds points based on first and last intervals, or to return NaNs. Default: True.

See also

Notes

Properties of Bernstein polynomials are well documented in the literature. Here’s a non-exhaustive list:

[R32]

[R33]

[R34]

E. H. Doha, A. H. Bhrawy, and M. A. Saker, Boundary Value Problems, vol 2011, article ID 829546, doi:10.1155/2011/829543

Examples

>>> x = [0, 1]
>>> c = [[1], [2], [3]]
>>> bp = BPoly(c, x)

This creates a 2nd order polynomial

$B(x) = 1 \times b_{0, 2}(x) + 2 \times b_{1, 2}(x) + 3 \times b_{2, 2}(x) \\ = 1 \times (1-x)^2 + 2 \times 2 x (1 - x) + 3 \times x^2$

Attributes

x	(ndarray) Breakpoints.
c	(ndarray) Coefficients of the polynomials. They are reshaped to a 3-dimensional array with the last dimension representing the trailing dimensions of the original coefficient array.

Methods

`__call__`(x[, nu, extrapolate])	Evaluate the piecewise polynomial or its derivative
`extend`(c, x[, right])	Add additional breakpoints and coefficients to the polynomial.
`derivative`([nu])	Construct a new piecewise polynomial representing the derivative.
`construct_fast`(c, x[, extrapolate])	Construct the piecewise polynomial without making checks.
`from_power_basis`(pp[, extrapolate])	Construct a piecewise polynomial in Bernstein basis from a power basis polynomial.
`from_derivatives`(xi, yi[, orders, extrapolate])	Construct a piecewise polynomial in the Bernstein basis, compatible with the specified values and derivatives at breakpoints.