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# scipy.interpolate.BPoly

class scipy.interpolate.BPoly(c, x, extrapolate=None, axis=0)

  Piecewise polynomial in terms of coefficients and breakpoints.

  The polynomial between `x[i]` and `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) = binom(k, a) * t**a * (1 - t)**(k - a),

  with `t = (x - x[i]) / (x[i+1] - x[i])` and `binom` is the binomial coefficient.

  Parameters:
    c ndarray, shape (k, m, …)

      Polynomial coefficients, order k and m intervals

    x ndarray, shape (m+1,)

      Polynomial breakpoints. Must be sorted in either increasing or decreasing order.

    extrapolate bool, optional

      If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If ‘periodic’, periodic extrapolation is used. Default is True.

    axis int, optional

      Interpolation axis. Default is zero.

  See also

  `PPoly`

    piecewise polynomials in the power basis

  Notes

  Properties of Bernstein polynomials are well documented in the literature, see for example [1] [2] [3].

  References

  [1]

  https://en.wikipedia.org/wiki/Bernstein_polynomial

  [2]

  Kenneth I. Joy, Bernstein polynomials, http://www.idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials.pdf

  [3]

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

  Examples

      >>> from scipy.interpolate import BPoly
      >>> x = [0, 1]
      >>> c = [[1], [2], [3]]
      >>> bp = BPoly(c, x)

  This creates a 2nd order polynomial

  \[\begin{split}B(x) = 1 \times b_{0, 2}(x) + 2 \times b_{1, 2}(x) + 3 \times b_{2, 2}(x) \\ = 1 \times (1-x)^2 + 2 \times 2 x (1 - x) + 3 \times x^2\end{split}\]
  Attributes:
    x ndarray

      Breakpoints.

    c ndarray

      Coefficients of the polynomials. They are reshaped to a 3-D array with the last dimension representing the trailing dimensions of the original coefficient array.

    axis int

      Interpolation axis.

  Methods

  `__call__`(x[, nu, extrapolate]) | Evaluate the piecewise polynomial or its derivative.
  `extend`(c, x) | Add additional breakpoints and coefficients to the polynomial.
  `derivative`([nu]) | Construct a new piecewise polynomial representing the derivative.
  `antiderivative`([nu]) | Construct a new piecewise polynomial representing the antiderivative.
  `integrate`(a, b[, extrapolate]) | Compute a definite integral over a piecewise polynomial.
  `construct_fast`(c, x[, extrapolate, axis]) | 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.