Skip to main content
`Ctrl`+`K`
scipy.interpolate.

# spalde

scipy.interpolate.spalde(x, tck)

  Evaluate a B-spline and all its derivatives at one point (or set of points) up to order k (the degree of the spline), being 0 the spline itself.

  Legacy

  This function is considered legacy and will no longer receive updates. While we currently have no plans to remove it, we recommend that new code uses more modern alternatives instead. Specifically, we recommend constructing a `BSpline` object and evaluate its derivative in a loop or a list comprehension.

  Parameters:
    x array_like

      A point or a set of points at which to evaluate the derivatives. Note that `t(k) <= x <= t(n-k+1)` must hold for each x.

    tck tuple

      A tuple (t,c,k) containing the vector of knots, the B-spline coefficients, and the degree of the spline whose derivatives to compute.

  Returns:
    results{ndarray, list of ndarrays}

      An array (or a list of arrays) containing all derivatives up to order k inclusive for each point x, being the first element the spline itself.

  See also

  `splprep`, `splrep`, `splint`, `sproot`, `splev`, `bisplrep`, `bisplev`
  `UnivariateSpline`, `BivariateSpline`

  Notes

  Array API Standard Support

  `spalde` is not in-scope for support of Python Array API Standard compatible backends other than NumPy.

  See Support for the array API standard for more information.

  References

  [1]

  de Boor C : On calculating with b-splines, J. Approximation Theory 6 (1972) 50-62.

  [2]

  Cox M.G. : The numerical evaluation of b-splines, J. Inst. Maths applics 10 (1972) 134-149.

  [3]

  Dierckx P. : Curve and surface fitting with splines, Monographs on Numerical Analysis, Oxford University Press, 1993.

  Examples

  To calculate the derivatives of a B-spline there are several aproaches. In this example, we will demonstrate that `spalde` is equivalent to calling `splev` and `splder`.

      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
      >>> from scipy.interpolate import BSpline, spalde, splder, splev

      >>> # Store characteristic parameters of a B-spline
      >>> tck = ((-2, -2, -2, -2, -1, 0, 1, 2, 2, 2, 2),  # knots
      ...        (0, 0, 0, 6, 0, 0, 0),  # coefficients
      ...        3)  # degree (cubic)
      >>> # Instance a B-spline object
      >>> # `BSpline` objects are preferred, except for spalde()
      >>> bspl = BSpline(tck[0], tck[1], tck[2])
      >>> # Generate extra points to get a smooth curve
      >>> x = np.linspace(min(tck[0]), max(tck[0]), 100)

  Evaluate the curve and all derivatives

      >>> # The order of derivative must be less or equal to k, the degree of the spline
      >>> # Method 1: spalde()
      >>> f1_y_bsplin = [spalde(i, tck)[0] for i in x ]  # The B-spline itself
      >>> f1_y_deriv1 = [spalde(i, tck)[1] for i in x ]  # 1st derivative
      >>> f1_y_deriv2 = [spalde(i, tck)[2] for i in x ]  # 2nd derivative
      >>> f1_y_deriv3 = [spalde(i, tck)[3] for i in x ]  # 3rd derivative
      >>> # You can reach the same result by using `splev`and `splder`
      >>> f2_y_deriv3 = splev(x, bspl, der=3)
      >>> f3_y_deriv3 = splder(bspl, n=3)(x)

      >>> # Generate a figure with three axes for graphic comparison
      >>> fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(16, 5))
      >>> suptitle = fig.suptitle(f'Evaluate a B-spline and all derivatives')
      >>> # Plot B-spline and all derivatives using the three methods
      >>> orders = range(4)
      >>> linetypes = ['-', '--', '-.', ':']
      >>> labels = ['B-Spline', '1st deriv.', '2nd deriv.', '3rd deriv.']
      >>> functions = ['splev()', 'splder()', 'spalde()']
      >>> for order, linetype, label in zip(orders, linetypes, labels):
      ...     ax1.plot(x, splev(x, bspl, der=order), linetype, label=label)
      ...     ax2.plot(x, splder(bspl, n=order)(x), linetype, label=label)
      ...     ax3.plot(x, [spalde(i, tck)[order] for i in x], linetype, label=label)
      >>> for ax, function in zip((ax1, ax2, ax3), functions):
      ...     ax.set_title(function)
      ...     ax.legend()
      >>> plt.tight_layout()
      >>> plt.show()