scipy.interpolate.

spalde

scipy.interpolate.spalde(x, tck)

对 B 样条函数及其所有导数值在某一点（或点集）上求值，最高至 k 阶（样条函数的次数），其中 0 为样条函数的值本身。

参数:

x类数组

要对导数值求值的点或点集。请注意，对于每个 x，必须满足 t(k) <= x <= t(n-k+1)。

tck元组

包含结点向量、B 样条函数系数以及要计算导数值的样条函数次数的元组 (t,c,k)。

返回:

results{ndarray, ndarray 列表}

包含对每个点 x 计算的最高至 k 阶的所有导数值的数组（或数组列表），第一个元素为样条函数的值本身。

另请参阅

splprep, splrep, splint, sproot, splev, bisplrep, bisplev

UnivariateSpline, BivariateSpline

引用

[1]

de Boor C：论使用 b 样条进行计算，J. Approximation Theory 6 (1972) 50-62。

[2]

Cox M.G.：b 样条的数值求解，J. Inst. Maths applics 10 (1972) 134-149。

[3]

Dierckx P.：使用样条进行曲线和曲面拟合，数值分析专著，牛津大学出版社，1993 年。

示例

在浏览器中尝试！

有几种方法来计算 B 样条的导数。在本示例中，我们将证明 spalde 等效于依次调用 splev 和 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 prefered, 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)

求解曲线及其所有导数

>>> # 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()

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