Spline interpolation requires two essential steps: (1) a spline representation of the curve is computed, and (2) the spline is evaluated at the desired points. In order to find the spline representation, there are two different ways to represent a curve and obtain (smoothing) spline coefficients: directly and parametrically. The direct method finds the spline representation of a curve in a two- dimensional plane using the function 

splrep. The first two arguments are the only ones required, and these provide the xx and yy components of the curve. The normal output is a 3-tuple, (t,c,k)(t,c,k) , containing the knot-points, tt , the coefficients cc and the order kk of the spline. The default spline order is cubic, but this can be changed with the input keyword, k.

For curves in NN -dimensional space the function splprep allows defining the curve parametrically. For this function only 1 input argument is required. This input is a list of NN -arrays representing the curve in NN -dimensional space. The length of each array is the number of curve points, and each array provides one component of the N

N -dimensional data point. The parameter variable is given with the keyword argument, u, which defaults to an equally-spaced monotonic sequence between 00 and 11 . The default output consists of two objects: a 3-tuple, (t,c,k)(t,c,k) , containing the spline representation and the parameter variable u.u.

The keyword argument, s , is used to specify the amount of smoothing to perform during the spline fit. The default value of ss is s=m−2m−−−√s=m−2m where mm is the number of data-points being fit. Therefore, if no smoothing is desired a value of s=0s=0 should be passed to the routines.

Once the spline representation of the data has been determined, functions are available for evaluating the spline (splev) and its derivatives (splev, spalde) at any point and the integral of the spline between any two points ( splint). In addition, for cubic splines ( k=3k=3 ) with 8 or more knots, the roots of the spline can be estimated ( sproot). These functions are demonstrated in the example that follows.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy import interpolate

Cubic-spline

>>> x = np.arange(0, 2*np.pi+np.pi/4, 2*np.pi/8)
>>> y = np.sin(x)
>>> tck = interpolate.splrep(x, y, s=0)
>>> xnew = np.arange(0, 2*np.pi, np.pi/50)
>>> ynew = interpolate.splev(xnew, tck, der=0)

>>> plt.figure()
>>> plt.plot(x, y, 'x', xnew, ynew, xnew, np.sin(xnew), x, y, 'b')
>>> plt.legend(['Linear', 'Cubic Spline', 'True'])
>>> plt.axis([-0.05, 6.33, -1.05, 1.05])
>>> plt.title('Cubic-spline interpolation')
>>> plt.show()

Derivative of spline

>>> yder = interpolate.splev(xnew, tck, der=1)
>>> plt.figure()
>>> plt.plot(xnew, yder, xnew, np.cos(xnew),'--')
>>> plt.legend(['Cubic Spline', 'True'])
>>> plt.axis([-0.05, 6.33, -1.05, 1.05])
>>> plt.title('Derivative estimation from spline')
>>> plt.show()

Integral of spline

>>> def integ(x, tck, constant=-1):
...     x = np.atleast_1d(x)
...     out = np.zeros(x.shape, dtype=x.dtype)
...     for n in range(len(out)):
...         out[n] = interpolate.splint(0, x[n], tck)
...     out += constant
...     return out

>>> yint = integ(xnew, tck)
>>> plt.figure()
>>> plt.plot(xnew, yint, xnew, -np.cos(xnew), '--')
>>> plt.legend(['Cubic Spline', 'True'])
>>> plt.axis([-0.05, 6.33, -1.05, 1.05])
>>> plt.title('Integral estimation from spline')
>>> plt.show()

Roots of spline

>>> interpolate.sproot(tck)
array([3.1416])

Notice that sproot failed to find an obvious solution at the edge of the approximation interval, 

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