# coding: utf-8 # # Interpolation Error # In[1]: import numpy as np import numpy.linalg as la import matplotlib.pyplot as pt # Let's fix a function to interpolate: # In[2]: if 1: def f(x): return np.exp(1.5*x) elif 0: def f(x): return np.sin(20*x) else: def f(x): return (x>=0.5).astype(np.int).astype(np.float) # In[3]: x_01 = np.linspace(0, 1, 1000) pt.plot(x_01, f(x_01)) # And let's fix some parameters. Note that the interpolation interval is just \$[0,h]\$, not \$[0,1]\$! # In[4]: degree = 1 h = 1 nodes = 0.5 + np.linspace(-h/2, h/2, degree+1) nodes # Now build the Vandermonde matrix: # In[5]: V = np.array([ nodes**i for i in range(degree+1) ]).T # In[6]: V # Now find the interpolation coefficients as `coeffs`: # In[7]: coeffs = la.solve(V, f(nodes)) # Here are some points. Evaluate the interpolant there: # In[8]: x_0h = 0.5+np.linspace(-h/2, h/2, 1000) # In[9]: interp_0h = 0*x_0h for i in range(degree+1): interp_0h += coeffs[i] * x_0h**i # Now plot the interpolant with the function: # In[10]: pt.plot(x_01, f(x_01), "--", color="gray", label="\$f\$") pt.plot(x_0h, interp_0h, color="red", label="Interpolant") pt.plot(nodes, f(nodes), "or") pt.legend(loc="best") # Also plot the error: # In[11]: error = interp_0h - f(x_0h) pt.plot(x_0h, error) print("Max error: %g" % np.max(np.abs(error))) # * What does the error look like? (Approximately) # * How will the error react if we shrink the interval? # * What will happen if we increase the polynomial degree?