Question

programming in python.

Write a python program that plots a given function and an interpolation polynomial. To test this, use the following cases: a) )xp(-4r2), tE-1,1]. As data, use equidistant nodes that include the endpoints and the function values in those nodes. The number of nodes should be 5, then 12.

Answer 1

import numpy as np
import matplotlib.pyplot as plt

nodes = 12;
#nodes = 5;

xmin = -1;xmax = 1;

x = np.linspace(xmin,xmax,nodes)
f = np.exp(-4*x*x)

# computing newton interpolating polynomial upto order 3

#coefficients upto 3rd order
a0 = f[0];
a1 = (f[1] - f[0])/(x[1] - x[0]);
f12 = (f[2] - f[1])/(x[2]-x[1]);
a2 = (f12 - a1)/(x[2]-x[0]);
f23 = (f[3] - f[2])/(x[3]-x[2]);
f123 = (f23-f12)/(x[3]-x[1]);
a3 = (f123 - a2)/(x[3]-x[0]);

#1st,2nd,and 3rd order polynomials pn(x)
p0 = a0;
p1 = lambda xx: p0 + a1*(xx-x[0]);
p2 = lambda xx: p1(xx) + a2*(xx-x[0])*(xx-x[1]);
p3 = lambda xx: p2(xx) + a3*(xx-x[0])*(xx-x[1])*(xx-x[2]);
p3x = p3(x)

plt.figure(1)
plt.plot(x,f,'o',x,p3x,':')
plt.gca().legend(('f(x)','p3(x)'))
plt.ylabel('f,p')
plt.xlabel('x')
plt.grid(True)
plt.title('plot of f(x) and p3(x)')
plt.show()

plot of f(x) and p3(x) 1.0 f(x) p3(x) 0.5 0.0 -0.5 -2.0 -2.5 -3.0 1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

plot of f(x) and p3(x) 101 f(x) p3(x) 8 4 2 0 1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

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