import numpy as np import matplotlib.pyplot as plt #Set the interval a, b = -1, 1 #This is number of points we evaluate the interpolation function on nplot = 1000 #This code is for the Runge's function runge = lambda x: 1 / (1 + 25 * x ** 2) x = np.linspace(a, b, nplot) y = runge(x) #Now we plot the corresponding function plt.plot(x, y) plt.text(0.2,0.7 , r'$y = \frac{1}{1+(5x)^2}=\frac{1}{1+25x^2}$', fontsize=15) plt.title ('Runge´s function') plt.xticks=[] plt.xlabel("x") plt.ylabel("y") plt.figtext(0.5, -0.04, "Figure 1", ha="center", fontsize=10)

from numpy import (zeros_like,prod) #In the code below we define the Lagrange(x_int,y_int,x) Interpolation function def lagrange(x_int, y_int, x): y_new = zeros_like(x) for xi, yi in zip(x_int, y_int): y_new += yi*prod([(x - xj)/(xi - xj) for xj in x_int if xi!=xj], axis=0) return y_new #x_int1/2/3 are the isometric nodes x_int1 = np.linspace(a, b, 6) #x=6 x_int2= np.linspace(a, b,11) #x=11 x_int3= np.linspace(a, b,16) #x=16 #y_new1/2/3 are Lagrange interpolation points y_new1 = lagrange(x_int1, runge(x_int1), x) y_new2= lagrange(x_int2,runge(x_int2),x) y_new3=lagrange(x_int3,runge(x_int3),x) #Code below is the exact solution for Runge's function plt.plot(x, runge(x), "k", label="Runge function") # Code below gives us the green graph for Lagrange Interpolation with nodes x = 6 plt.plot(x, y_new1,'g--',label="Lagrange Interpolation x=6") #Code below gives us the yellow graph for Lagrange Interpolation with nodes x = 11 plt.plot(x, y_new2,'y--',label="Lagrange Interpolation x=11") #Code below gives us the blue graph for Lagrange Interpolation with nodes x = 16 plt.plot(x, y_new3,'b--',label="Lagrange Interpolation x=16") plt.legend(bbox_to_anchor=(1.1,1)) plt.title('Example of Runges Theorem') plt.xlabel("x") plt.ylabel("y") plt.figtext(0.5, -0.04, "Figure 2", ha="center", fontsize=10) #The figure shows the Runges Phenomenon plt.show()

#Here we select Chebyshev nodes with x=6 x_Cheby = np.linspace(-1,1,6) for i in range(6): x_Cheby[i] = np.cos((2*(i+1) - 1)/(2*6)*np.pi) y_Cheby = 1/(1 + 25*x_Cheby**2) y_newCheby = lagrange(x_Cheby, y_Cheby, x) #plt under is exact solution for runge plt.plot(x, runge(x), "k", label="Runge function") #green graph - Lagrange interpolation plt.plot(x,y_new1,'g--',label="Lagrange Interpolation x=6") #Cheby plot plt.plot(x,y_newCheby,'r--',label="Chebyshev nodes x=6 ") plt.legend(bbox_to_anchor=(1.6,1)) plt.title('Lagrange Interpolation and Chebyshev nodes') plt.xlabel("x") plt.ylabel("y") plt.figtext(0.5, -0.04, "Figure 3", ha="center", fontsize=10) plt.show()

#Here we select Chebyshev nodes with x=11 x_Cheby = np.linspace(-1,1,11) for i in range(11): x_Cheby[i] = np.cos((2*(i+1) - 1)/(2*11)*np.pi) y_Cheby = 1/(1 + 25*x_Cheby**2) y_newCheby2 = lagrange(x_Cheby, y_Cheby, x) #plt under is exact solution for runge plt.plot(x, runge(x), "k", label="Runge function") #Yellow graph - Lagrange interpolation plt.plot(x, y_new2,'y--',label="Lagrange Interpolation x=11") #Chebyshev plt.plot(x, y_newCheby2,'r--',label="Chebyshev nodes x=11") plt.legend(bbox_to_anchor=(1.1,1)) plt.title('Lagrange Interpolation and Chebyshev nodes') plt.xlabel("x") plt.ylabel("y") plt.figtext(0.5, -0.04, "Figure 4", ha="center", fontsize=10) plt.show()

#Here we select Chebyshev nodes with x=16 x_Cheby = np.linspace(-1,1,16) for i in range(16): x_Cheby[i] = np.cos((2*(i+1) - 1)/(2*16)*np.pi) y_Cheby = 1/(1 + 25*x_Cheby**2) y_newCheby3 = lagrange(x_Cheby, y_Cheby, x) #plt under is exact solution for rhunge plt.plot(x, runge(x), "k", label="Runge function") #Blue graph - Lagrange interpolation plt.plot(x,y_new3,'b--',label="Lagrange Interpolation x=16") #cheby plot plt.plot(x,y_newCheby3,'r--',label="Chebyshev nodes x=16") plt.legend(bbox_to_anchor=(1.1,1)) plt.title('Lagrange Interpolation and Chebyshev nodes') plt.xlabel("x") plt.ylabel("y") plt.figtext(0.5, -0.04, "Figure 5", ha="center", fontsize=10) plt.show()