Runge's Phenomenon

import numpy as np import matplotlib.pyplot as plt def demonstrate_runges_phenomenon(n): #Create the equidistant data points x=np.linspace(-1,1,n) #Create the interpolating polynomial y=[] for i in x: y.append(1/(1+(5*i)**2)) coefficients=np.polyfit(x,y,len(x)) #Create the inputs for the graph x_high_res=np.linspace(-1,1,1000) y_high_res=1/(1+(5*x_high_res)**2) y_interpo=np.polyval(coefficients,x_high_res) #Create the graph plt.plot(x_high_res,y_high_res,label='Real Function') plt.plot(x,y,'o',label='Data Points') plt.plot(x_high_res,y_interpo,label='Interpolating Polynomial') plt.legend() plt.show() demonstrate_runges_phenomenon(11)

# Start writing code here... import numpy as np import matplotlib.pyplot as plt def demonstrate_effectiveness_of_chebyshev_nodes_ex1(n): #Crate the equidistant data points x=np.linspace(-1,1,n) #Crate the interpolating polynomial y=[] for i in x: y.append(1/(1+(5*i)**2)) coefficients=np.polyfit(x,y,len(x)) #Create the inputs for the graph x_high_res=np.linspace(-1,1,1000) y_high_res=1/(1+(5*x_high_res)**2) y_interpo=np.polyval(coefficients,x_high_res) #Create the Chebyshev nodes x_cheb=np.cos((2*np.arange(1,n+1)-1)*np.pi/(2*n)) #Create the interpolating polynomial for the Chebyshev nodes y_cheb=[] for i in x_cheb: y_cheb.append(1/(1+(5*i)**2)) coefficients=np.polyfit(x_cheb,y_cheb,len(x)) #Create the inputs for the graph of the polynomial using the Chebyshev nodes y_interpo_cheb=np.polyval(coefficients,x_high_res) #Create the graph plt.plot(x_high_res,y_high_res,label='Real Function') plt.plot(x_cheb,y_cheb,'o',label='Chebchev Nodes') plt.plot(x_high_res,y_interpo,label='Interpolating Polynomial') plt.plot(x_high_res,y_interpo_cheb,label='Interpolating Polynomial Using the Chebchev Nodes') plt.legend() plt.show() demonstrate_effectiveness_of_chebyshev_nodes_ex1(11)

import numpy as np import matplotlib.pyplot as plt def demonstrate_effectiveness_of_chebyshev_nodes_ex2(n): #Crate the equidistant data points x=np.linspace(-1,1,n) #Crate the interpolating polynomial y=[] for i in x: y.append(1/(1+(10*i)**-2)) coefficients=np.polyfit(x,y,len(x)) #Create the inputs for the graph x_high_res=np.linspace(-1,1,1000) y_high_res=1/(1+(10*x_high_res)**-2) y_interpo=np.polyval(coefficients,x_high_res) #Create the Chebyshev nodes x_cheb=np.cos((2*np.arange(1,n+1)-1)*np.pi/(2*n)) #Create the interpolating polynomial for the Chebyshev nodes y_cheb=[] for i in x_cheb: y_cheb.append(1/(1+(10*i)**-2)) coefficients=np.polyfit(x_cheb,y_cheb,len(x)) #Create the inputs for the graph of the polynomial using the Chebyshev nodes y_interpo_cheb=np.polyval(coefficients,x_high_res) #Create the graph plt.plot(x_high_res,y_high_res,label='Real Function') plt.plot(x_cheb,y_cheb,'o',label='Chebchev Nodes') plt.plot(x_high_res,y_interpo,label='Interpolating Polynomial') plt.plot(x_high_res,y_interpo_cheb,label='Interpolating Polynomial Using the Chebchev Nodes') plt.legend() plt.show() demonstrate_effectiveness_of_chebyshev_nodes_ex2(11)

import numpy as np import matplotlib.pyplot as plt def demonstrate_effectiveness_of_chebyshev_nodes_ex3(n): #Crate the equidistant data points x=np.linspace(-1,1,n) #Crate the interpolating polynomial y=[] for i in x: y.append(np.sin(10*i)) coefficients=np.polyfit(x,y,len(x)) #Create the inputs for the graph x_high_res=np.linspace(-1,1,1000) y_high_res=np.sin(10*x_high_res) y_interpo=np.polyval(coefficients,x_high_res) #Create the Chebyshev nodes x_cheb=np.cos((2*np.arange(1,n+1)-1)*np.pi/(2*n)) #Create the interpolating polynomial for the Chebyshev nodes y_cheb=[] for i in x_cheb: y_cheb.append(np.sin(10*i)) coefficients=np.polyfit(x_cheb,y_cheb,len(x)) #Create the inputs for the graph of the polynomial using the Chebyshev nodes y_interpo_cheb=np.polyval(coefficients,x_high_res) plt.plot(x_high_res,y_high_res,label='Real Function') plt.plot(x_cheb,y_cheb,'o',label='Chebchev Nodes') plt.plot(x_high_res,y_interpo,label='Interpolating Polynomial') plt.plot(x_high_res,y_interpo_cheb,label='Interpolating Polynomial Using the Chebchev Nodes') plt.legend() plt.show() demonstrate_effectiveness_of_chebyshev_nodes_ex3(11)