p12: Accuracy of Chebyshev spectral differentiation

%config InlineBackend.figure_format='svg'
from numpy import zeros,pi,inf,linspace,arange,abs,dot,exp
from scipy.linalg import toeplitz,norm
from matplotlib.pyplot import figure,subplot,semilogy,loglog,title,xlabel,ylabel,axis,grid
from chebPy import *

Nmax = 50
E = zeros((4,Nmax))
for N in range(1,Nmax+1):
    D,x = cheb(N)
    
    v = abs(x)**3          # 3rd deriv in BV
    vprime = 3.0*x*abs(x)
    E[0][N-1] = norm(dot(D,v)-vprime,inf)
    
    v = exp(-(x+1.0e-15)**(-2))   # C-infinity
    vprime = 2.0*v/(x+1.0e-15)**3
    E[1][N-1] = norm(dot(D,v)-vprime,inf)
    
    v = 1.0/(1.0+x**2)     # analytic in a [-1,1]
    vprime = -2.0*x*v**2
    E[2][N-1] = norm(dot(D,v)-vprime,inf)
    
    v = x**10
    vprime = 10.0*x**9   # polynomial
    E[3][N-1] = norm(dot(D,v)-vprime,inf)


titles = ["$|x|^3$", "$\\exp(-x^{-2})$", \
          "$1/(1+x^2)$", "$x^{10}$"]
figure(figsize=(10,10))
for iplot in range(4):
    subplot(4,2,2*iplot+1)
    semilogy(arange(1,Nmax+1,),E[iplot][:],'.-')
    title(titles[iplot]+", semilogy")
    xlabel('N'), ylabel('error'), grid(True)
    subplot(4,2,2*iplot+2)
    loglog(arange(1,Nmax+1,),E[iplot][:],'.-')
    title(titles[iplot]+", loglog")
    xlabel('N'), ylabel('error'), grid(True)

• \(|x|^3\): loglog plot is straight line indicating algebraic decay of the form \(N^{-p}\).

• \(\exp(-x^{-2})\): loglog plot goes down faster than a straight line, indicating faster than algebraic decay.

• \(1/(1+x^2)\): semilogy plot is straight line, indicating exponential decay of the form \(\exp(-\alpha N)\).

• \(x^{10}\): for \(N \ge 10\) we get exact answer.