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interp.py
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interp.py
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import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as plt
def driver():
f = lambda x: np.sinc(5*x)
N = 17
''' interval'''
a = -1
b = 1
''' create equispaced interpolation nodes'''
xint = np.array(chebyshev_nodes(N+1,a,b))
''' create interpolation data'''
yint = f(xint)
''' create points for evaluating the Lagrange interpolating polynomial'''
Neval = 1000
xeval = np.array(chebyshev_nodes(Neval+1,a,b))
yeval_l= np.zeros(Neval+1)
yeval_dd = np.zeros(Neval+1)
'''Initialize and populate the first columns of the
divided difference matrix. We will pass the x vector'''
y = np.zeros( (N+1, N+1) )
for j in range(N+1):
y[j][0] = yint[j]
y = dividedDiffTable(xint, y, N+1)
''' evaluate lagrange poly '''
for kk in range(Neval+1):
yeval_l[kk] = eval_lagrange(xeval[kk],xint,yint,N)
yeval_dd[kk] = evalDDpoly(xeval[kk],xint,y,N)
''' create vector with exact values'''
fex = f(xeval)
plt.figure()
plt.plot(xeval,fex,'ro-')
plt.plot(xeval,yeval_l,'bs--')
plt.plot(xeval,yeval_dd,'c.--')
plt.legend()
plt.figure()
err_l = abs(yeval_l-fex)
err_dd = abs(yeval_dd-fex)
plt.semilogy(xeval,err_l,'ro--',label='lagrange')
plt.semilogy(xeval,err_dd,'bs--',label='Newton DD')
plt.legend()
plt.show()
def chebyshev_nodes(n, a=-1, b=1):
i = np.arange(n)
x = np.cos((2*i + 1) / (2*n) * np.pi)
# Map from [-1, 1] to [a, b]
return 0.5 * (a + b) + 0.5 * (b - a) * x
def eval_lagrange(xeval,xint,yint,N):
lj = np.ones(N+1)
for count in range(N+1):
for jj in range(N+1):
if (jj != count):
lj[count] = lj[count]*(xeval - xint[jj])/(xint[count]-xint[jj])
yeval = 0.
for jj in range(N+1):
yeval = yeval + yint[jj]*lj[jj]
return(yeval)
''' create divided difference matrix'''
def dividedDiffTable(x, y, n):
for i in range(1, n):
for j in range(n - i):
y[j][i] = ((y[j][i - 1] - y[j + 1][i - 1]) /
(x[j] - x[i + j]));
return y;
def evalDDpoly(xval, xint,y,N):
''' evaluate the polynomial terms'''
ptmp = np.zeros(N+1)
ptmp[0] = 1.
for j in range(N):
ptmp[j+1] = ptmp[j]*(xval-xint[j])
'''evaluate the divided difference polynomial'''
yeval = 0.
for j in range(N+1):
yeval = yeval + y[0][j]*ptmp[j]
return yeval
driver()