# p38 - solve u_xxxx = exp(x), u(-1)=u(1)=u'(-1)=u'(1)=0

# compare with p13

#

from chebdif import *

import numpy as np

from scipy.linalg import solve

from matplotlib import pyplot as plt

# Construct dscrete biharmonic operator:

N = 15

x,DM = chebdif(N+1,4) # Weideman&Reddy function for differentiation matrix

v = np.zeros(N+1) # Work array

v[1:N] = 1./(1.-x[1:N]**2)

S = np.diag(v)

D2 = DM[1,:,:] # Python counting starts from zero...

D3 = DM[2,:,:]

D4 = DM[3,:,:]

print D4.shape

D4 = np.dot((np.dot(np.diag(1.-x**2),D4) - 8*np.dot(np.diag(x),D3) - 12*D2),S)

D4 = D4[1:N,1:N]

# Solve boundary-value problem and plot results:

f=np.exp(x[1:N])

u=solve(D4,f)

u1=np.zeros(N+1)

u1[0]=0.0

u1[N]=0.0

u1[1:N]=u

# Exact solution and max err:

A = np.array([ [1,-1,1,-1],

[0,1,-2,3],

[1,1,1,1],

[0,1,2,3] ])

V = np.vander(x)

V = V[:,N-3:N+1]

V = (V.T[::-1]).T # transpose - reverse up-down - transpose

v = np.array([-1,-1,1,1])

c = solve(A,np.exp(v))

exact=(np.exp(x)-np.dot(V,c))

maxerr=np.max(u1-exact)

# Interpolate to more grid points

xx=np.linspace(-1.,1.,100)

uu = (1.-xx**2)*np.polyval(np.polyfit(x,np.dot(S,u1),N),xx)

plt.title('max err = %e' % maxerr)

plt.plot(xx,uu,'b',x,exact,'o')

plt.show()