for i,j in enumerate(n):
h = 1/(j+1)
xx = np.linspace(0,1,j)
# The forcing function
fx = 100*np.exp(-10*xx)
# Analytic solution
ux = 1 - (1 - np.exp(-10))*xx - np.exp(-10*xx)
# Thomas algorithm setup
a = -1*np.ones([j,1])
b = 2*np.ones([j,1])
c = -1*np.ones([j,1])
d = (h**2)*fx.reshape(len(fx),1)
# Thomas algorithm for matrix inverse
th_ux = Functions.Thomas_algorithm_solution(a,b,c,d)
plt.subplot(len(n),1,i+1)
plt.plot(xx,th_ux)
plt.plot(xx,ux)
plt.title('n={}'.format(j))
plt.ylabel('u(x)')
plt.xlabel('x')
plt.legend(['Analytic','Thomas Algorithm'],loc='upper right')
plt.grid()
a = np.concatenate([a1, a2])
A = toeplitz(a, a)
# Normalize to get orthonormal matrix for Jacobi
V = A.copy()
for j in range(i):
V[:, j] = V[:, j] / np.linalg.norm(V[:, j])
##############
lmbda_an = np.zeros(i)
for j in range(i):
lmbda_an[j] = A[j, j] + 2 * A[0, 1] * np.cos(j * np.pi / i)
time_start_B = time.clock()
eigB = Functions.bisect_eig(A, n_pol=i)
eigB = Functions.sort_eig_val(eigB)
time_elapsed_B = (time.clock() - time_start_B)
B.append(time_elapsed_B)
time_start_Nu = time.clock()
eigNP, eigNPv = np.linalg.eig(A)
eigNP = Functions.sort_eig_val(eigNP)
time_elapsed_Nu = (time.clock() - time_start_Nu)
Nu.append(time_elapsed_Nu)
time_start_J = time.clock()
eigJ, eigJv = Functions.jacobi(A, V, n=i, conv=10**-8)
eigJ, eigJv = Functions.sort_eig(eigJ, eigJv)
eigJ = Functions.sort_eig_val(eigJ)
time_elapsed_J = (time.clock() - time_start_J)
for i,j in enumerate(n):
h = 1/(j+1)
xx = np.linspace(0,1,j)
# The forcing function
fx = 100*np.exp(-10*xx)
# Analytic solution
ux = 1 - (1 - np.exp(-10))*xx - np.exp(-10*xx)
# Gauss algorithm setup
a = -1*np.ones([j,1])
b = 2*np.ones([j,1])
d = (h**2)*fx.reshape(len(fx),1)
# Gaus elimination for matrix inverse
#th_ux = Functions.gaus_elim(b,a,d)
# special case
th_ux = Functions.special_case(d)
plt.subplot(len(n),1,i+1)
plt.plot(xx,th_ux)
plt.plot(xx,ux)
plt.title('n={}'.format(j))
plt.ylabel('u(x)')
plt.xlabel('x')
plt.legend(['Analytic','Gauss elimination'],loc='upper right')
plt.grid()
h = 1/(j+1)
xx = np.linspace(0,1,j)
# The forcing function
fx = 100*np.exp(-10*xx)
# Analytic/closed form solution
ux = 1 - (1 - np.exp(-10))*xx - np.exp(-10*xx)
# Special case setup
d = (h**2)*fx.reshape(len(fx),1)
time_start_th = time.clock()
# special case
th_ux = Functions.special_case(d)
time_elapsed_th = (time.clock() - time_start_th)
th.append(time_elapsed_th)
# Tridiagonal matrix setup
a1 = np.array([2,-1])
a2 = np.zeros([j-2])
a = np.concatenate([a1,a2])
# Define the Toeplitz matrix
A = toeplitz(a, a)
time_start_lu = time.clock()
# LU inverse
lu_ux = Functions.LU_decomp_inverse(A,d).T
time_elapsed_lu = (time.clock() - time_start_lu)
rho = (j + 1 * h)
V[j] = rho**2
d = 2.0 / h**2 + V[j]
e = -1.0 / h**2
A[j, j] = d
if j > 0:
A[j, j - 1] = e
if j < i - 1:
A[j, j + 1] = e
##############
W = A.copy()
for j in range(i):
W[:, j] = W[:, j] / np.linalg.norm(W[:, j])
eigJ, eigJv = Functions.jacobi(A, W, n=i, conv=10**-8)
eigJ, eigJv = Functions.sort_eig(eigJ, eigJv)
eigJ = Functions.sort_eig_val(eigJ)
eigNP, eigNPv = np.linalg.eig(A)
eigNP = Functions.sort_eig_val(eigNP)
#plt.figure()
ax = plt.figure().gca()
ax.xaxis.set_major_locator(MaxNLocator(integer=True))
plt.title('N={}'.format(i))
plt.scatter(np.arange(4), lmbd_an[0:4], marker='x')
plt.scatter(np.arange(4), eigJ[i - 4:][::-1], marker='v')
plt.scatter(np.arange(4), eigNP[i - 4:][::-1], marker='o')
plt.legend(['Analytical', 'Jacobi', 'Numpy'])
plt.ylabel('Eigenvalue')
# The forcing function
fx = 100*np.exp(-10*xx)
# Analytic solution
ux = 1 - (1 - np.exp(-10))*xx - np.exp(-10*xx)
# Tridiagonal matrix setup
a1 = np.array([2,-1])
a2 = np.zeros([n-2])
a = np.concatenate([a1,a2])
# Define the Toeplitz matrix
A = toeplitz(a, a)
d = (h**2)*fx.reshape(len(fx),1)
x = np.random.randn(n).reshape(n,1)
v = Functions.Jacobi_it(A,x,d,eps=0.001)
plt.figure()
plt.plot(xx,ux)
plt.plot(xx,v)
plt.ylabel('u(x)')
plt.xlabel('x')
plt.legend(['Analytic','Jacobi'],loc='upper right')
plt.grid()