def gs_iter(x_0, epsilon, max_iter): # 2a
A = np.matrix([[1.0, 1.0/2, 1.0/3], [1.0/2, 1.0, 1.0/4], [1.0/3, 1.0/4, 1.0]])
B = np.array([.1, .1, .1]).reshape(3, 1)
S_inv = np.zeros((3, 3))
a, b, c, d, e, f = A[0, 0], A[1, 0], A[1, 1], A[2, 0], A[2, 1], A[2, 2]
S_inv[0, 0] = 1.0/a
S_inv[1, 0] = -b/(a*c)
S_inv[1, 1] = 1.0/c
S_inv[2, 0] = (-c*d+b*e)/(a*c*f)
S_inv[2, 1] = -e/(c*f)
S_inv[2, 2] = 1.0/f
S = np.tril(A)
U = A - S
iteration = 0
error = 1000
x_n = x_0
while iteration < max_iter:
x_1 = pascal.mult(pascal.mult(S_inv, -U), x_n)
x_1 += pascal.mult(S_inv, B)
error = pascal.norm_inf(x_1 - x_n)
x_n = x_1
if epsilon > error:
return x_n, iteration
iteration += 1
else:
return None, None
def jacobi_iter(x_0, epsilon, max_iter): # 2a
A = np.matrix([[1.0, 1.0/2, 1.0/3], [1.0/2, 1.0, 1.0/4], [1.0/3, 1.0/4, 1.0]])
b = np.array([.1, .1, .1]).reshape(3, 1)
iteration = 0
error = 1000
x_n = x_0
while iteration < max_iter:
d = np.diag(1 / np.diag(A))
alu = A - np.diag(np.diag(A))
x_1 = pascal.mult(d, b) + pascal.mult(pascal.mult(d, -alu), x_n)
error = pascal.norm_inf(x_1 - x_n)
x_n = x_1
if epsilon > error:
return x_n, iteration
iteration += 1
else:
return None, None
jacobi, gs = iterative.generate_data()
j_vals = [i[1] for i in jacobi]
g_vals = [i[1] for i in gs]
jacobi_approx = sum(j_vals) / len(j_vals) # 2c
gs_approx = sum(g_vals) / len(g_vals)
print("Averages, jacobi then gauss-seidel, then iterations")
print(jacobi_approx)
print(gs_approx)
print(float(sum(j[2] for j in jacobi))/sum(g[2] for g in gs))
exact = np.array([9.0/190, 28.0/475, 33.0/475]).reshape(3,1)
errs_jacobi = [pascal.norm_inf(j-exact) for j in j_vals]
errs_gs = [pascal.norm_inf(g-exact) for g in g_vals]
plt.plot([j[2] for j in jacobi], errs_jacobi, 'ko', [g[2] for g in gs], errs_jacobi, 'bo')
plt.savefig('./iterative_err.png')
plt.clf()
powers = power.generate_data()
ds = [p[0] for p in powers if p[0] is not None]
ts = [p[1] for p in powers if p[1] is not None]
tis = [p[2] for p in powers if p[2] is not None]
maxs = [p[3] for p in powers if p[3] is not None]
mins = [p[4] for p in powers if p[4] is not None]
big = max(maxs)
small = max(mins)
