def delta_x(x):
mat = Matrix.from_list(jacobi(x))
B = Vector.from_list([-f1(x), -f2(x)])
LU, P = lu.LU_decomposition(mat)
new_B = lu.get_new_B(B, P)
delta = lu.LU_solve(LU, new_B)
return delta.get_data()
def get_c(x, f, h):
n = len(f)
a = [0] + [h[i - 1] for i in range(3, n)]
b = [2 * (h[i - 1] + h[i]) for i in range(2, n)]
c = [h[i] for i in range(2, n - 1)] + [0]
d = [
3 * ((f[i] - f[i - 1]) / h[i] - ((f[i - 1] - f[i - 2]) / h[i - 1]))
for i in range(2, n)
]
x = tma(TridiagonalMatrix.from_lists(a, b, c), Vector.from_list(d))
res = [0, 0] + x.get_data()
return res
def mls(n, x, y):
N = len(x)
mat = [[sum([x_j**(i + j) for x_j in x]) for i in range(n + 1)]
for j in range(n + 1)]
mat[0][0] = N + 1
b = [sum([x_j**i * y_j for x_j, y_j in zip(x, y)]) for i in range(n + 1)]
mat = Matrix.from_list(mat)
B = Vector.from_list(b)
LU, P = lu.LU_decomposition(mat)
new_B = lu.get_new_B(B, P)
coefs = lu.LU_solve(LU, new_B)
return coefs.get_data()
def finite_difference_method(f, p, q, a, b, alpha, beta, delta, gamma, y0, y1,
h):
n = int((b - a) / h)
x = [i for i in np.arange(a, b + h, h)]
A = [0] + [1 - p(x[i]) * h / 2 for i in range(0, n - 1)] + [-gamma]
B = [alpha * h - beta] + [q(x[i]) * h**2 - 2
for i in range(0, n - 1)] + [delta * h + gamma]
C = [beta] + [1 + p(x[i]) * h / 2 for i in range(0, n - 1)] + [0]
D = [y0 * h] + [f(x[i]) * h**2 for i in range(0, n - 1)] + [y1 * h]
y = tma(TridiagonalMatrix.from_lists(A, B, C), Vector.from_list(D))
return x, y.get_data()
def seidel_method(alpha, beta, eps):
sz = len(alpha)
np_alpha = np.array(alpha.get_data())
np_beta = np.array(beta.get_data())
B = np.tril(np_alpha, -1)
C = np_alpha - B
tmp1 = inv(np.eye(sz, sz) - B) @ C
tmp2 = inv(np.eye(sz, sz) - B) @ np_beta
x = tmp2
while True:
x_i = tmp2 + tmp1 @ x
if finish_iter_process(x_i, x, norm(tmp1), norm(C), eps, zeidel=True):
break
x = x_i
return Vector.from_list(x_i.tolist())