def test(eigenvalue, v_matrix, matrix):
print('Проверка')
vectors = []
for vector in zip(*v_matrix.matrix):
v = Vector(v_matrix.size)
v.vector = vector
vectors.append(v)
for i in range(0, len(vectors) - 1):
print('v_{0}: '.format(i + 1), end='')
vectors[i].print_vector()
print('v_{0}: '.format(len(vectors)), end='')
v.print_vector()
print('(v_{0}, v_{1}): '.format(i + 1, len(vectors)), end='')
res = sum([vectors[i][j] * v[j] for j in range(0, v.size)])
print('{0:8.20f}'.format(res))
print()
print('Проверка A * x = a_k * x')
for i in range(0, len(vectors)):
print('a_k = ', eigenvalue[i])
print('x = ', end='')
vectors[i].print_vector()
print('A * x = ', end='')
(matrix * vectors[i]).print_vector()
print('a_k * x = ', end='')
(vectors[i] * eigenvalue[i]).print_vector()
print('-------------------------------------------------------------')
def method_jacobi_rotations(a, eps):
n_iter = 0
a_k = copy.copy(a)
v = Matrix(a.size, single=True)
while True:
i_max, j_max = find_max(a_k)
fi = 0.5 * math.atan(2 * a_k[i_max][j_max] /
(a_k[i_max][i_max] - a_k[j_max][j_max]))
u = Matrix(a.size, single=True)
u[i_max][i_max] = math.cos(fi)
u[i_max][j_max] = -math.sin(fi)
u[j_max][i_max] = math.sin(fi)
u[j_max][j_max] = math.cos(fi)
u_t = copy.copy(u)
u_t.transpose()
a_k = u_t * a_k * u
v = v * u
n_iter += 1
if t(a_k) < eps:
eigenvalue = Vector(a_k.size)
eigenvalue.vector = [a_k[i][i] for i in range(0, a_k.size)]
print('Итераций: ', n_iter)
return eigenvalue, v
def quadratures_method(points, h, lambda_exp, k_exp, g_exp):
x = sympy.Symbol('x')
t = sympy.Symbol('t')
k = sympy.lambdify([x, t], parse_expr(k_exp))
g = sympy.lambdify(x, parse_expr(g_exp))
lambda_ = float(parse_expr(lambda_exp))
A = get_matrix_system(points, h, lambda_, k)
G = Vector(len(points))
G.vector = get_values(points, g)
return lu_solve(A, G).vector
def get_poly_1_deg(points, values):
sum_points = sum(points)
sum_points_2 = sum([point**2 for point in points])
sum_values = sum(values)
sum_v_p = sum([values[i] * points[i] for i in range(0, len(points))])
a = Matrix(2)
a.matrix = [[len(points), sum_points], [sum_points, sum_points_2]]
b = Vector(2)
b.vector = [sum_values, sum_v_p]
l, u, p = lup(a)
coefficients = lup_solve(l, u, p, b).vector
return parse_expr(f'{coefficients[0]} + {coefficients[1]} * x')
def get_c(x, f, h):
a_diag = [0.0] + [h[i - 1] for i in range(3, len(x))]
b_diag = [2 * (h[i - 1] + h[i]) for i in range(2, len(x))]
c_diag = [h[i] for i in range(2, len(x) - 1)] + [0.0]
d = [
3 * ((f[i] - f[i - 1]) / h[i] - ((f[i - 1] - f[i - 2]) / h[i - 1]))
for i in range(2, len(x))
]
matrix = TridiagonalMatrix(len(x) - 2)
matrix.A = a_diag
matrix.B = b_diag
matrix.C = c_diag
vec = Vector(len(x) - 2)
vec.vector = d
c = [0.0, 0.0] + rtm(matrix, vec)
return c
def get_poly_2_deg(points, values):
sum_points = sum(points)
sum_points_2 = sum([point**2 for point in points])
sum_points_3 = sum([point**3 for point in points])
sum_points_4 = sum([point**4 for point in points])
sum_values = sum(values)
sum_v_p = sum([values[i] * points[i] for i in range(0, len(points))])
sum_v_p2 = sum([values[i] * points[i]**2 for i in range(0, len(points))])
a = Matrix(3)
a.matrix = [[len(points), sum_points, sum_points_2],
[sum_points, sum_points_2, sum_points_3],
[sum_points_2, sum_points_3, sum_points_4]]
b = Vector(3)
b.vector = [sum_values, sum_v_p, sum_v_p2]
l, u, p = lup(a)
coefficients = lup_solve(l, u, p, b).vector
return parse_expr(
f'{coefficients[0]} + {coefficients[1]} * x + {coefficients[2]} * x ** 2'
)
def finite_difference(pt, coeffs, alpha, beta, delta, gamma, step, cond_0, cond_1):
x = sympy.Symbol('x')
p_exp = parse_expr(f'({coeffs[1]}) / ({coeffs[0]})')
q_exp = parse_expr(f'({coeffs[2]}) / ({coeffs[0]})')
p = sympy.lambdify(x, p_exp)
q = sympy.lambdify(x, q_exp)
sz = len(pt) - 1
a = [0.0] + [1 - p(pt[i]) * step / 2 for i in range(0, sz - 1)] + [- gamma]
b = [alpha * step - beta] + [-2 + q(pt[i]) * step ** 2 for i in range(0, sz - 1)] + [delta * step + gamma]
c = [beta] + [1 + p(pt[i]) * step / 2 for i in range(0, sz - 1)] + [0.0]
d = [cond_0 * step] + [0 for _ in range(0, sz - 1)] + [cond_1 * step]
matr = TridiagonalMatrix(sz + 1)
matr.A = a
matr.B = b
matr.C = c
vec = Vector(sz + 1)
vec.vector = d
return rtm(matr, vec)