Solve the system 81x1 == - cos(x1)+9x+27 sin(x3) 3x2 sin(x1) + 3 cos(x3) = 18x3 = -2 cos(x1) + 6x2 + 3 sin(x3) using Newton-Raphson method. Use appropriate initial iterate, tolerance 10-15 and maximum iter- ation of 100. Include in your documentation the solution, relative error, and the number of iteration. should contain the code you used to solve the problem using the Newton Method.
import numpy as np
def f(x):
"""
Defines the
Parameters:
- x: The vector of unknowns.
Returns:
- f_x: The value of f(x).
"""
f1 = 81 * x[0] + np.cos(x[0]) - 9 * x[1]**2 - 27 * np.sin(x[2])
f2 = 3 * x[0] - np.sin(x[0]) - 3 * np.cos(x[2]) - 3 * x[1]
f3 = 18 * x[0] + 2 * np.cos(x[0]) - 6 * x[1] - 3 * np.sin(x[2])
return np.array([f1, f2, f3])
def jacobian_f(x):
"""
Defines the Jacobian matrix of f(x) for the given system of equations.
Parameters:
- x: The vector of unknowns.
Returns:
- J: The Jacobian matrix of f(x).
"""
J = np.zeros((3, 3))
J[0, 0] = 81 - np.sin(x[0])
J[0, 1] = -18 * x[1]
J[0, 2] = -27 * np.cos(x[2])
J[1, 0] = 3 - np.cos(x[0])
J[1, 1] = -3
J[1, 2] = 3 * np.sin(x[2])
J[2, 0] = 18 - 2 * np.sin(x[0])
J[2, 1] = -6
J[2, 2] = -3 * np.cos(x[2])
return J
def newton_raphson_method(x0, tol=1e-15, max_iter=100):
"""
Solves the system of equations using the Newton-Raphson method.
Parameters:
- x0: The initial guess for the solution.
- tol: The tolerance for convergence.
- max_iter: The maximum number of iterations.
Returns:
- x: The solution vector.
- rel_error: The relative error.
- num_iter: The number of iterations performed.
"""
x = x0.copy()
for num_iter in range(1, max_iter + 1):
f_x = f(x)
J = jacobian_f(x)
delta_x = np.linalg.solve(J, -f_x)
x += delta_x
if np.linalg.norm(delta_x) / np.linalg.norm(x) < tol:
rel_error = np.linalg.norm(f(x)) / np.linalg.norm(f(x0))
return x, rel_error, num_iter
raise ValueError("Newton-Raphson method did not converge.")
# Set the initial guess and tolerance
x0 = np.array([0.5, 0.5, 0.5])
tol = 1e-15
max_iter = 100
# Solve the system using the Newton-Raphson method
x, rel_error, num_iter = newton_raphson_method(x0, tol, max_iter)
# Print the solution, relative error, and number of iterations
print("Solution:")
print("x1 =", x[0])
print("x2 =", x[1])
print("x3 =", x[2])
print("Relative Error:", rel_error)
print("Number of Iterations:", num_iter)
Step by step
Solved in 2 steps with 1 images
