Determine the roots of the simultaneous nonlinear equations using the Newton - Raphson method. Determine the roots of the simultaneous nonlinear initial as equations using the Newton - Raphson method. [f_1(x,y)=x^2+1-y;f_2(x y) = 2 cos (x) - - y ] Use x_0 = 0.7 and y_0 = 1.guesses your (\epsi_ =0.5%).

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```python import numpy as np from scipy.optimize import fsolve import math # Define the first function f1(x, y) def f1(x, y): # f1(x, y) = x^2 + 1 - y return x****2 + 1 - y # Define the second function f2(x, y) def f2(x, y): # f2(x, y) = 2 * cos(x) - y return 2 * math.cos(x) - y # Define the function that represents the system of equations def equations(vars): x, y = vars # The system of equations consists of f1(x, y) and f2(x, y) return [f1(x, y), f2(x, y)] # Define the initial guesses for x and y x0 = 0.7 y0 = 1 # Solve the system of equations using fsolve from scipy.optimize solution = fsolve(equations, (x0, y0)) # Print the solution print(f"The roots of the simultaneous nonlinear equations are: x = {solution[0]}, y = {solution[1]}") ``` ```python # main function that tests the Newton-Raphson method for solving the system of equations def main_function_name(): # test the equations function with initial guesses # use case 1 x0 = 0.7 y0 = 1 solution = fsolve(equations, (x0, y0)) assert abs(f1(*solution)) < 0.005 and abs(f2(*solution)) < 0.005 # use case 2 with different initial guesses x0 = -0.7 y0 = 1 solution = fsolve(equations, (x0, y0)) assert abs(f1(*solution)) < 0.005 and abs(f2(*solution)) < 0.005 # Call the main function to test the solutions main_function_name() ``` ∻Explanation∻ ⚹The code defines two functions, f1 and f2, representing the given nonlinear equations. It then defines a function called equations that takes a tuple of variables and returns a list of the results of f1 and f2. The fsolve function from scipy.optimize is used to find the roots of the system of equations given the initial guesses. The main function tests the solution with two use cases to ensure the roots satisfy the equations within the specified epsilon.⚹ ∻Step-by-step Instruction∻ ‖ ▪ ⋮ Define the functions f1 and f2 representing the given nonlinear equations. ‖ ‖ ▪ ⋮ Define the equations function that combines f1 and f2 into a system to be solved. ‖ ‖ ▪ ⋮ Use fsolve from scipy.optimize to find the roots of the system starting from the initial guesses. ‖ ‖ ▪ ⋮ Define the main function to test the solution with different use cases. ‖ ∻Time Complexity∻ ⚹The time complexity of the fsolve function depends on the complexity of the functions being solved and the convergence criteria. It typically involves iterative methods that can vary in performance based on the specific system of equations.⚹ ∻Space Complexity∻ ⚹The space complexity is generally low and depends on the storage of the function definitions and the variables used in the fsolve algorithm. It does not grow significantly with the size of the input for this problem⚹

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