#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Thu Apr  3 11:53:24 2025

@author: christian
"""

from scipy.optimize import fsolve, root, minimize
from sympy import symbols, Eq, nsolve
import numpy as np

### fsolve ###
def equations(vars):
    x, y = vars
    eq1 = x**2 + y**2 - 25
    eq2 = x**2 - y
    return [eq1, eq2]

initial_guess = [1, 1]
solution_fsolve= fsolve(equations, initial_guess)


### root ###
def equations(vars):
    x, y = vars
    eq1 = x**2 + y**2 - 25
    eq2 = x**2 - y
    return [eq1, eq2]

initial_guess = [1, 1]
solution_root = root(equations, initial_guess)


### minimize ###
# Define the equations
def equation1(x, y):
    return x**2 + y**2 - 25

def equation2(x, y):
    return x**2 - y

# Define the objective function for optimization
def objective(xy):
    x, y = xy
    return equation1(x, y)**2 + equation2(x, y)**2

# Initial guess
initial_guess = [1, 1]

# Perform optimization
result = minimize(objective, initial_guess)
solution_optimization = result.x


### nsolve ###
# Define the variables
x, y = symbols('x y')

# Define the equations
eq1 = Eq(x**2 + y**2, 25)
eq2 = Eq(x - y, 0)

# Initial guess for the solution
initial_guess = [1, 1]

# Use nsolve to find the solution
solution_nsolve = nsolve([eq1, eq2], [x, y], initial_guess)


### newton_method ###
def equations(vars):
    x, y = vars
    eq1 = x**2 + y**2 - 25
    eq2 = x**2 - y
    return np.array([eq1, eq2])

def newton_method(initial_guess, tolerance=1e-6, max_iter=100):
    vars = np.array(initial_guess, dtype=float)
    for _ in range(max_iter):
        J = np.array([[2 * vars[0], 2 * vars[1]], [2 * vars[0], -1]])
        F = equations(vars)
        delta = np.linalg.solve(J, -F)
        vars += delta
        if np.linalg.norm(delta) < tolerance:
            return vars

initial_guess = [1, 1]
solution_newton = newton_method(initial_guess)