# Solving non-linear equations

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## Introduction


## Prerequisites

```python
import numpy
import scipy
import sympy
```


## fsolve from SciPy 

```python
from scipy.optimize import 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(equations, initial_guess)
print("Solution:", solution)
```

```python
from scipy.optimize import fsolve

f = lambda x: x**3-100*x**2-x+100

fsolve(f, [2, 80])
```

## root from SciPy
```python
from scipy.optimize import 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(equations, initial_guess)
print("Solution:", solution.x)
```


## minimize from SciPy
```python
from scipy.optimize import 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

print("Optimization Method Solution:", solution_optimization)

```

## nsolve from SymPy
```python
from sympy import symbols, Eq, 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([eq1, eq2], [x, y], initial_guess)
print("Solution:", solution)
```

## newton_method from NumPy

```python
import numpy as np

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_method(initial_guess)
print("Solution:", solution)
```