Root Finding Methods

 1def newton(f, df, x0, tol=1e-6, max_iter=100):
 2    """Newton's method for root finding"""
 3    x = x0
 4    for i in range(max_iter):
 5        fx = f(x)
 6        if abs(fx) < tol:
 7            return x
 8        x = x - fx / df(x)
 9    return x
10
11# Example: find sqrt(2)
12f = lambda x: x**2 - 2
13df = lambda x: 2*x
14root = newton(f, df, x0=1.0)
15print(f"√2 ≈ {root:.10f}")

 1def bisection(f, a, b, tol=1e-6):
 2    """Bisection method (slower but guaranteed)"""
 3    while (b - a) / 2 > tol:
 4        c = (a + b) / 2
 5        if f(c) == 0:
 6            return c
 7        elif f(a) * f(c) < 0:
 8            b = c
 9        else:
10            a = c
11    return (a + b) / 2

1from scipy.optimize import fsolve, brentq
2
3# Newton-type
4root = fsolve(f, x0=1.0)[0]
5
6# Brent's method (robust)
7root = brentq(f, a=0, b=2)