Animated: Newton’s tangents against bisection’s halving

Bisection keeps a bracket around the root and halves it every step, so it gains one binary digit per iteration: linear convergence. Newton’s method slides down the tangent line at the current guess and, near a simple root, doubles the number of correct digits per iteration: quadratic convergence. This animation runs both on Newton’s own cubic \(f(x) = x^3 - 2x - 5\), one iteration per frame, with the error of each growing in the bottom panel.

import matplotlib.pyplot as plt

from mathematicskit.numerical_analysis import Bisection, NewtonRaphson
from mathematicskit.numerical_analysis.visualizers import animate_root_finding

# In Jupyter or JupyterLite, show animations as an HTML/JavaScript player
# (no ffmpeg needed).
plt.rcParams["animation.html"] = "jshtml"


def f(x):
    return x**3 - 2.0 * x - 5.0


def fprime(x):
    return 3.0 * x**2 - 2.0


bisection = Bisection(f, 1.0, 3.0, tol=1e-12).solve()
newton = NewtonRaphson(f, fprime, x0=3.0, tol=1e-15).solve()
print(f"bisection: {bisection.iterations} iterations, Newton: {newton.iterations} iterations")

bisection: 41 iterations, Newton: 7 iterations

One iteration per frame

The orange band is bisection’s current bracket; the red segment is Newton’s tangent from \((x_{k-1}, f(x_{k-1}))\) down to \(x_k\). Newton reaches machine precision while bisection is still a few digits in, then holds still as bisection keeps halving.

anim = animate_root_finding(f, [bisection, newton], x_range=(0.8, 3.2))
anim

Once Loop Reflect

plt.show()