Newton's Method
Numerical root finding
Finds a root of f(x)=0 numerically
About This Calculator
Newton's method (Newton-Raphson) is an iterative algorithm that finds roots of a differentiable function f(x) by repeatedly improving an initial guess. Starting from x₀, each iteration moves along the tangent line to the x-axis, rapidly converging to a root. It typically converges quadratically — doubling the correct digits with each step.
Formula
xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)
Repeat until |xₙ₊₁ − xₙ| < tolerance
Convergence is quadratic: errors roughly square each iteration
Example Calculation
Find √2 using Newton's method (solve x²−2=0, start x₀=1).
- f(x)=x²−2, f'(x)=2x
- x₁ = 1 − (1−2)/(2) = 1 + 0.5 = 1.5
- x₂ = 1.5 − (2.25−2)/3 = 1.5 − 0.0833 = 1.4167
- x₃ = 1.4167 − (2.0070−2)/2.8333 ≈ 1.4142
Converges to √2 ≈ 1.41421 in 4 iterations
Newton's Method Convergence for √2
| Iteration | x_n | f(x_n) = x²−2 | Error |
|---|---|---|---|
| 0 | 1.0000 | −1.0000 | 0.4142 |
| 1 | 1.5000 | 0.2500 | 0.0858 |
| 2 | 1.4167 | 0.0069 | 0.0025 |
| 3 | 1.4142 | 0.000006 | 0.000002 |
| 4 | 1.41421356 | ~0 | < 10⁻¹² |
Frequently Asked Questions
Does Newton's method always converge?
No. It can diverge or cycle if the initial guess is poor, the function has inflection points near the root, or the derivative is zero. Choosing a starting point close to the expected root and checking convergence each iteration is important.
What is the advantage over bisection method?
Newton's method converges quadratically (doubles correct digits each step) versus linear convergence of bisection. It reaches very high precision in far fewer iterations, but requires computing the derivative and a good starting guess.
What is the Newton-Raphson method used for in practice?
It is used in computer graphics (ray tracing, rendering), optimization, solving nonlinear systems of equations, computing square roots in hardware, calibration in engineering, and financial option pricing.
What happens when f'(x) = 0?
If the derivative at xₙ is zero, the formula has a division by zero and fails. This typically happens at local maxima or minima. A small perturbation to the guess or a modified method (e.g. Halley's method) can work around this.