Newton's Method Square-Root Calculator
Apply Newton's iterative method to compute √a. Each step uses x_{n+1} = x_n − f(x_n)/f'(x_n) for f(x) = x² − a, so x_{n+1} = (x_n + a/x_n) / 2.
How to Use the Newton's Method Square-Root Calculator
- Enter the non-negative value a.
- Enter an initial guess x₀ (try 1 if unsure).
- Pick the number of iterations.
- Click Calculate.
Use Cases
- •Teaching numerical analysis.
- •Embedded systems without a hardware √.
- •Demonstrating quadratic convergence.
Formula
x_{n+1} = x_n − (x_n² − a) / (2·x_n). Quadratic convergence — 3–5 iterations are typically enough for double precision.
Frequently Asked Questions
Why Newton's method?
For well-behaved functions Newton's method converges quadratically — the number of correct digits roughly doubles each step. For √a it works for any positive initial guess.
What happens if the guess is bad?
A bad initial guess (too small or zero) slows convergence; starting with x₀ ≈ a/2 or x₀ = 1 is usually sufficient to reach machine precision in ~5 iterations.