Root Finding Calculator
Find a root of f(x) = 0 by bisection, regula falsi, Newton–Raphson, the secant method or fixed-point iteration, with the full iteration table.
About the Root Finding Calculator
Bisection halves an interval [a, b] where f changes sign, keeping the half that still has the sign change: slow but sure. Regula falsi uses the chord through (a, f(a)) and (b, f(b)) instead of the midpoint. Newton–Raphson follows the tangent, xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ): very fast near a simple root, but it needs a good start and f′ ≠ 0. The secant method replaces f′ by the slope through the last two points. Fixed-point iteration rewrites f(x) = 0 as x = g(x) and repeats xₙ₊₁ = g(xₙ), which converges when |g′| < 1 near the root.
Each answer is substituted back to confirm that f is (very nearly) 0.
The formula
Newton–Raphson: xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ); bisection: c = (a + b)/2
- Regula falsi
- c = (a f(b) − b f(a))/(f(b) − f(a))
- Secant
- xₙ₊₁ = xₙ − f(xₙ)(xₙ − xₙ₋₁)/(f(xₙ) − f(xₙ₋₁))
Worked example
x³ − x − 1 = 0 on [1, 2]
- f(1) = −1, f(2) = 5: a root lies between
- Bisection: 1.5, 1.25, 1.375, 1.3125, …
- Root ≈ 1.3247 (4 d.p.)
Questions people ask
Why does Newton's method sometimes fail?
If f′(x) is 0 or tiny near a guess, the tangent shoots far away; and a poor start can circle or diverge. Bisection always works once the root is bracketed.
How many bisection steps do I need?
Each step halves the interval, so about log₂((b − a)/tolerance) steps.