Root finding, step by step

Set a bracket [a, b] (where f changes sign), a starting value x₀ and, for fixed-point iteration, a rearrangement x = g(x). Step ▶ takes one step with every method:

The error plot shows |xₙ − root| on a log scale (bisection’s straight line, Newton’s plunge, the secant’s golden-ratio order 1.618, fixed point’s constant ratio |g′(r)|), and reports the first step at which each method gets below the tolerance (10⁻¹² unless the link sets tol). The table lists every iterate; pick one method to see its full table — a, b, m and f(m) for bisection; xₙ, f(xₙ), f′(xₙ) and the correction for Newton; the error ratio for fixed point.

Drag x₀ (or click the curve) and see where Newton’s method goes, with each tangent drawn and a table of xₙ, f(xₙ), f′(xₙ) and the correction. The verdict says whether it converges, cycles, diverges, stops at a horizontal tangent, or wanders. The colour strip under the graph colours every starting value in view by the root it reaches — zoom in on x³ − x near ±0.447 to see the bands interleave. Animate replays the tangents one at a time; race bisection adds bisection on [a, b] to the error plot. For arctan x the page prints the exact divergence threshold, x₀ ≈ 1.3917452, where Newton bounces between x₀ and −x₀ for ever.

Newton’s method works unchanged on complex numbers. Every pixel is a starting value z₀, coloured by the root of z³ − 1 it reaches (darker: more steps). Click to zoom in ×2. Try z³ − 2z + 2 to see black islands of starting points trapped in the same 0 ↔ 1 cycle as on the real line.

The Babylonian (Heron’s) method: average your guess g with a/g. Drag the side of a square until its area matches a, then step: g and a/g close in on √a from opposite sides, and the table shows the correct digits of each guess in green — 1, 3, 6, 12, then all 16 a double can hold.

Related: Graphing calculator (roots, extrema, intersections), Where floating point bites (cancellation, the V-shaped error of numerical derivatives), Algebra step by step (the quadratic formula, factoring), Symbolic algebra (exact derivatives to paste into f′).

Companion to Mathematics, Reconnected: Powers and roots (the root finder), Quadratics (bisection versus Newton), Limits (the bisection stepper, Heron’s method), Applications of derivatives (Newton’s method explorer) and Numerical methods (root finding compared, Newton’s basins).

Every setting can be written by hand in the address, so a link can open an exact state, for example /tools/root-finding/?f=x^3-2x%2B2&x0=0&n=6#newton. The part after # picks the tab: #steps, #newton, #fractal or #sqrt. Settings a link leaves out take their defaults.

Write ` in an expression as `%2B` to be safe. A bare ` also works: a URL turns it into a space, and the tool reads a space between two terms as plus (x^2+1 → x^2 1 → x^2+1). Write products without spaces (2x, 2*x).

Book examples, as ?example=<name> (the tab is chosen for you):