Concept

Newtons method

The iteration that replaces a guess at a root by where the tangent there crosses the axis. Its derivative at a simple root is exactly zero, so the error is squared at each step and the number of correct digits doubles — which is why it converges in a handful of steps and why it fails badly at a multiple root.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The basins of Newton's method on z³ = 1. The complex plane coloured by which cube root of one Newton's method converges to from each starting point.

Where Newton's method goes instead

An algorithm designed to find roots, run from every starting point at once. Three roots, three basins, and a boundary at which all three are arbitrarily close — so a rule with no randomness in it has starting points whose answer cannot be predicted.

dynamics · newton basins

How fast an iteration arrives. The distance from the fixed point plotted against the step number on a logarithmic vertical axis, for two ordinary iterations and for Newton's method, whose curve bends downward.

How fast the staircase arrives

The slope at a crossing decides whether an orbit reaches it. The same number decides how fast — and when the slope is zero the arithmetic changes kind, from a fixed factor per step to a doubling of the correct digits.

dynamics · iteration

Named alongside it

The objects these essays reach for when they reach for this one.

Derivative Iteration Basin of attraction Complex numbers Convergence Convergence rate Error analysis Fixed point Fractal Linear convergence Multiplier Quadratic convergence

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