Concept

Quadratic convergence

The behaviour of an iteration whose error is roughly squared at each step, so the number of correct digits doubles. Newton's method has it near a simple root, and it is what lets a correction outrun losses that would defeat a slower method.

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

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

The coordinate change that makes the map a rotation. Graphs of the departure from the identity of the conjugacy between the circle map and the golden rotation, at three nonlinearities, each smooth and growing steeper with K.

How smooth the disguise is

An unlocked circle map is a rigid rotation in different coordinates, and whether those coordinates are smooth is decided by arithmetic. Solving for the change of coordinates means dividing, harmonic by harmonic, by numbers that come arbitrarily close to nought — and the golden rotation number keeps them far enough away that the result is not merely smooth but analytic, while a number lying close to a fraction makes it buckle.

dynamics · Mode locking

Named alongside it

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

Newtons method Analytic function Circle map Conjugacy Continued fractions Convergence rate Derivative Diophantine approximation Error analysis Fixed point Fourier series Iteration

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