What does the relaxation parameter ω do?

ω scales the Newton correction. ω = 1 is classical Newton; ω ≠ 1 is relaxed Newton. Changing ω alters convergence speed and can reshape basin boundaries, sometimes creating or destroying attractive cycles.

Why are the boundaries so intricate?

Near the basin boundaries, tiny changes in the starting guess send orbits to different roots. That sensitive dependence produces self-similar fractal separators rather than smooth Voronoi cells.

Does every starting point converge to a root?

Not necessarily. Some seeds diverge, wander, or fall into non-root attracting cycles, especially for altered ω. The colored basins highlight successful attraction to the n-th roots of unity within the iteration limit.

Newton Fractal

Basins of attraction for Newton iteration on zⁿ−1 with adjustable relaxation ω.

About this model

Newton’s method for roots of a complex polynomial iterates z_{n+1} = z_n − ω f(z_n)/f'(z_n). For f(z) = zⁿ − 1 the roots are the n-th roots of unity, and each attracts a basin of starting points; boundaries between basins form fractal Julia-like sets. This simulator colors pixels by which root the iteration converges to (and often by speed), with an adjustable relaxation parameter ω that can over-relax or under-relax the classical Newton step (ω = 1). Assumptions include exact arithmetic idealization, a fixed polynomial family zⁿ − 1, and no safeguarding against periodic cycles that are not roots. Vary n and ω to watch basins rearrange and fractal boundaries thicken or shift.

Who it's for: Numerical analysis, complex dynamics, and undergrad courses on root-finding and fractals.

Key terms

Newton fractal
Basins of attraction
Newton's method
Roots of unity
Relaxation parameter
Complex iteration

How it works

Newton’s method for p(z) = zⁿ − 1 colours each starting point by which root the iteration zₙ₊₁ = zₙ − ω·p(zₙ)/p′(zₙ) converges to. The boundary between the basins of attraction is a fractal with the property that near any point of three different colours all three colours meet — Newton himself never imagined this. Try n = 4 or 5 for the most photogenic basins; ω ≠ 1 turns it into a relaxed scheme with very different boundary structure.

Key equations

zₙ₊₁ = zₙ − ω · p(zₙ) / p′(zₙ)

p(z) = zⁿ − 1, p′(z) = n z^{n−1}

roots: ζₖ = exp(2πi k / n), k = 0…n−1

Frequently asked questions

What does the relaxation parameter ω do?: ω scales the Newton correction. ω = 1 is classical Newton; ω ≠ 1 is relaxed Newton. Changing ω alters convergence speed and can reshape basin boundaries, sometimes creating or destroying attractive cycles.
Why are the boundaries so intricate?: Near the basin boundaries, tiny changes in the starting guess send orbits to different roots. That sensitive dependence produces self-similar fractal separators rather than smooth Voronoi cells.
Does every starting point converge to a root?: Not necessarily. Some seeds diverge, wander, or fall into non-root attracting cycles, especially for altered ω. The colored basins highlight successful attraction to the n-th roots of unity within the iteration limit.

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How it works

Key equations

Frequently asked questions

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Frequently asked questions