Why do Newton basins have complicated boundaries?

Newton steps are local linear solves, but far from a root those steps can send nearby initial guesses to different roots. Repeating this nonlinear map creates sensitive, often fractal-like basin boundaries.

What is a Jacobian failure?

The Newton step requires solving J delta = F. If the Jacobian is singular or nearly singular, the step is undefined or extremely large, so the iteration may stall, escape, or fail the convergence test.

Newton-Raphson Basins in 2D Systems

Map Newton basins for nonlinear F(x,y)=0 systems: initial-guess sensitivity, iteration counts, root attraction, and singular-Jacobian failures.

Each pixel is a different initial guess. Newton can converge quadratically near a simple root, but far away the same formula can jump across basins, hit a nearly singular Jacobian, or fail to converge within the iteration limit.

Live graphs

About this model

Newton-Raphson in two variables updates u=(x,y) with u_{k+1}=u_k-J(u_k)^{-1}F(u_k). This simulator treats z^n-1=0 as the real 2D system Re(z^n-1)=0 and Im(z^n-1)=0, then colors every initial guess by the root it reaches. Brightness shows iteration count, dark pixels mark failures, and the probe path reveals how a single starting point can jump between basins or hit a nearly singular Jacobian.

Who it's for: Numerical methods, nonlinear equations, applied mathematics, complex dynamics, and scientific computing courses.

Key terms

Newton-Raphson method
Basin of attraction
Jacobian
Nonlinear system
Initial guess
Quadratic convergence

How it works

Newton-Raphson basin visualizer for nonlinear 2D systems F(x,y)=0, showing convergence basins, initial-guess sensitivity, and Jacobian failures.

Key equations

Newton step: u_{k+1}=u_k − J(u_k)^{-1}F(u_k), u=(x,y)

Demo system: z^n−1=0, equivalent to Re(z^n−1)=0 and Im(z^n−1)=0

Frequently asked questions

Why do Newton basins have complicated boundaries?: Newton steps are local linear solves, but far from a root those steps can send nearby initial guesses to different roots. Repeating this nonlinear map creates sensitive, often fractal-like basin boundaries.
What is a Jacobian failure?: The Newton step requires solving J delta = F. If the Jacobian is singular or nearly singular, the step is undefined or extremely large, so the iteration may stall, escape, or fail the convergence test.

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Newton-Raphson Basins in 2D Systems

Live graphs

About this model

How it works

Key equations

Frequently asked questions

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Newton-Raphson Basins in 2D Systems

Live graphs

About this model

How it works

Key equations

Frequently asked questions