r/AppliedMath • u/Active_Television_10 • May 03 '26

Stability vs. Divergence: A Computational Study of Parameter Space for Nonlinear Root-Finding

I wanted to share a visual output from my latest preprint, focused on the Parameter Space Study for a family of Chebyshev-like iterative methods for solving nonlinear equations.

The construction of this space is based on analysis of critical point dynamics. As shown in the attached snippet, we analyze how the map S(z) behaves for a wide range of complex values for the parameter K.

Parameter space

I would love to get your thoughts or feedback on this computational approach to evaluating root-finding methods.

Preprint

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u/New-Economy123 20d ago

Nice work. The basin-of-attraction analysis is exactly what's missing from most root-finding implementations — people assume convergence without characterizing where the method actually fails.

Two things I found particularly useful:

  1. The Möbius reduction to a single parameter K is elegant. It means you can precompute the safe initialization regions once and reuse them across all polynomials with the same multiplicity ratio.

  2. The strange fixed point stability analysis — knowing exactly when false attractors become trapping — is directly relevant to hardware implementations where you can't retry or recover from divergence. We're building a balanced ternary ALU (MathLoom) that does iterative division, and your convergence basin maps would define the safe starting regions if we extend it to root finding. No room for chaotic orbits on silicon.

Have you considered extending this to balanced ternary arithmetic? The truncation-equals-rounding property changes the error propagation in interesting ways.