DEV Community: Rat Wolf (ratwolf)

A Triplet-Based Parameterization for the Local Asymptotic Characterization of Polynomial Roots

Rat Wolf (ratwolf) — Tue, 31 Mar 2026 08:24:01 +0000

Abstract

This paper introduces a compact three-parameter framework for characterizing the local geometry of polynomial roots. For each root, the framework records its position, its algebraic multiplicity, and a newly defined quantity called the characteristic deflection distance 𝛿. This third parameter acts as a natural geometric scale: it measures how sharply or gradually the polynomial departs from zero in the immediate vicinity of the root, and it encodes the collective influence of all other roots through their distances from the one being analyzed.

The characteristic deflection distance generalizes the classical condition number of a simple root to roots of arbitrary multiplicity, and it allows direct geometric comparison across roots of different degrees. A key finding is that multiplicity alone does not determine geometric dominance — a lower-multiplicity root can have a larger spatial footprint than a higher-multiplicity one, depending on the global root configuration.

Symbolic and numerical Python implementations are provided, along with a worked example. The framework extends naturally to polynomials over the complex numbers.

DOI (Zenodo)

To visualize the spatial competition between these triplets, we map the 𝛿-Normalized Distance Field and the Newton Flow. This provides a "topographical" view of the polynomial's geometry where every root is evaluated against its own intrinsic scale.

GitHub

High-Precision Companion Matrix Root Finder

Rat Wolf (ratwolf) — Fri, 13 Mar 2026 06:53:50 +0000

High-Precision Companion Matrix Root Finder

GitHub
Medium

High-Precision Companion Matrix Root Finder

A robust polynomial solver leveraging arbitrary-precision floating-point arithmetic to compute roots of ill-conditioned polynomials. This tool primarily addresses rounding and finite-precision issues present in standard 64-bit floating-point implementations by constructing and solving a companion matrix within the mpmath environment.

The solver cannot eliminate the intrinsic mathematical ill-conditioning of certain polynomials, but it can significantly reduce the numerical errors caused by rounding, making the computed roots far more reliable.

It features modular code structure, residual checks, and robust input validation to improve usability and reliability.

Key Idea

The solver does not attempt to solve the fundamental conditioning problem of polynomial root finding.

Instead, it focuses on reducing numerical rounding errors using high‑precision arithmetic, which in practice leads to far more reliable root approximations for difficult polynomials.

Fast Chladni Pattern Simulation with a Simplified Membrane Model Using Vectorization and Stochastic Particles View

Rat Wolf (ratwolf) — Thu, 05 Mar 2026 06:25:48 +0000

Chladni Figures Simulation

This Python code simulates the nodal line patterns of Chladni figures, approximating Ernst Chladni’s 1787 experiments with vibrating plates by modeling the resonant modes of a square membrane. It captures the visual essence of these wave patterns without simulating the complex physics of elastic bending and the dynamics of particle motion.

Python Code, detailed simulation description, physics and mathematics see:

GitHub

Magnitude View: A 2D static plot showing the time-independent displacement magnitude ∣Z(x,y,f)∣ of the steady-state response at a driving frequency f matching the degenerate modes (3,5) and (5,3). A small damping factor (γ = 0.01) ensures these modes dominate. The magnitude highlights nodal lines and makes positive and negative antinodes indistinguishable, mimicking experimental Chladni patterns.

Phase View: This view displays the signed spatial amplitude of the steady-state response at a given frequency. The sign indicates whether a region oscillates in phase or π out of phase with a reference. Nodal lines appear where the amplitude vanishes. The membrane motion remains oscillatory in time; this view represents the spatial structure and phase relationships of that motion.

Sand View: The Sand View acts as a visually intuitive “nodal detector” derived from the same resonance-weighted field used in the magnitude and phase views.

Relation to Physical Experiments

In real Chladni experiments:

Grains bounce dynamically due to plate acceleration. Migration occurs through complex frictional and collisional processes.
Accumulation is influenced by time evolution and nonlinear interactions.

The Sand View abstracts these dynamics into a stationary probability model. It captures the final accumulation pattern without modeling transient motion.

Limitations

Field-based simulation; does not model particle dynamics.

Note: This simulation visualizes the nodal patterns of a unit square membrane L_x = L_y with fixed edges, governed by the wave equation, capturing the essence of Chladni figures. In real experiments, elastic plates follow complex bending physics, but the membrane model approximates their nodal patterns using simpler wave mechanics. Focusing on nodal patterns avoids the unresolved complexity of grain motion, highlighting the universal physics of resonance, independent of material properties.

Only finite max_mode included.
Damping gamma is uniform; real plates have non-uniform damping.
Geometry is idealized unit square membrane.
Uses a membrane model for computational efficiency, which simplifies the physics of a true plate with bending stiffness.

Simulation–Experiment Match Fidelity

Mode Category	Typical Example(s)	Match Fidelity	Notes
Exact Symmetric Modes	(2,2), (3,3), (4,4)	100%	Perfect match: Clean axial nodal lines align exactly with experiments.
Low-order Degenerate Pairs	(1,2)+(2,1), (1,3)+(3,1)	~50–60%	Partial match: Shows tilted/blended nodals instead of sharp crosses seen in some low-order plate patterns.
High-order Degenerate Pairs	(3,5)+(5,3), (1,6)+(6,1)	~90–95%	Excellent match: Complex, interior nodals closely mimic historical figures; diagonals may flip orientation.

Notes:

Symmetric modes m,m are exact (100%) and visually identical to experiments.

Asymmetric modes m ≠ n always form degenerate pairs in this square model, blending into symmetric patterns—both contribute fully e.g., (1,2) and (2,1) dominate together. No isolated "single" asymmetric modes appear due to symmetry.

Low-order pairs can look less crisp; high-order excel at reproducing rich, experimental-like topology, including accidental degeneracies where symmetric modes blend with pairs (e.g., (10,10) + (2,14)/(14,2)). Larger gamma adds asymmetry from nearby modes, enhancing realism.

Ratings prioritize nodal patterns (positions and shapes); the sim's blending captures the visual spirit of Chladni figures effectively, even if idealized.

Historical Context

Ernst Chladni (1756–1827), known as the father of acoustics, pioneered the study of vibrational modes by observing how elastic plates, sprinkled with sand or powder, formed Chladni figures—geometric patterns along nodal lines where the surface remains stationary. These patterns reveal standing waves and remain a subject of research today.

In his 1787 experiments:

An elastic plate (thin brass plate) was fixed at its center or edges.
It was vibrated with a violin bow at various frequencies.
Particles migrated from high-vibration areas to nodal lines, creating striking patterns.

Chladni figures laid foundations for acoustics, wave physics, and applications like mechanical engineering, with their nodal patterns inspiring both historical and modern scientific inquiry.

References

Chladni, Ernst Florens Friedrich. Entdeckungen über die Theorie des Klanges. Leipzig: Weidmanns Erben und Reich, 1787. ETH-Bibliothek Zürich,
Rar 5284. https://doi.org/10.3931/e-rara-4235. Public Domain Mark.
Herman, Russell. Vibrations of Rectangular Membranes. (2024, September 4) University of North Carolina Wilmington. CC BY‑NC‑SA. https://math.libretexts.org/@go/page/90264
Wikipedia contributors. “Ernst Chladni.” Wikipedia. https://en.wikipedia.org/wiki/Ernst_Chladni

An old Classic with new Perspective in 2D and 3D

Rat Wolf (ratwolf) — Wed, 04 Dec 2024 14:31:34 +0000

Historical Context

The Hopalong* attractor, authored by Barry Martin of Aston University in Birmingham, England [2], was popularized by A.K. Dewdney in the September 1986 issue of Scientific American. In Germany, the Hopalong attractor gained further popularity through an algorithm called HÜPFER in the article Psychotapeten published in Spektrum der Wissenschaft. [3].

*Nicknamed by A.K. Dewdney.

Python Code, detailed description and mathematics see:

GitHub

The Hopalong Attractor Functions

The mathematical definition of the Hopalong attractor is given by the following system of recursive functions, defined in [2]:

Where:

The sequence starts from the initial point (x0 , y0) = (0 , 0).
xn and yn represent the coordinates at the n-th iteration of the attractor.
a, b, and c are parameters influencing the attractor's dynamics.
sgn is the signum function. However, the programs use math.copysign(), which is defined as follows:

Computational Approach

The available Python programs calculate and visualize the attractor by iterating the described function system.

Goal:

Visual Representation:
The attractor is rendered as a density heatmap, with color intensity reflecting areas of high or low point concentration.
Efficiency:
- Large numbers of iterations are computed with low memory usage and high processing speed.
- Just-in-time (JIT) compilation is applied and supported by a low-complexity code structure.

Core Algorithm:

The algorithm consists of two main passes:

First Pass: Determines the spatial extent of the attractor trajectory.
Second Pass: Maps the sequentially generated trajectory points in continuous space directly to a discrete pixel grid, keeping track of the number of pixel hits used to generate the density heatmap matrix.

References

[1] J. Lansdown and R. A. Earnshaw (eds.), Computers in Art, Design and Animation. New York: Springer-Verlag, 1989.
e-ISBN-13: 978-1-4612-4538-4.

[2] Barry Martin, "Graphic Potential of Recursive Functions," in Computers in Art, Design and Animation [1], pp. 109–129.

[3] A.K. Dewdney, Psychotapeten, algorithm HÜPFER, in Spektrum der Wissenschaft: Computer Kurzweil.
Spektrum der Wissenschaft Verlagsgesellschaft mbH & Co., Heidelberg, 1988.
(German version of Scientific American).
ISBN-10: 3922508502, ISBN-13: 978-3922508502.