Chebyshev Cyclic System
[MAL-Seeker Test Case] Chebyshev Cyclic System of Equations (8-Dimensional)
8-Variable Chebyshev Cyclic System of Equations
Roots: 256 Points
1. The Target Equation (A Fiercely Oscillating Challenge)
For the second test case of the all-solution search solver "MAL-Seeker," we tackled the 8-dimensional version of the "Chebyshev cyclic system of equations."
This equation is formed by cyclically coupling "Chebyshev polynomials," which are known to oscillate violently between -1 and 1. Because the variables are chained together in a ring, mapping this out creates a highly complex terrain crowded with numerous "valleys" and "peaks." There are exactly 256 true solutions in total.
2. A Terrain Where Local Search Gets "Lost"
If a local search solver like Newton's method is applied alone to such a wavy terrain, even a slight shift in the initial value can blast the solver into an adjacent valley, or the calculation might halt in a flat area where the gradient becomes zero. In short, the solver quickly gets "lost."
However, in the case of MAL-Seeker, the "Sense of Taste" (gradient test) possessed by the 100,000 "Marus" (search seeds) unleashed into the forest proves highly effective here. Before spinning the drill, they lick the slope at their feet to check the gradient, pre-detecting excessively flat singular points or steep cliffs that might blow them away, thereby enabling a remarkably stable search.
3. Execution Results
Here is the final terminal log of running MAL-Seeker in a local MacBook environment.
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Total Valid Roots Found: 256 (Out of 256 local minima)
Execution Time: 63.086525 seconds
=========================================From within this complex, oscillating terrain, we successfully discovered all 256 solutions without missing a single one. The computation time was approximately 63.1 seconds (about 1 minute).
Problems riddled with local minima and maxima, such as those involving Chebyshev polynomials, are notorious for causing calculations to diverge, making them a nightmare for solvers.
However, the pre-processing by Maru's "5 senses" (especially the gradient checks and fine-tuning via steepest descent) functioned beautifully. They reliably captured the valley bottoms while eliminating futile calculations. Being able to retrieve all 256 solutions in about 1 minute confirms the solid stability of MAL-Seeker as a search engine, which is a huge relief.
4. Analysis Environment
- Analysis Engine: MAL-Seeker (Antares Ver 3.1.1 / C Language)
- PC: MacBook
- OS: macOS Monterey 12.7.6
- CPU: 1.2GHz Dual-Core Intel Core m5
- Memory: 8GB