Equations Archive
Nonlinear Equation Benchmark Problems
Optimization & Complementarity
Rosenbrock Function
Famous 'valley' function used as a standard test case for optimization problems.
Himmelblau's System
A fusion of SPICE Monte Carlo and Python data science. A hacking record of overcoming the "amnesia" of OP analysis via spatial parallelization to capture all four unknown roots simultaneously.
Rastrigin Function
Conquering the "devil's function", the Rastrigin function. Breaking through the limits of continuous mathematics (SPICE-oriented analysis) by building a powerful hybrid method with discrete mathematics. The genesis of our new core engine, "MAL-Seeker".
Ackley Function (20D)
Global optimization of the 20-variable Ackley function, a "bed of nails" where sharp local minima guard the center. A record of achieving a perfect landing at the origin by bypassing high-dimensional stagnation using V4.0.0's "Sixth Sense."
Schwefel Function (5D)
Global optimization of the notorious 5D Schwefel function, known for its "deceptive" terrain. A record of capturing the true solution hidden at the periphery by excluding central traps using "Deflation (Sense of Smell)."
Homotopy Global Search
Finding all 15 solutions for Branin's Six-Hump Camel function. Demonstrating continuous trajectory tracking using the Newton-Homotopy method.
Linear Programming
Analysis of the Klee-Minty problem. Exploration of global optimum using convex formulation with Augmented Lagrangian and Homotopy method applied to KKT conditions.
Linear Complementarity Problem
Analysis of LCP, a general framework encompassing Linear and Quadratic Programming. Reduction to nonlinear equations using the Fischer-Burmeister function and verification of solution search via Newton/Homotopy methods.
Nonlinear & Polynomial Systems
Broyden Tridiagonal Function
Tridiagonal structure problem for testing large-scale and high-dimensional scalability.
Powell's Singular Function
A problem where the Jacobian matrix becomes singular, targeting weaknesses in Newton's method.
Wilkinson's Polynomial
The monster of "ill-conditioned problems" in numerical analysis. Exploring the defeat of fixed-point homotopy and the magic of "self-scaling" via Newton homotopy to find all 20 roots.
Chebyshev Cyclic System
The second test case for MAL-Seeker. Challenging the fiercely oscillating 8-dimensional Chebyshev cyclic system of equations, a terrain where local search solvers easily get "lost". Maru's "Sense of Taste" (gradient test) effectively avoided futile searches, conquering all 256 solutions in about 1 minute.
10-Variable Nonlinear System
MAL-Seeker's first battle. Challenging a complex 10-variable nonlinear system of equations. A record of conquering all 1,024 solutions scattered across a 10-dimensional labyrinth in about 4 minutes, using extreme pre-processing with 100,000 "Marus".
Coupled Exp-Cos (10D)
A 10D fixed point problem with intricate exponential and cosine coupling. In a landscape so dense with local traps that the total root count is unknown, MAL-Seeker endured 5 minutes of stress testing to extract at least 11 true roots.
Katsura-4 Equations
An all-solution search challenging a global computational algebra benchmark using pure numerical computation. A proof of MAL-Seeker's perfection, flawlessly capturing all mathematically proven "12 real roots" in just 66 seconds.
MQ Problem (5D)
Analysis of a toy model for the Multivariable Quadratic (MQ) problem, the foundation of post-quantum cryptography. A record of absolute robustness, enduring 11.5 minutes without crashing in a brutal terrain of 1,168 local minima (traps) to extract two real roots.
Electronic Circuits
Bistable Latch Circuit
A bistable latch made from two cross-coupled cubic nonlinear inverters. The search expands from three physical operating points to five real roots and finally all nine roots over the complex domain.
Five-Node Tunnel-Diode Ladder
A five-node nonlinear ladder whose negative-resistance characteristic is represented by a cubic polynomial. Direct real search and total-degree homotopy both capture all 3⁵ = 243 DC operating points. Includes the reconstructed circuit and complete result data.
Differential Equations & Mechanics
1D Heat Equation
Numerical simulation of the 1D heat equation. Discretization process and equivalent circuit analysis using SPICE B-sources.
Double Pendulum
Double pendulum following Lagrangian mechanics. Visualization of chaotic behavior and verification of numerical accuracy based on energy conservation.
Lorenz Attractor
The gold standard of chaos theory found in weather models. Visualization of the 'Butterfly Effect' from deterministic equations.
Van der Pol Oscillator
Self-excited oscillation equation proposed as a vacuum tube model. Convergence to limit cycles and transition to relaxation oscillations.
Restricted Three-Body Problem
Search for the 5 equilibrium points (Lagrange points) in the restricted three-body problem (Earth-Moon system).
Coupled Phase Oscillator
The third test case for MAL-Seeker. A 6-dimensional Coupled Phase Oscillator Model with infinite periodic solutions. By combining a human-operated "Scouter" (dynamic bounding) with Maru's search, we filtered 729 true roots from 2048 local minima in 54 seconds.
Discrete Boundary Value Problem
Analysis of the discrete boundary value problem for a fixed beam's deflection curve. A record of suppressing gradient explosion and accurately determining the existence of a single unique root, proving its robustness as a simulation engine.
Stewart Platform
Forward kinematics of a 6-DOF parallel link mechanism. Numerical solution using SPICE and robust measures against singularities.
Physical, Biological & Economic Models
Chandrasekhar's H-equation
A nonlinear integral equation in radiative transfer theory. Discretized into a system of 100 equations using Gauss-Legendre quadrature, and successfully solved via SPICE DC analysis (homotopy-like approach).
Neurophysiology Model (10D)
Analysis of a 10-variable nonlinear neurophysiology model. In a harsh "Dense Matrix" environment where all variables interact, MAL-Seeker (Project Bellatrix) successfully extracted all 3 real roots with 7-digit theoretical precision.
Non-Isothermal CSTR
Analysis of steady-state multiplicity in a non-isothermal CSTR, a standard chemical engineering benchmark. A record of capturing all three states, including the difficult-to-find unstable saddle point, in just 1.9 seconds using MAL-Seeker (Project Bellatrix V1.0.0).
Multiphase Equilibrium
Solving the Rachford-Rice equation for Multiphase Equilibrium. Used Pseudo-Transient Analysis and the Penalty Method to strictly constrain the solution and track continuous phase transitions.
Scarf's Economic Equilibrium Problem
A benchmark problem in General Equilibrium Theory. Resolved the singularity caused by Walras' Law by introducing an auxiliary term, and successfully located the equilibrium solution via the Homotopy Method.
PHCpack Benchmarks
PHCpack Benchmark Test 4
Results for 18 previously unpublished PHCpack systems in Bellatrix V1.0.4, including Katsura systems, H-equations and a 461/462 checkpoint for pole34sys.
PHCpack Benchmark Test 3
The closing chapter of the PHCpack main collection. All 45 additional systems from cyclic7 through rose reach their official isolated-solution counts. Resume, reduction, scaling, and total-degree and multihomogeneous homotopies complete a first pass through the main 80 problems.
PHCpack Benchmark Test 2
The second PHCpack chapter. Complex arithmetic, symmetry closure, checkpoint resume, and reformulation conquer ten more systems from redcyc7 through caprasse. Every system reaches the official isolated-solution count as MAL-Seeker evolves from brute force into a structure-aware search.
PHCpack Benchmark Test
An application experiment on the "PHCpack" test suite, a notorious benchmark in algebraic geometry. While successfully overpowering 25 formidable problems using only pure Newton's method and 5SS, we also observed an unexplored limit: "gradient explosion" triggered by extreme high-degree polynomials. An interim report documenting the engine's robustness and its unfinished challenges exactly as they stand.