Equations Archive
Nonlinear Equation Benchmark Problems
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.
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.
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).
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.
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.
1D Heat Equation
Numerical simulation of the 1D heat equation. Discretization process and equivalent circuit analysis using SPICE B-sources.
Restricted Three-Body Problem
Search for the 5 equilibrium points (Lagrange points) in the restricted three-body problem (Earth-Moon system).
Double Pendulum
Double pendulum following Lagrangian mechanics. Visualization of chaotic behavior and verification of numerical accuracy based on energy conservation.
Stewart Platform
Forward kinematics of a 6-DOF parallel link mechanism. Numerical solution using SPICE and robust measures against singularities.
Van der Pol Oscillator
Self-excited oscillation equation proposed as a vacuum tube model. Convergence to limit cycles and transition to relaxation oscillations.
Lorenz Attractor
The gold standard of chaos theory found in weather models. Visualization of the 'Butterfly Effect' from deterministic equations.
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.
Rosenbrock Function
Famous 'valley' function used as a standard test case for optimization problems.