OCCT/src/FoundationClasses/TKMath/MathSys

MathSys Package

Overview

The MathSys package provides algorithms for solving systems of nonlinear equations. These solvers are essential for geometric computations involving curve/surface intersections, constraint solving, and optimization problems in CAD/CAM applications.

Available Algorithms

Newton-Raphson Method (MathSys_Newton.hxx)

The Newton-Raphson method is an iterative algorithm for finding roots of systems of equations F(X) = 0. At each iteration, it solves the linear system J(X) * dX = -F(X), where J is the Jacobian matrix.

Features:

• Fast quadratic convergence near the solution
• Supports per-variable tolerance specification
• Bounded variant with box constraints

Functions:

• MathSys::Newton() - Standard Newton-Raphson iteration
• MathSys::NewtonBounded() - Newton method with box constraints

Optimized 2D Newton Solver (MathSys_Newton2D.hxx)

A highly optimized Newton solver specifically designed for 2D problems (systems of 2 equations in 2 unknowns). Common use cases include point-surface extrema, curve intersections, and surface-surface intersection curves.

Features:

• Zero allocation overhead (no math_Vector/math_Matrix)
• Cramer's rule for 2x2 linear system (faster than LU decomposition)
• Squared norm comparisons (avoids sqrt in convergence check)
• Step limiting to prevent wild oscillations
• Gradient descent fallback for singular Jacobian
• Symmetric Jacobian variant for gradient-based problems

Functions:

• MathSys::Newton2D() - General 2D Newton solver
• MathSys::Newton2DSymmetric() - Optimized for symmetric Jacobians (e.g., gradient of scalar function)

Usage Example (Point-Surface Extrema):

// Function with symmetric Jacobian: F = [dD/dU, dD/dV] where D = ||S(u,v) - P||^2
auto aResult = MathSys::Newton2DSymmetric(myDistFunc, u0, v0,
                                           uMin, uMax, vMin, vMax,
                                           tolerance, maxIter);
if (aResult.IsDone())
{
  double u = aResult.U;
  double v = aResult.V;
}

Levenberg-Marquardt Algorithm (MathSys_LevenbergMarquardt.hxx)

The Levenberg-Marquardt algorithm is designed for nonlinear least squares problems, minimizing ||F(X)||^2. It combines Gauss-Newton (fast convergence near minimum) with gradient descent (robust far from minimum) using adaptive damping.

Features:

• Robust convergence even with poor initial guesses
• Adaptive damping parameter for stability
• Configurable convergence criteria
• Bounded variant with box constraints

Functions:

• MathSys::LevenbergMarquardt() - Standard LM algorithm
• MathSys::LevenbergMarquardtBounded() - LM with box constraints

Usage

All solvers in this package operate on function sets that provide:

• NbVariables() - Number of variables in the system
• NbEquations() - Number of equations
• Values(X, F, J) - Compute function values and Jacobian at point X

Results are returned as VectorResult structures containing:

• Status - Convergence status (OK, MaxIterations, NumericalError, etc.)
• Solution - The computed solution vector
• NbIterations - Number of iterations performed

Dependencies

• MathUtils - Core utilities, types, and configuration
• MathLin - Linear algebra solvers for the Newton step