Math & Kinematics

Conjugate Gradient Method

The conjugate gradient method is an iterative algorithm for solving large systems of linear equations with symmetric positive-definite matrices, and by extension for unconstrained optimization, by searching along mutually conjugate directions. Developed by Hestenes and Stiefel (1952), it converges in at most n iterations for an n-dimensional exact problem and requires only matrix-vector products, making it memory efficient. Nonlinear variants extend it to general smooth objectives.

Why it matters for physical AI

Conjugate gradient solvers appear inside physics engines for contact dynamics, in trajectory optimization, and in natural-gradient reinforcement learning methods like TRPO, quietly powering both simulation and policy optimization.