The Conjugate Gradients Method (CG)

This lecture introduces the Conjugate Gradients method, a technique for solving linear systems with a positive definite operator. Starting with the definition and properties of CG, the instructor explains the iterative process, the computation of residuals, and the concept of conjugate directions. The lecture covers the convergence properties and the necessary conditions for quadratic convergence, emphasizing the importance of linear independence among the conjugate directions. The instructor also discusses the representation of Hessians and the Riemannian distance on a manifold, providing insights into the practical implementation of CG.