Conjugate Gradient Methods: Overview

This lecture covers the review of steepest descent and conjugate gradient methods, preconditioning, generalized conjugate gradient method, and nonlinear conjugate gradient methods. It discusses the line search using the Newton-Raphson method, singular value decomposition, and the convergence of the two algorithms. The instructor explains the remaining problems and solutions, such as preconditioning, generalized CG method, and nonlinear conjugate method. The concept of Jacobi preconditioning is introduced, along with the idea of solving Ax = b indirectly. The lecture also delves into the nonlinear conjugate gradient method, highlighting the differences from the ordinary CG method. Additionally, it explores the line search goal and the geometric interpretation of singular value decomposition (SVD) and eigenvalue decomposition (EVD).