The Conjugate Gradient Method: Bridging Gradient Descent and Newton's Method
The Conjugate Gradient (CG) method serves as an intermediate approach between Steepest Descent and Newton's Method. It leverages only first-order derivative information while overcoming the slow convergence of Steepest Descent and avoiding the computational burden of storing, computing, and inverting the Hessian matrix required by Newton's Method. The CG method is not only one of the most useful techniques for solving large linear systems but also stands as one of the most efficient algorithms for large-scale nonlinear optimization problems. In implementation, CG typically uses iterative updates with conjugate directions computed through recurrence relations rather than matrix operations.