Conjugate Gradient

Abstract

The conjugate gradient method is also aimed at accelerating the methods of Chap. 2. Its first motivation is to solve in n iterations a linear system with symmetric positive definite matrix (or, equivalently, to minimize in n iterations a quadratic strongly convex function on ℝⁿ), without storing an additional matrix, without even storing the matrix of the system. In fact, to solve Ax + b = 0 (A symmetric positive definite), the conjugate gradient method just needs a “black box” (a subroutine) which, given the vector u, computes the vector v = Au. Naturally, this becomes particularly interesting when, while n is large, A is sparse and/or enjoys some structure allowing automatic calculations. Typical examples come from the discretization of partial differential equations.