5 Conjugate Gradient Methods

• 5.1 The Linear Conjugate Gradient Method
  • Conjugate Direction Methods
• 5.2 Nonlinear Conjugate Gradient Method

Our interest in conjugate gradient methods is twofold. First, they are among the most useful techniques for solving large linear systems of equations. Second, they can be adapted to solve nonlinear optimization problems.

The performance of the linear conjugate gradient method is determined by the distribution of the eigenvalues of the coefficient method. By transforming, or preconditioning the linear system, we can improve the convergence of the method.

The key features of the nonlinear conjugate gradient methods are that they require no matrix storage and are faster than the steepest descent method.