Chapter 6: The Conjugate Gradient method
Abstract
In this chapter, we discuss the Conjugate Gradient (CG) method for solving\[\text{A}x = b,\](6.1)where \(\text{A} \in \mathbb{R}^{n\times n}\) is a symmetric positive definite matrix. Such linear systems arise commonly when considering inverse problems. For example, when using Tikhonov regularization for linear inverse problems, we need to solve a linear system with a symmetric positive definite coefficient matrix; see, e.g., (2.17). As another example, when using the Gauss–Newton method for solving inverse problems, a system with a symmetric positive definite coefficient matrix is solved in each iteration to find a search direction; see Chapter 7.
Fields of Study & Subjects
Matrix Theory and Algorithms
Numerical methods in inverse problems
Advanced Optimization Algorithms Research
Publication details
- Journal / source
- Society for Industrial and Applied Mathematics eBooks
- Pages
- 73–81
- Publication date
- 27 February 2026
- Language
- EN
- Document type
- Book chapter