Conjugate Gradient and Preconditioners in MATLAB

This repository contains the implementation of the conjugate gradient (CG) method and three preconditioners (Jacobi, Incomplete LU, and Incomplete Cholesky) in MATLAB.

Conjugate Gradient Method

The conjugate gradient (CG) method is an iterative algorithm for solving linear systems of equations of the form Ax=b, where A is a symmetric positive definite matrix. It is particularly useful when the matrix A is large and sparse, as it can converge faster than direct methods such as LU decomposition.

Preconditioners

Preconditioners are techniques used to improve the convergence of iterative methods such as the conjugate gradient (CG) method. They operate by transforming the original system Ax=b into a new system M^{-1}Ax=M^{-1}b, where M is a preconditioner matrix. The choice of M can significantly affect the convergence of the CG method.

There are many different types of preconditioners, and the best one to use depends on the properties of the matrix A and the problem at hand. The preconditioners included are:

• Jacobi preconditioner: Uses the diagonal elements of the matrix A to construct the preconditioner matrix M.
• Incomplete LU (ILU) preconditioner: Uses an incomplete factorization of the matrix A to construct the preconditioner matrix M.
• Incomplete Cholesky (IC) preconditioner: Also uses an incomplete factorization of the matrix A to construct the preconditioner matrix M.