https://github.com/giuseppegrieco/conjugate_gradient
Preconditioned Conjugate Gradient implementation in MATLAB
https://github.com/giuseppegrieco/conjugate_gradient
conjugate-gradient conjugate-gradient-matlab preconditioned-conjugate-gradient preconditioner-matlab
Last synced: 13 days ago
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Preconditioned Conjugate Gradient implementation in MATLAB
- Host: GitHub
- URL: https://github.com/giuseppegrieco/conjugate_gradient
- Owner: giuseppegrieco
- Created: 2020-03-11T11:00:50.000Z (about 5 years ago)
- Default Branch: master
- Last Pushed: 2022-12-25T10:06:50.000Z (over 2 years ago)
- Last Synced: 2025-05-07T12:13:17.102Z (13 days ago)
- Topics: conjugate-gradient, conjugate-gradient-matlab, preconditioned-conjugate-gradient, preconditioner-matlab
- Language: MATLAB
- Homepage:
- Size: 45.9 KB
- Stars: 4
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# 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`.