Jacobi Method

The Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm used for solving a system of linear equations ( $Ax=b$ ).

Practice implementing it here:

• https://www.deep-ml.com/problems/11

Resources

• https://en.wikipedia.org/wiki/Jacobi_method

This method is particularly useful for large systems where direct methods, such as Gaussian Elimination, are computationally expensive.

1. Initialization: Start with an initial guess for ( $x$ ).
2. Iteration: For each equation ( i ), update ( $x[i$ ) using:

$x[i]=\frac{1}{a_{ii}}\left(b[i]-{\sum }_{j\ne i}{a}_{ij}x[j]\right)$

• where ( $a_{ii}$ ) are the diagonal elements of ( A ), and ( $a_{ij}$} ) are the off-diagonal elements.

1. Convergence: Repeat the iteration until the changes in $x$ are below a certain tolerance or until a maximum number of iterations is reached.