## Franklin's Notes

### Jacobi method

The Jacobi method approximates a solution to the linear system $Ax=b$ iteratively as follows. We start with an initial guess $x^{(0)}$ for the solution, and calculate successive approximations $x^{(k)}$ using the following recurrence: where $D$ is the diagonal matrix whose diagonal entries are the diagonal entries of $A$, and $r^{(k)}=b-Ax^{(k)}$ is the residual of $x^{(k)}$. This method relies on the assumption that $D^{-1}r^{(k)}\approx A^{-1}r^{(k)}$.