## Franklin's Notes

### Iterative methods for matrices

An iterative method for solving a linear system or an eigenvalue problem is the opposite of a direct solver, and it produces a sequence of approximations converging to the exact solution. For instance, a direct solver for the linear system $Ax=b$ might make use of the LU Factorization , which yields an exact solution but has the high cost of $\mathcal{O}(m^3)$ for an $m\times m$ matrix. Iterative methods, like GMRES and conjugate gradient , rely on a mat-vec multiply as their core operation. This is particularly effective for sparse matrices: if the number of nonzero entries of $A$ per row is bounded, then the cost of calculating $Ax$ is $\mathcal{O}(m)$.

Here is a list of iterative methods: