First, we split A into two matrices and in the following manner so that we can solve the system easily. Let's now look at how the errors are related from step to . Let be the exact solution, the estimate of at the iteration and our initial guess for the iterative method. The error at the step satisfies which leads to The iterate will converge if as . This will happen if every eigenvalue of is less than 1 in magnitude This is equivalent to saying that the spectral norm of , .
Many iterative methods are linearly convergent with constant , ie.
There are many iterative methods each depending on the way the coefficient matrix is decomposed or split . In Jacobi method matrix is diagonal, so is trivially solved. In Gauss-Sidel and SOR matrix is triangular, so can be solved much more efficiently than . Nonstationary methods differ from stationary methods since the computations involve information that changes from one step to another, that is, changes as the iteration index changes. For a general matrix algebra, iterative schemes, etc.