Dynamic Iteration Error Analysis

We investigate the relationship between the order of magnitude of parameter values underpinning given differential systems and the propagation of the errors of the corresponding Jacobi dynamic iterative schemes. We do so by applying them to a test problem involving two-dimensional linear differential systems and by deriving exact formulas for these errors. The formulas are provided for any iteration index k and are written in terms of given parameters and an initial error at \(k=0\) . These formulas allow to deduce the principle that the smaller the magnitude of the particular parameters that are multiplied by previous iterates, the faster the rate of convergence of the dynamic iterative schemes. Another principle resulting from the formulas is that the smaller the value of the parameters multiplied by present iterates, the faster the convergence of the dynamic iterative schemes. We also derive relationships between the errors of Jacobi dynamic iterative schemes applied to differential systems that differ by off-diagonal parameters. We finish with numerical examples demonstrating the theoretical findings.