$L$-derivative of an approximate solution to $\Phi '=\bold A(t)\Phi $: series and product formulae for left corrections

Igor Najfeld, William Lakin
2003 Quarterly of Applied Mathematics  
Given an initial approximation <&0 to the fundamental matrix of solutions for = A(t), it is shown that a left correction, T^o, is locally more accurate than a right correction, ^oT. For each relative error function considered, there is a left correction r and the associated differential equation. The common feature is the same integrable part whose forcing function is the difference between L-derivatives of the exact and the initial solution. Upon transformation into a Volterra integral
more » ... , fixed point iterations generate infinite series of a lacunary type which converge globally whenever an integral equation is linear. Alternatively, when the integrable solution is used for iterative refinement, the outcomes are infinite product representations. Necessary conditions for the absolute convergence are given.
doi:10.1090/qam/1999836 fatcat:u4lojhopujfz3jgjlyfnhyh3aq