On solving nonlinear simultaneous equations arising from the double-exponential Sinc-collocation method for initial value problems
For researchers using the double-exponential Sinc-collocation method, this work provides theoretical justification for the observed fast convergence of the fixed-point iteration, but the analysis is incremental as it builds on existing methods.
The paper analyzes the convergence of Gauss-Seidel type fixed-point iteration for solving nonlinear equations arising from the double-exponential Sinc-collocation method for initial value problems, providing a sufficient condition for global convergence and an upper bound on the convergence factor that explains its efficiency.
The double-exponential Sinc-collocation method is known as a super-accurate method for solving initial value problems of ordinary differential equations, for which the error decreases almost exponentially as a function of the number of sample points in the temporal direction, $N$. However, this method requires solving nonlinear simultaneous equations in $nN$ variables when the problem dimension is $n$. Recently, Ogata pointed out that Gauss-Seidel type fixed-point iteration works surprisingly well for solving these equations, typically reducing the error by one or two orders of magnitude at each iteration. In this paper, we analyze the convergence of this iteration and give a sufficient condition for its global convergence. We also provide an upper bound on its convergence factor, which explains the efficiency of this iteration. Some numerical examples that illustrate the validity of our analysis are also provided.