Two-Sided Approximations

Monotone approximations are interesting from not only a theoretical but also a practical point of view. In particular, two-sided approximations can be efficiently used as a posteriori estimations for the desired solution, which means that one can control the error at each iteration step. The iteration parameter in Newton-type methods plays an essential role in the convergence. It can expand the convergence domain and control the convergence behaviour. In particular, we established that in some cases \(\tau _k\) from the interval (0,1) enables the monotonicity of iterations, and the \(\tau \) -region of two-sided convergence is completely included in the interval (1,2).