Optimal-Order Parameter Identification in Solving Nonlinear Systems in a Banach Space

We study the sufficient and necessary conditions of the convergence for parameter-based rational methods in a Banach space. We derive a closed form of error bounds in terms of a real parameter $\lambda$ ($1 \leq \lambda < 2$). We also discuss some behaviors when the family is applied to abstract quadratic functions on a Banach space for $ \lambda = 2 $.