Root Finding by High Order Iterative Methods Based on Quadratures
For numerical analysts seeking efficient root-finding algorithms, this provides a theoretical framework for generating higher-order methods, though it is an incremental extension of known quadrature-based approaches.
This paper proposes a recursive family of high-order iterative methods for root finding, based on Newton-Cotes quadrature rules, and proves that using an n+1 node quadrature yields convergence order at least n+2, with n=0 recovering Newton's method.
We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with $n+1$ nodes is used the resulting iterative method has convergence order at least $n+2$, starting with the case $n=0$ (which corresponds to the Newton's method).