A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm

In this paper, we aimed to investigate the error inequality of the open method, known as Euler-Maclaurin's inequality, which is similar to Simpson's rule. We intended to explore some novel Maclaurin-like inequalities involving functions having convexity properties. To further accomplish this task, we built an identity and demonstrated new inequalities. With the help of a new auxiliary result and some well-known ones, like Hölder's, the power mean, improved Hölder, improved power mean, convexity, and bounded features of the function, we obtained new bounds for Euler-Maclaurin's inequality. From an applicable perspective, we developed several intriguing applications of our results, which illustrated the relationship between the means of real numbers and the error bounds of quadrature schemes. We also included a graphical breakdown of our outcomes to demonstrate their validity. Additionally, we constructed a new iterative scheme for non-linear equations that is cubically convergent. Afterwards, we provided a comparative study between the proposed algorithm and standard methods. We also discussed the proposed algorithm's impact on the basins of attraction.