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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).

For the class of polynomial quadrature rules we show that conveniently chosen bases allow to compute both the weights and the theoretical error expression of a n-point rule via the undetermined coefficients method. As an illustration, the framework is applied to some classical rules such as Newton-Cotes, Adams-Bashforth, Adams-Moulton and Gaussian rules.

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In order to approximate the integral I( f ) = b a f (x)dx, where f is a sufficiently smooth function, models for quadrature rules are developed using a given panel of n (n ≥ 2) equally spaced points. These models arise from the undetermined coefficients method, using a Newton's basis for polynomials. Although part of the final product is algebraically equivalent to the well known closed Newton-Cotes rules, the algorithms obtained are not the classical ones.In the basic model the most simple quadrature rule Q n is adopted (the so-called left rectangle rule) and a correctioñ E n is constructed, so that the final rule S n = Q n +Ẽ n is interpolatory. The correctionẼ n , depending on the divided differences of the data, might be considered a realistic correction for Q n , in the sense thatẼ n should be close to the magnitude of the true error of Q n , having also the correct sign. The analysis of the theoretical error of the rule S n as well as some classical properties for divided differences suggest the inclusion of one or two new points in the given panel. When n is even it is included one point and two points otherwise. In both cases this approach enables the computation of a realistic errorĒ S n for the extended or corrected rule S n . The respective output (Q n ,Ẽ n , S n ,Ē S n ) contains reliable information on the quality of the approximations Q n and S n , provided certain conditions involving ratios for the derivatives of the function f are fulfilled. These simple rules are easily converted into composite ones. Numerical examples are presented showing that these quadrature rules are useful as a computational alternative to the classical Newton-Cotes formulas.

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Mann's sequences are difficult to accelerate in the presence of a nonhyperbolic fixed point. New accelerators are constructed for Mann's sequences which are useful even for other sets of very slowly convergent sequences.