Some variant of Newton's method with third-order convergence.
Newton's method for the approximation of the root \(x\) of a system of nonlinear equations can be interpreted as a computation of the indefinite integral arising from Newton's theorem. Using various quadrature formulas one can obtain various iterative processes, the so-called modifications of Newton method. An error analysis providing the higher order of convergence is proposed and the best efficiency, in the term of function evaluations, of two of these methods is provided. It is proved that if the order of the quadrature formula is at last one and if \(x\) is a simple root, then the order of the method is always three, independently of the order quadrature formula. Among this class of methods the most efficient ones are the iterative methods using the mid/point and trapezoidal quadrature rules.
scientific article; zbMATH DE number 1978093