On Some Applications to the Newton--Raphson Method
Newton's method is classically viewed as an iterative procedure for approximating a zero of a nonlinear function. In this paper, we consider the Newton iterates from a quadrature perspective: the iterates generate a partition of the interval between the initial point and the root, which in turn induces a composite trapezoidal rule. We establish an exact decomposition of the integral associated with this Newton-generated quadrature and characterize the resulting quadrature error. We also obtain an explicit bound for this error in terms of the initial Newton decrement and give a condition under which it is smaller than the error of the trapezoidal rule on the original interval. These results provide a connection between Newton's method and the quadrature rule induced by its iterates.