一个阶为2+sqrt{6}的Newton改进算法

针对非线性方程求单根问题,提出了一种新的Newton预测-校正格式.通过每步迭代增加计算一个函数值和一阶导数值,使得每步迭代需要估计两个函数值和两个一阶导数值.与标准的Newton算法的二阶收敛速度相比,新算法具有更高阶的收敛速度2+sqrt{6}.通过测试函数对新算法进行测试, 与相关算法比较,表明算法在迭代次数、运算时间及最优值方面都具有较明显的优势. 最后,将这种新格式推广到多维向量值函数, 采用泰勒公式证明了其收敛性,并给出了两个二维算例来验证其收敛的有效性.

A modification of Newton method withconvergence of order 2+sqrt{6}

In this paper, a new modification of the standard Newton method for approximating the root of a univariate function is introduced. Two evaluations of function and two evaluations of its first derivative are required at a cost of one additional function and first derivative evaluations per iteration. The modified method converges faster with the order of convergence 2+sqrt{6} compared with 2 for the standard Newton method. Numerical examples demonstrate the newalgorithm has advantages in the iteration number, computation time and optimal value compared with the current algorithms. At last, the predictor-corrector improvement is generalized to multi-dimensional vector-valued functions, its convergence is proved using Taylor formula, and two two-dimensional examples are given to illustrate its convergence.