We generalize an optimal fourth order Kung–Traub method to Banach spaces and study its local convergence to approximate a locally-unique solution of a system of nonlinear equations. New analysis also provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of higher order derivatives. Furthermore, based on the Kung–Traub method a $p+1$-step scheme of Kung–Traub-like methods with increasing convergence order $2p+2$ is proposed. The novelty of the scheme is that in each step the order of convergence is increased by an amount two at the cost of only one additional function evaluation. Numerical examples are provided to verify the theoretical results and to show the convergence behavior.

我们将最佳四阶Kung-Traub方法推广到Banach空间，并研究其局部收敛性，以近似非线性方程组的局部唯一解。新的分析还提供了收敛半径，误差范围以及对解决方案唯一性的估计。使用高阶导数的泰勒展开式的方法未提供此类估计。此外，基于Kung-Traub方法a$p+\mathrm{1个}$收敛阶递增的Kung–Traub类方法的逐步方案 $2p+2$被提议。该方案的新颖之处在于，在每个步骤中，收敛顺序都增加了两个，而仅需进行一次附加功能评估。数值例子验证了理论结果并表明了收敛性。