Two numerical algorithms for analyzing planar central and balanced configurations in the $(n+1)$-body problem with a small mass are presented. The first one relies on a direct solution method of the $(n+1)$-body problem by using a stochastic optimization approach, while the second one relies on an analytic-continuation method, which involves the solutions of the $n$-body and the restricted $(n+1)$-body problem, and the application of a local search procedure to compute the final $(n+1)$-body configuration in the neighborhood of the configuration obtained at the first two steps. Some exemplary central and balanced configurations in the cases $n=4,5,6$ are shown.

翻译：提出了两个数字算法,用于分析(n+1)美元-体积问题中小质量的(n+1)美元-体积问题,第一个数字算法采用随机优化方法,依靠直接解决$(n+1)美元-体积问题的方法,第二个数字算法则依赖分析持续方法,包括解决(n+1)美元-体积问题和限定(n+1)美元-体积问题,以及采用本地搜索程序在配置前两步获得的周边计算(n+1)美元-体积最终配置。