We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in $\mathbb{R}^n$. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has R-quadratic convergence locally near simple zeros of the system. We also analyze the temporal complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to three that the method is both fast and accurate on a wide range of problems. The algorithm should also work well in higher dimensions. Our tests show that the algorithm outperforms other standard methods on this problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available.

翻译：我们提出一种新算法，用于求解$\mathbb{R}^n$中有界区间内实值函数系统的孤立零点。该算法结合切比雪夫代理方法，融合细分、约简法及利用切比雪夫多项式特殊性质的消元检验。我们证明该方法在系统单根附近具有R-二次收敛性。同时分析算法的时间复杂度与数值稳定性，并在三维及以下维度提供数值证据，表明该方法在广泛问题上兼具快速性与准确性。该算法亦应适用于更高维度问题。实验表明，在求解有界域内所有实零点这一问题上，本算法优于其他标准方法。我们公开了该算法的Python实现。