In this paper, we first introduce a new concept on matrices which we call annihilator vectors. Then using the shifted form of these vectors, we propose new algorithms for (1) computing the inverse of a Vandermonde matrix, (2) solving some Vandermonde systems of linear equations, both algorithms in \(O(n^2)\) arithmetic operations. Our methods are based on a new technique which lead to novel algorithms, need less storage and, from experimental results, are more efficient than the previous usual methods. Finally, comparisons with other methods are broached and some numerical examples are supplied.

中文翻译：

---

关于Vandermonde矩阵和系统的新研究

在本文中，我们首先介绍关于矩阵的新概念，我们称其为an灭向量。然后使用这些向量的移位形式，我们提出了新的算法，用于（1）计算Vandermonde矩阵的逆，（2）解决一些线性方程式的Vandermonde系统，这两种算法都采用\（O（n ^ 2）\）算法操作。我们的方法基于一种导致新算法的新技术，需要更少的存储，并且从实验结果来看，它比以前的常用方法更有效。最后，提出了与其他方法的比较，并提供了一些数值示例。