We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle. The nodes are “off the grid,” pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance. Such growth rates are known in greater generality if all nodes collide or for groups of colliding nodes. For pairs of nodes, we provide reasonable sharp constants that are independent of the number of nodes as long as non-colliding nodes are well-separated.

中文翻译：

---

关于具有几乎冲突节点对的Vandermonde矩阵的条件数

我们证明了在复单位圆上具有节点的矩形范德蒙德矩阵的光谱条件数的上下界。节点处于“离网”状态，成对的节点几乎发生碰撞，并且所研究的条件数随反间隔距离线性增长。如果所有节点都发生冲突，或者对于成组的冲突节点，则这种增长速度更为普遍。对于成对的节点，只要非冲突节点之间的分隔良好，我们将提供与节点数量无关的合理的尖锐常数。