求解l维空间的一点到二次曲面的距离是基于聚类的模式识别的一个难题。后者在航天器激光威胁告警中具有重要的应用。本文证明在A的特征根相同的情况下如此的代数方程仅有一个根,并给出求解此根的迭代算法。在A的特征根同号的情形下给出Lagrange乘子的上下界,对于采用二分法求解Lagrange乘子提供了可能和便利。该问题的解决为激光探测和激光告警提供了一种新的途径。

It is a difficult problem of pattern recognition based on clustering to solve the distance between a point in l-dimension space and a quadric surface.The latter has an important application in spacecraft laser-warning.The paper shows that the algebra equation stated above exits single root if the eigenvalues of A are all equal,and gives a recursion algorithm searching for the root.The upper or down boundaries of a Lagrangian multiplier are given if the eigenvalues of A take the same sign,which provides possibility and convenience for dichotomy solving Lagrangian multiplier.Resolving the problem will provide a new method for laser detecting and laser-warning.