线性Hankel矩阵秩亏的实根求法
Real root finding for rank defects in linear Hankel matrices
摘要：
令$H\_0, ..., H\_n$是具有$\QQ$和Hankel结构中的元素的$m \times m$矩阵，即常数斜对角。我们考虑线性Hankel矩阵$H(\vecx)=H\_0+\X\_1H\_1+...+\X\_nH\_n$和计算样本点的问题，在每个连接的组件的实代数集的秩约束${\sf rank}(H(\vecx))\leq r$，为一个给定的整数$r \leq m-1$。计算由线性矩阵秩亏定义的实代数集合中的样本点是一个普遍的问题，在控制理论、计算几何、优化等许多领域都有应用。此外，Hankel矩阵还出现在工程科学的许多领域。此外，由于汉克尔矩阵是对称的，因此这个问题的任何算法开发都可以被视为解决半定规划问题（即线性矩阵不等式）的专用精确算法的第一步。在输入的一些泛型假设（如平滑的发病率品种）下，我们设计了一个概率算法来解决这个问题。它是所谓的临界点法的一种改进，它利用了问题的特殊结构。它的复杂性反映了这一点：它基本上是二次在特定的程度范围内的发病率品种。我们报告的实际实验和分析算法如何利用这种特殊的结构。第一种实现优于现有的实现，用于计算一般实代数集合中的样本点：它解决了现有技术无法达到的示例。
英文摘要：
Let $H\_0, ..., H\_n$ be $m \times m$ matrices with entries in $\QQ$ and Hankel structure, i.e. constant skew diagonals. We consider the linear Hankel matrix $H(\vecx)=H\_0+\X\_1H\_1+...+\X\_nH\_n$ and the problem of computing sample points in each connected component of the real algebraic set defined by the rank constraint ${\sf rank}(H(\vecx))\leq r$, for a given integer $r \leq m-1$. Computing sample points in real algebraic sets defined by rank defects in linear matrices is a general problem that finds applications in many areas such as control theory, computational geometry, optimization, etc. Moreover, Hankel matrices appear in many areas of engineering sciences. Also, since Hankel matrices are symmetric, any algorithmic development for this problem can be seen as a first step towards a dedicated exact algorithm for solving semi-definite programming problems, i.e. linear matrix inequalities. Under some genericity assumptions on the input (such as smoothness of an incidence variety), we design a probabilistic algorithm for tackling this problem. It is an adaptation of the so-called critical point method that takes advantage of the special structure of the problem. Its complexity reflects this: it is essentially quadratic in specific degree bounds on an incidence variety. We report on practical experiments and analyze how the algorithm takes advantage of this special structure. A first implementation outperforms existing implementations for computing sample points in general real algebraic sets: it tackles examples that are out of reach of the state-of-the-art.
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