用基于ZNN的路径跟踪法构造可分解矩阵和一般矩阵的值域
Constructing the Field of Values of Decomposable and General Matrices Using the ZNN Based Path Following Method
摘要：
本文描述和开发了一种快速准确的路径跟踪算法，该算法可以计算每个可能的复数或实数方阵$A$的值边界曲线域。它依赖于一个矩阵流分解方法，找到一个适当的块对角流表示相关联的厄米矩阵流${\cal F}_A(t) = \cos(t) H + \sin(t) K$。这里，${\cal F}_A(t)$是$A$的实部矩阵$H = (A+A^*)/2$和斜部矩阵$K = (A-A^*)/(2i)$的单参数变化线性组合。为了分解流${\cal F}_A(t)$，该算法将给定的稠密通用矩阵$A$统一分解为块对角形式$U^*AU = \text { diag} (A_j)$，其中$j > 1$对角块$A_j$的各个大小相加等于$A$的大小。然后，它使用路径跟踪ZNN特征值方法分别为每个对角块$A_j$计算值边界的域。然后，所有子域的值边界点的凸包，确定域的值边界曲线正确分解和非分解矩阵$A$。该算法消除了路径跟随FoV方法的标准限制，这些方法通常不能处理由于${\cal F}_A(t)$的可能特征曲线交叉而导致的分解矩阵$A$。测试和数值比较包括在内。我们的ZNN为基础的方法编码的顺序和并行计算和两个版本运行非常准确，非常快相比，约翰逊的弗朗西斯QR特征值和Bendixon矩形为基础的方法，计算完整的特征分析${\cal F}_A(t_k)$为每个选择的$t_k \in {[} 0,2π{]}$更慢。
英文摘要：
This paper describes and develops a fast and accurate path following algorithm that computes the field of values boundary curve for every conceivable complex or real square matrix $A$. It relies on a matrix flow decomposition method that finds a proper block-diagonal flow representation for the associated hermitean matrix flow ${\cal F}_A(t) = \cos(t) H + \sin(t) K$. Here ${\cal F}_A(t)$ is a 1-parameter-varying linear combination of the real and skew part matrices $H = (A+A^*)/2$ and $K = (A-A^*)/(2i)$ of $A$. For decomposing flows ${\cal F}_A(t)$, the algorithm decomposes a given dense general matrix $A$ unitarily into block-diagonal form $U^*AU = \text { diag} (A_j)$ with $j > 1$ diagonal blocks $A_j$ whose individual sizes add up to the size of $A$. It then computes the field of values boundaries separately for each diagonal block $A_j$ using the path following ZNN eigenvalue method. The convex hull of all sub-fields of values boundary points then determines the field of values boundary curve correctly for decomposing and non-decomposing matrices $A$. The algorithm removes standard restrictions for path following FoV methods that generally cannot deal with decomposing matrices $A$ due to possible eigencurve crossings of ${\cal F}_A(t)$. Tests and numerical comparisons are included. Our ZNN based method is coded for sequential and parallel computations and both versions run very accurately and very fast when compared with Johnson's Francis QR eigenvalue and Bendixon rectangle based method that computes complete eigenanalyses of ${\cal F}_A(t_k)$ for every chosen $t_k \in {[} 0,2π{]}$ more slowly.
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