论文标题
使用基于ZNN的路径以下方法构建可分解和一般矩阵的值字段
Constructing the Field of Values of Decomposable and General Matrices Using the ZNN Based Path Following Method
论文作者
论文摘要
本文描述并开发了遵循算法的快速准确的路径，该算法计算每个可能的复合物或真实方形矩阵$ a $的值边界曲线字段。它依赖于矩阵流量分解方法，该方法找到了相关的Hermitean矩阵流量$ {\ cal f} _a（t）= \ cos（t）h + \ sin（t）k $的适当块对基流量表示。这里$ {\ cal f} _a（t）$是真实和偏斜零件矩阵$ h =（a+a^*）/2 $ and $ k =（a-a^*）/（2i）$ a $ a $的1-参数变化的线性组合。 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 $A$.然后，它使用Znn特征值方法的路径分别计算每个对角线块$ a_j $的值边界字段。然后，所有值边界点的子场的凸壳，然后确定值正确分解和非分解矩阵$ a $ a $的值边界曲线的场。该算法删除了遵循FOV方法的标准限制，因为FOV方法通常无法处理分解矩阵$ a $，这是由于可能的特征库交叉点为$ {\ cal f} _a（t）$。包括测试和数值比较。与Johnson的Francis Qr eigenvalue和Bendixon矩形方法相比，我们基于ZNN的方法进行了顺序和并行计算的编码，并且这两个版本都非常准确，非常快，该方法基于$ {\ cal f} _a（t_k）$ ofers $ t_k $ t_k \ in contose $ {\ cal f} _a（t_k）$的完整eigenanalyses $ {\ cal f} _a（t_k）$
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.
