Conditioning of Rectangular Vandermonde Matrices with Nodes in the Unit Disk
Conditioning of Rectangular Vandermonde Matrices with Nodes in the Unit Disk
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节点位于单位圆盘中的矩形 Vandermonde 矩阵的条件化
DOI：
10.1137/s0895479898336021
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
F. Bazán
中科院分区：
文献类型：
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作者：
F. Bazán
令WN=WN（z1，z2，. . . z1）是阶为n × N，$N\geq n，$的矩形范德蒙矩阵，其不同节点zj在单位圆盘中，$z_j^{k-1}$作为其（j，k）项。这种类型的矩阵经常出现在频率估计和系统识别问题中。本文分析了WN的条件性，并给出了谱条件数$\kappa_2（W_N）$的界.边界取决于n、N和节点的间隔。通过分析作为N的函数的边界的行为，我们得出结论，这些矩阵可以变得良好的条件，提供节点接近单位圆，但不是非常接近彼此，并提供WN的列数是足够大的。分析了条件本身和边界的渐近性态，并通过数值例子验证了理论结果。
Let WN=WN(z1,z2, . . . z1) be a rectangular Vandermonde matrix of order n × N, $N\geq n,$ with distinct nodes zj in the unit disk and $z_j^{k-1}$ as its (j,k) entry. Matrices of this type often arise in frequency estimation and system identification problems. In this paper, the conditioning of WN is analyzed and bounds for the spectral condition number $\kappa_2(W_N)$ are derived. The bounds depend on n, N, and the separation of the nodes. By analyzing the behavior of the bounds as functions of N, we conclude that these matrices may become well conditioned, provided the nodes are close to the unit circle but not extremely close to each other and provided the number of columns of WN is large enough. The asymptotic behavior of both the conditioning itself and the bounds is analyzed and the theoretical results arising from this analysis verified by numerical examples.