关于初等对称多项式的封闭形式及范德蒙矩阵的逆
On the Closed Form Expression of Elementary Symmetric Polynomials and the Inverse of Vandermonde Matrix
摘要：
逆范德蒙矩阵的计算是求解非奇异线性系统$Vc=b$的一个长期存在的问题，其中方阵$V$的行由幂多项式的级数构成。它在科学计算中有许多应用，包括插值，超分辨率，以及在密码学中应用的特殊矩阵的构造。尽管其众多的应用，矩阵是高度病态的，其中专门的治疗被认为是近似，如转换为柯西矩阵，谱分解，和算法的定制的数值解。在本文中，我们提出了一个广义算法，采取任意成对（非重复）的样本节点求解逆范德蒙矩阵。这是通过两个步骤完成的：首先，引入具有$\mathcal{O}(N)$复杂度的高度平衡的递归算法，以求解基本对称多项式的组合求和;其次，为逆Vandermonde定制封闭形式的解，其中矩阵的元素利用该递归求和进行逆计算。通过对与Vandermonde矩阵相关联的Frobenius友矩阵进行谱分解，分析了该方法的数值稳定性和精度.结果表明，使用特定节点（如复平面上定义的单位根$N$）的最新解决方案有了显着的改进。一维插值问题的一个基本应用被认为是证明所提出的方法的超分辨信号的效用。
英文摘要：
Inverse Vandermonde matrix calculation is a long-standing problem to solve nonsingular linear system $Vc=b$ where the rows of a square matrix $V$ are constructed by progression of the power polynomials. It has many applications in scientific computing including interpolation, super-resolution, and construction of special matrices applied in cryptography. Despite its numerous applications, the matrix is highly ill-conditioned where specialized treatments are considered for approximation such as conversion to Cauchy matrix, spectral decomposition, and algorithmic tailoring of the numerical solutions. In this paper, we propose a generalized algorithm that takes arbitrary pairwise (non-repetitive) sample nodes for solving inverse Vandermonde matrix. This is done in two steps: first, a highly balanced recursive algorithm is introduced with $\mathcal{O}(N)$ complexity to solve the combinatorics summation of the elementary symmetric polynomials; and second, a closed-form solution is tailored for inverse Vandermonde where the matrix' elements utilize this recursive summation for the inverse calculations. The numerical stability and accuracy of the proposed inverse method is analyzed through the spectral decomposition of the Frobenius companion matrix that associates with the corresponding Vandermonde matrix. The results show significant improvement over the state-of-the-art solutions using specific nodes such as $N$th roots of unity defined on the complex plane. A basic application in one dimensional interpolation problem is considered to demonstrate the utility of the proposed method for super-resolved signals.
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