论文标题
Bernstein-Vandermonde矩阵的结构反转
Structured inversion of the Bernstein-Vandermonde Matrix
论文作者
论文摘要
伯恩斯坦多项式是近似理论和计算几何形状的主食，在有限元方法中也越来越引起人们的关注。插值和近似中的许多基本问题引起了有趣的线性代数问题。当试图找到边界或初始数据的多项式近似时，人们会遇到Bernstein-Vandermonde矩阵，该矩阵被认为是严重的条件。以前，我们使用了单一的bezout矩阵与汉克尔矩阵倒数之间的关系来获得伯恩斯坦质量基质的倒数，这是汉克尔，toeplitz和对角线矩阵的分解。在本文中，我们使用Bernstein-Bezout矩阵的特性来将Bernstein-Vandermonde矩阵的倒数分解为Hankel，Toeplitz和对角线矩阵的产物的差异。我们还使用非标准矩阵标准来研究Bernstein-Vandermonde矩阵的调理，表明在这种情况下的调节要比标准2-Norm更好。此外，我们使用多元伯恩斯坦多项式的属性来推导block $ lu $ $ $ $ $ $ $ $ $ $分解，该伯恩斯坦 - 范德尔蒙德矩阵对应于$ d $ simplex上的equispaced节点。
Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to interesting linear algebra questions. When attempting to find a polynomial approximation of boundary or initial data, one encounters the Bernstein-Vandermonde matrix, which is found to be highly ill-conditioned. Previously, we used the relationship between monomial Bezout matrices and the inverse of Hankel matrices to obtain a decomposition of the inverse of the Bernstein mass matrix in terms of Hankel, Toeplitz, and diagonal matrices. In this paper, we use properties of the Bernstein-Bezout matrix to factor the inverse of the Bernstein-Vandermonde matrix into a difference of products of Hankel, Toeplitz, and diagonal matrices. We also use a nonstandard matrix norm to study the conditioning of the Bernstein-Vandermonde matrix, showing that the conditioning in this case is better than in the standard 2-norm. Additionally, we use properties of multivariate Bernstein polynomials to derive a block $LU$ decomposition of the Bernstein-Vandermonde matrix corresponding to equispaced nodes on the $d$-simplex.
