Bernstein-Vandermonde矩阵的结构逆
Structured inversion of the Bernstein-Vandermonde Matrix
摘要：
伯恩斯坦多项式，长期以来的主食逼近论和计算几何，也越来越成为兴趣的有限元法。插值和逼近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逼近时，会遇到Bernstein-Vandermonde矩阵，发现它是高度病态的。在此之前，我们利用单项Bezout矩阵和Hankel矩阵的逆之间的关系，得到了Bernstein质量矩阵的逆在Hankel、Toeplitz和对角矩阵方面的分解。本文利用Bernstein-Bezout矩阵的性质，将Bernstein-Vandermonde矩阵的逆分解为Hankel矩阵、Toeplitz矩阵和对角矩阵乘积的差.我们还使用非标准矩阵范数来研究Bernstein-Vandermonde矩阵的条件，表明在这种情况下的条件比标准2-范数更好。此外，我们使用多元伯恩斯坦多项式的属性，推导出块$LU$分解的伯恩斯坦-范德蒙矩阵对应于均匀节点上的$d$ -单纯形。
英文摘要：
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.
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