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. Previously, we gave block-structured algorithms for inverting the Bernstein mass matrix on simplicial cells, but did not study fast alorithms for the univariate case. Here, we give several approaches to inverting the univariate mass matrix based on exact formulae for the inverse; decompositions of the inverse in terms of Hankel, Toeplitz, and diagonal matrices; and a spectral decomposition. In particular, the eigendecomposition can be explicitly constructed in $\mathcal{O}(n^2)$ operations, while its accuracy for solving linear systems is comparable to that of the Cholesky decomposition. Moreover, we study conditioning and accuracy of these methods from the standpoint of the effect of roundoff error in the $L^2$ norm on polynomials, showing that the conditioning in this case is far less extreme than in the standard 2-norm.

中文翻译：

---

Bernstein 质量矩阵的结构化反演

伯恩斯坦多项式长期以来一直是近似理论和计算几何的主要内容，在有限元方法中也越来越受到关注。插值和逼近中的许多基本问题引发了有趣的线性代数问题。以前，我们给出了块结构算法，用于在单纯细胞上反转 Bernstein 质量矩阵，但没有研究单变量情况下的快速算法。在这里，我们给出了几种基于逆的精确公式来求逆单变量质量矩阵的方法；根据 Hankel、Toeplitz 和对角矩阵对逆矩阵进行分解；和频谱分解。特别地，特征分解可以在 $\mathcal{O}(n^2)$ 操作中显式构造，而其求解线性系统的精度可与 Cholesky 分解相媲美。此外，我们从 $L^2$ 范数对多项式的舍入误差影响的角度研究了这些方法的条件和准确性，表明这种情况下的条件远没有标准 2 范数那么极端。