# Structured inversion of the Bernstein mass matrix

## TL;DR

This paper develops fast, accurate algorithms for inverting the univariate Bernstein mass matrix using explicit formulas, matrix decompositions, and spectral methods, with analysis of conditioning and numerical stability.

## Contribution

It introduces several novel approaches for inverting the univariate Bernstein mass matrix, including explicit inverse formulas and spectral decomposition, improving computational efficiency and accuracy.

## Key findings

- Eigendecomposition can be constructed in O(n^2) operations.
- The spectral method's accuracy is comparable to Cholesky decomposition.
- Conditioning in the L^2 norm is less severe than in the standard 2-norm.

## Abstract

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.