  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.
