  The Bernstein-B\'ezier form of a polynomial is widely used in the fields of
computer aided geometric design, spline approximation theory and, more
recently, for high order finite element methods for the solution of partial
differential equations. However, if one wishes to compute the classical
Lagrange interpolant relative to the Bernstein basis, then the resulting
Bernstein-Vandermonde matrix is found to be highly ill-conditioned.
  In the univariate case of degree $n$, Marco and Martinez showed that using
Neville elimination to solve the system exploits the total positivity of the
Bernstein basis and results in an $\mathcal{O}(n^2)$ complexity algorithm.
Remarkable as it may be, the Marco-Martinez algorithm has some drawbacks: The
derivation of the algorithm is quite technical; the interplay between the ideas
of total positivity and Neville elimination are not part of the standard
armoury of many non-specialists; and, the algorithm is strongly associated to
the univariate case.
  The present work addresses these issues. An alternative algorithm for the
inversion of the univariate Bernstein-Vandermonde system is presented that has:
The same complexity as the Marco-Martinez algorithm and whose stability does
not seem to be in any way inferior; a simple derivation using only the basic
theory of Lagrange interpolation (at least in the univariate case); and, a
natural generalisation to the multivariate case.
