  In 1848 Ch.~Hermite asked if there exists some way to write cubic
irrationalities periodically. A little later in order to approach the problem
C.G.J.~Jacobi and O.~Perron generalized the classical continued fraction
algorithm to the three-dimensional case, this algorithm is called now the
Jacobi-Perron algorithm. This algorithm is known to provide periodicity only
for some cubic irrationalities.
  In this paper we introduce two new algorithms in the spirit of Jacobi-Perron
algorithm: the heuristic algebraic periodicity detecting algorithm and the
$\sin^2$-algorithm. The heuristic algebraic periodicity detecting algorithm is
a very fast and efficient algorithm, its output is periodic for numerous
examples of cubic irrationalities, however its periodicity for cubic
irrationalities is not proven. The $\sin^2$-algorithm is limited to the
totally-real cubic case (all the roots of cubic polynomials are real numbers).
In the recent paper~\cite{Karpenkov2021} we proved the periodicity of the
$\sin^2$-algorithm for all cubic totally-real irrationalities. To our best
knowledge this is the first Jacobi-Perron type algorithm for which the cubic
periodicity is proven. The $\sin^2$-algorithm provides the answer to Hermite's
problem for the totally real case (let us mention that the case of cubic
algebraic numbers with complex conjugate roots remains open).
  We conclude this paper with one important application of Jacobi-Perron type
algorithms to computation of independent elements in the maximal groups of
commuting matrices of algebraic irrationalities.

Number Theory