  We propose a new algorithm for simulating noncommutative phi-four theory on
the fuzzy sphere based on, i) coupling the scalar field to a U(1) gauge field,
in such a way that in the commutative limit N\longrightarrow \infty, the two
modes decouple and we are left with pure scalar phi-four on the sphere, and ii)
diagonalizing the scalar field by means of a U(N) unitary matrix, and then
integrating out the unitary group from the partition function. The number of
degrees of freedom in the scalar sector reduces, therefore, from N^2 to the N
eigenvalues of the scalar field, whereas the dynamics of the U(1) gauge field,
is given by D=3 Yang-Mills matrix model with a Myers term. As an application,
the phase diagram, including the triple point, of noncommutative phi-four
theory on the fuzzy sphere, is reconstructed with small values of N up to N=10,
and large numbers of statistics.
