Numerical resolution of algebraic equations with symmetries
Studied here is the effect of the presence of symmetry groups in a system of algebraic equations on the numerical resolution with fixed-point algorithms. It is proved that the symmetries imply two important properties of the system: the solutions are not isolated, but distributed in orbits by the symmetry group and zero is an eigenvalue of the Jacobian evaluated at any of the solutions, being the multiplicity at least the dimension of the symmetry group. From the point of view of the numerical resolution, the concept of orbital convergence is introduced and a corresponding convergence theorem is proved. This establishes the conditions under which a fixed-point algorithm converges to some element of the orbit of a solution of the system. Two numerical examples illustrate these results.