Algebraic Computation of the Hierarchical Renormalization Group Fixed Points and their $ϵ$-Expansions

Nontrivial fixed points of the hierarchical renormalization group are computed by numerically solving a system of quadratic equations for the coupling constants. This approach avoids a fine tuning of relevant parameters. We study the eigenvalues of the renormalization group transformation, linearized around the non-trivial fixed points. The numerical results are compared with $ϵ$-expansion.