Nontrivial solutions of Δu+μu + -νu - =0 in Ω R n with zero Dirichlet condition on Ω are studied. The collection of the pairs (μ,ν) is the so called Fucik spectrum σ F . A new variational formulation for parts of σ F is presented and analyzed. Based on this formulation a minimization algorithm for the computation of a part of σ F is developed. Alternatively, an approach using an implicit function argument is discussed analytically and, via Newton's method, also numerically. By combining the variational minimization method with Newton's method a new bifurcation result is first observed numerically and then proved rigorously. By replacing the variational minimization method by the Mountain Pass Algorithm higher curves in σ F are found numerically. Several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of Fucik curves originating from different eigenvalues.