Entanglement in multipartite systems is a key resource for quantum information and communication protocols, making its verification in complex systems a necessity. Because an exact calculation of arbitrary entanglement probes is impossible, we derive and implement a numerical method to construct multipartite witnesses to uncover entanglement in arbitrary systems. Our technique is based on a substantial generalization of the power iteration, an essential tool for computing eigenvalues, and it is a solver for the separability eigenvalue equations, enabling the general formulation of optimal entanglement witnesses. Beyond our rigorous derivation and direct implementation of this method, we also apply our approach to several examples of complexly quantum-correlated states and benchmark its general performance. Consequently, we provide an generally applicable numerical tool for the identification of multipartite entanglement.