Computing numerically the functional derivative of an effective action

The functional derivative of the effective action with respect to an external field is part of the equation of motion of this field if one-loop effects induced by quantum fluctuations or thermal fluctuations are included when minimizing the action of this field. Examples occur in all field theories displaying classical solutions or - as in the Nambu-Jona-Lasinio model - self consistent field configurations. We describe here a numerical method for computing such functional derivatives; we use a fermion field with Yukawa interaction to an external field as an example which is sufficiently simple and sufficiently general. We compare the computed functional derivative to analytical estimates.