Based on the observation that iterative learning control (ILC) can be based on the inverse plant but that the approach can be degraded by modelling errors, particularly at high frequencies, this article investigates the construction and properties of a multi-parameter parameter-optimal ILC algorithm that uses an approximate polynomial representation of the inverse with natural high-frequency attenuation. In its simplest form, the algorithm replicates the original work of Owens and Feng but, more generally, it is capable of producing significant improvements to the observed convergence rate. As the number of parameters increases, convergence rates approach that of the ideal plant inverse algorithm. Introducing compensation into the algorithm provides a formal link to previously published gradient and norm-optimal ILC algorithms and indicates that the polynomial approach can be regarded as approximations to those control laws. Simulation examples verify the theoretical performance predictions.