In this paper, we present an analytical solution to the nonlinear optimization problem encountered in the context of the filtered canonical polyadic decomposition. It is shown that the originally proposed alternating least squares (ALS) approach can be significantly accelerated through a thorough analysis of one of the matrix factors, that is, the factor that destroys the multilinearity. The main contribution consists of analytical partial derivates with respect to the matrix factor in question. Making use of these derivatives, the nonlinear optimization is framed as a damped Gauss-Newton algorithm. The proposed method is then compared to the original approach in point of computational time and accuracy, both for simulation and experimental data sets. The results suggest that, on average, a speedup of factor three is obtained when compared to the original method. Furthermore, this acceleration is achieved while preserving the original numerical accuracy. The authors conclude that the presented method provides a promising approach for solving filtered canonical polyadic decompositions in the framework of function decoupling.