In this paper, we propose an algorithm for computing an approximate polynomial matrix eigenvalue decomposition (PEVD). The PEVD of a para-Hermitian matrix yields a factorisation into a polynomial matrix product consisting of a spectrally majorised diagonal matrix that is preand post- multiplied by paraunitary (PU) matrices. All current PEVD algorithms, such as the second order sequential best rotation (SBR2) algorithm, perform a factorisation whereby diagonalisation and spectral majorisation are only achieved in approximation. The purpose of this paper is to present a new iterative approach which constitutes a "Householder-like" version of SBR2 and is akin to Tkacenko's approximate EVD (AEVD); however, unlike the AEVD, the proposed method carries out the diagonalisation successively by applying arbitrary-degree, finite impulse response PU matrices. We show an application of our algorithm to the design of signal-adapted PU filter banks for subband coding. Simulation results for the proposed approach show very close agreement with the behaviour of the infinite order principal component filter banks and demonstrate its superiority compared to state-of-the-art algorithms in terms of strong decor- relation and spectral majorisation.