We present a novel algorithm for the factorization of triangular polynomial 2x2 matrices. Our algorithm offers flexibility in choosing between 'exact' and 'natural' pairs of partial indices. Integrated into the general Winer-Hopf factorization algorithm for 2x2 matrices, as described in https://doi.org/10.1098/rspa.2020.0027, the approach offers a versatile method applicable to practical scenarios. Specifically, given a matrix function S, our method enables the construction of a Wiener-Hopf factorization for an approximate matrix with manageable factors, even when the exact factorization of S may involve large factors. These large factors of S are presumed to arise due to inaccuracies in the construction process of S, rendering its exact factorization impractical. The cases where S or its approximation have unstable partial indices are not excluded. Thus, selecting a reasonable factorization of the approximated matrix function S emerges as a natural choice, facilitating an automatic determination of the partial indices, which in turn, could serve as a novel regularization procedure. 
(Co-Authors: G. Mishuris, I. Spitkovsky)