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There are fundamentally two different cases to consider: the scalar and the matrix Wiener–Hopf factorizations.

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By constructive factorizations it is meant that there is an algorithmic way to obtain the factorizations.

Due to its complexity, different classes of matrix functions have to be treated separately.

Keywords: Wiener-Hopf; Asymptotic factorization; Riemann-Hilbert Many problems in mathematics, physics and engineering give rise to partial differential equations (PDEs); hence the solution of these problems relies on the ability to solve different types of PDEs.

The study of PDEs is a vast and expanding area of research [1-15].

Approximating matrix Wiener– Hopf equations is complex due to instabilities.

This paper concerns reliable approximate matrix Wiener–Hopf factorizations and its implementation.

The existence of matrix factorizations under general assumptions has been known for a long time [12,22].

Nevertheless, to date constructive matrix factorizations remains a formidable challenge.

Below we outline some of the few classes of matrix functions with known constructive factorizations and give references that an interested reader can follow.

Constructive factorizations algorithms exist for classes of Veitch and Peake [25]. An algorithm for factorization has been known to exist for a long time. What complicates the application of algorithms further is that, given a matrix, it is difficult to determine which class it belongs to, if any.

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