Wiener-Hopf matrix factorization using ordinary differential equations in the commutative case

A. V. Shanin

doi:10.1093/qjmam/hbt017

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arXiv · 1301.4000

Wiener-Hopf matrix factorization using ordinary differential equations in the commutative case

A. V. Shanin

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Abstract

A matrix factorization problem is considered. The matrix to be factorized is algebraic, has dimension 2 X 2 and belongs to Moiseev's class. A new method of factorization is proposed. First, the matrix factorization problem is reduced to a Riemann-Hilbert problem using the Hurd's method. Secondly, the Riemann-Hilbert problem is embedded into a family of Riemann-Hilbert problems indexed by a variable b taking values on a half-line. A linear ordinary differential equation (ODE1) with respect to b is derived. The coefficient of this equation remains unknown at this step. Finally, the coefficient of the ODE1 is computed. For this, it is proven that it obeys a non-linear ordinary differential equation (ODE2) on a half-line. Thus, the numerical procedure of matrix factorization becomes reduced to two runs of solving of ordinary differential equations on a half-line: first ODE2 for the coefficient of ODE1, and then ODE1 for the unknown function. The efficiency of the new method is demonstrated on some examples.

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A. V. Shanin. 2013-01-17. Wiener-Hopf matrix factorization using ordinary differential equations in the commutative case. https://doi.org/10.1093/qjmam/hbt017

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