  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.
