The Riemann problem on the factorization of a matrix function \(r(\lambda)\) on the unit circle is formulated to construct functions \(\phi(\lambda)\) and \(\psi(\lambda)\) which admit a nonsingular analytic continuation such that \(\phi(\lambda)r(\lambda)=\psi(\lambda)\) for \(| \lambda | =1\), \(\phi (0)=1\). An algorithm is presented to solve this problem in the class of functions \(r(\lambda)\) with Re\(r(\lambda)>0\) for all \(\lambda\), \(| \lambda | =1\).