Abstract <p> The paper is in line with the research initiated and developed in the papers of I. I. Eremin, V. V. Vasin, L. D. Popov, E. A. Berdnikova, I. M. Sokolinskaya, A. V. Ershova, E. A. Nurminskii, and others. The main result is a new version of a Fejér-type mapping designed for finding a non-negative solution to a system of linear algebraic equations. This mapping combines the operation of orthogonal projection onto the linear variety of solutions to a system of linear algebraic equations with the operation of projection onto a non-negative orthant, without using the traditional positive slice operation; instead, it uses an elementwise operation of calculating the absolute value. The global linear convergence of the proposed algorithm is proved, and its asymptotic convergence constant is estimated. The algorithm, its theoretical justification, and numerical experiments confirming the significantly faster convergence of the proposed mapping compared to that based on the positive slice operation are presented. </p>