Improving an estimate of the convergence rate of the Seidel method by selecting the optimal order of equations in the system of linear algebraic equations

The Seidel method for solving a system of linear algebraic equations and an estimate of its convergence rate are considered. It is proposed to change the order of equations. It is shown that the method described in Faddeevs' book Computational Methods of Linear Algebra can deteriorate the convergence rate estimate rather than improve it. An algorithm for establishing the optimal order of equations is proposed, and its validity is proved. It is shown that the computational complexity of the reordering is $2n^2$ additions and $(12)n^2$ divisions. Numerical results for random matrices of order $100$ are presented that confirm the proposed improvement.