Faster Convergence for Iterative Solutions to Systems via Three-Part Splitting

For the linear system $Ax = y_0 $, we explore linear stationary second degree methods, or so-called three-part splittings (which include the first-degree methods of Jacobi, Gauss–Seidel and SOR) for defining the sequence $\{ x_n \} $ where $x_n \to x$. By measuring the asymptotic rates of convergence of the sequence, we are able to determine when the second-degree method is superior to a corresponding first-degree method. In fact, if B is the iteration matrix of the first degree splitting, then this improvement is analyzed if the spectrum of B is real (§ 5) and when the spectrum of B is purely imaginary (§ 6).