Optimal 2-cyclic MSOR for “bowtie” spectra and the “continuous” manteuffel algorithm

For the numerical solution of a linear system whose matrix coefficient is block 2-cyclic consistently ordered, with the eigenvalues of the associated block Jacobi matrix lying in a ‘bowtie” region, several efficient stationary iterative methods have been proposed—among others, by Chin and Manteuffel (1988), Elman and Golub (1990), de Pillis (1991) and Eiermann, Niethammer, and Varga (1992). We propose as an alternative the stationary modified successive overrelaxation (MSOR) method or an “equivalent” two-step method applied to the cyclically reduced linear system. It is shown both theoretically and experimentally that the application of a “continuous” version of Manteuffelʹs algorithm to derive the optimal parameters of the two-step method produces an iterative method that is asymptotically faster than all the aforementioned methods.