On the Convergence of Two-Stage Iterative Processes for Solving Linear Equations

Nichols, N. ORCID: https://orcid.org/0000-0003-1133-5220 (1973) On the Convergence of Two-Stage Iterative Processes for Solving Linear Equations. SIAM Journal on Numerical Analysis (SINUM), 10 (3). pp. 460-469. ISSN 0036-1429 doi: 10.1137/0710040

Abstract/Summary

This paper considers two-stage iterative processes for solving the linear system $Af = b$. The outer iteration is defined by $Mf^{k + 1} = Nf^k + b$, where $M$ is a nonsingular matrix such that $M - N = A$. At each stage $f^{k + 1} $ is computed approximately using an inner iteration process to solve $Mv = Nf^k + b$ for $v$. At the $k$th outer iteration, $p_k $ inner iterations are performed. It is shown that this procedure converges if $p_k \geqq P$ for some $P$ provided that the inner iteration is convergent and that the outer process would converge if $f^{k + 1} $ were determined exactly at every step. Convergence is also proved under more specialized conditions, and for the procedure where $p_k = p$ for all $k$, an estimate for $p$ is obtained which optimizes the convergence rate. Examples are given for systems arising from the numerical solution of elliptic partial differential equations and numerical results are presented.