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

Nichols, N. (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

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To link to this item 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.

Item Type: Article Yes Science > School of Mathematical, Physical and Computational Sciences > Department of MeteorologyScience > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics 27521 Society for Industrial and Applied Mathematics

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