Any decreasing cycle-convergence curve is possible for restarted GMRES

Abstract

Given a matrix order $n$, a restart parameter $m$ ($m < n$), a decreasing positive sequence $f(0) > f(1) > ... > f(q) \geq 0$, where $q < n/m$, it is shown that there exits an $n$-by-$n$ matrix $A$ and a vector $r_0$ with $\|r_0\|=f(0)$ such that $\|r_k\|=f(k)$, $k=1,...,q$, where $r_k$ is the residual at cycle $k$ of restarted GMRES with restart parameter $m$ applied to the linear system $Ax=b$, with initial residual $r_0=b-Ax_0$. Moreover, the matrix $A$ can be chosen to have any desired eigenvalues. We can also construct arbitrary cases of stagnation; namely, when $f(0) > f(1) > ... > f(i) = f(i+1) \geq 0 $ for any $i <q $. The restart parameter can be fixed or variable.