Any Nonincreasing Convergence Curve is Possible for GMRES

Authors:
Anne Greenbaum;Vlastimil Pták;Zdenuek Strakous
Affiliations:
-;-;-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
1996

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Abstract

Given a nonincreasing positive sequence $f(0) \geq f(1) \geq \cdots \geq f(n-1) 0$, it is shown that there exists an $n$ by $n$ matrix $A$ and a vector $r^0$ with $\| r^0 \| = f(0)$ such that $f(k) = \| r^k \|$, $k=1, \ldots , n-1$, where $r^k$ is the residual at step $k$ of the GMRES algorithm applied to the linear system $Ax=b$, with initial residual $r^0 = b - A x^0$. Moreover, the matrix $A$ can be chosen to have any desired eigenvalues.