Any decreasing cycle-convergence curve is possible for restarted GMRES

2009/07/21 by Eugene Vecharynski, Vecharynski, Eugene, Julien Langou +1
Computer Science · Mathematics · #65F10 #Advanced Topics in Algebra #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.0907.3573

openalex publication_date 2009/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a matrix order n, a restart parameter m (m < n), a decreasing positive sequence f(0) > f(1) > ... > f(q) ≥ 0, where q < n/m, it is shown that there exits an n-by-n matrix A and a vector r₀ with ‖r₀‖=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 r₀=b-Ax₀. 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) ≥ 0 for any i

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