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Some properties of GMRES in Hilbert spaces. (English) Zbl 1167.65023
The authors consider the linear problem \(Ax=b\), where \(x,b\) belong to a separable Hilbert space, and \(A\) is a bounded linear operator on \(H\). The goal is to apply GMRES (Generalized Minimal RESidual method). The finite dimensional case has been considered for example by [P. N. Brown and H. F. Walker, SIAM J. Matrix Anal. Appl. 18, No. 1, 37–51 (1997; Zbl 0876.65019); D. Calvetti, B. Lewis and L. Reichel, Linear Algebra Appl. 316, No. 1–3, 157–169 (2000; Zbl 0963.65042); I. C. F. Ipsen and C. D. Mayer, Am. Math. Mon. 105, No. 10, 889–899 (1998; Zbl 0982.65034); L. Reichel and Q. Ye, SIAM J. Matrix Anal. Appl. 26, No. 4, 1001–1021 (2005; Zbl 1086.65030)].
The goal of the paper under review is to consider the infinite dimensional case. The authors show that results from the papers cited above extend to the infinite dimensional case under the additional assumption that the operator \(A\) is algebraic, i.e. there exists a polynomial \(p\) such that \(p(A)=0\).

MSC:
65J10 Numerical solutions to equations with linear operators
47A10 Spectrum, resolvent
47A50 Equations and inequalities involving linear operators, with vector unknowns
Software:
IPSEN
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References:
[1] DOI: 10.1137/S0895479894262339 · Zbl 0876.65019
[2] DOI: 10.1016/S0024-3795(00)00064-1 · Zbl 0963.65042
[3] DOI: 10.1007/s002110100339 · Zbl 1022.65044
[4] DOI: 10.2307/2589281 · Zbl 0982.65034
[5] DOI: 10.1080/01630568708816235 · Zbl 0633.47001
[6] DOI: 10.1137/S0895479803437803 · Zbl 1086.65030
[7] Taylor A.E., Introduction to Functional Analysis., 2. ed. (1980) · Zbl 0501.46003
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