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Improvement by iteration for compact operator equations

Author: Ian H. Sloan
Journal: Math. Comp. 30 (1976), 758-764
MSC: Primary 65J05
MathSciNet review: 0474802
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Abstract: The equation $ y = f + Ky$ is considered in a separable Hilbert space H, with K assumed compact and linear. It is shown that every approximation to y of the form $ {y_{1n}} = {\Sigma ^n}{a_{ni}}{u_i}$ (where {$ {u_i}$} is a given complete set in H, and the $ {a_{ni}},1 \leqslant i \leqslant n$, are arbitrary numbers) is less accurate than the best approximation of the form $ {y_{2n}} = f + {\Sigma ^n}{b_{ni}}K{u_i}$, if n is sufficiently large. Specifically it is shown that if $ {y_{1n}}$ is chosen optimally (i.e. if the coefficients $ {a_{ni}}$ are chosen to minimize $ \left\Vert {y - {y_{1n}}} \right\Vert$), and if $ {y_{2n}}$ is chosen to be the first iterate of $ {y_{1n}}$, i.e. $ {y_{2n}} = f + K{y_{1n}}$, then $ \left\Vert {y - {y_{2n}}} \right\Vert \leqslant {\alpha _n}\left\Vert {y - {y_{1n}}} \right\Vert$, with $ {\alpha _n} \to 0$. A similar result is also obtained, provided the homogeneous equation $ x = Kx$ has no nontrivial solution, if instead $ {y_{1n}}$ is chosen to be the approximate solution by the Galerkin or Galerkin-Petrov method. A generalization of the first result to the approximate forms $ {y_{3n}},{y_{4n}}, \ldots $ obtained by further iteration is also shown to be valid, if the range of K is dense in H.

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  • [1] N. I. AHIEZER &. I. M. GLAZMAN, Theory of Linear Operators in Hilbert Space, vol. 1, GITTL, Moscow, 1950; English transl., Ungar, New York, 1961, Chapter 1. MR 13, 358; 41 #9015a. MR 0044034 (13:358b)
  • [2] C. T. H. BAKER, "Expansion methods," Numerical Solution of Integral Equations, L. M. Delves and J. Walsh (Editors), Clarendon Press, Oxford, 1974, Chapter 7. MR 0464624 (57:4551)
  • [3] M. A. KRASNOSEl'SKIĬ, G. M. VAĬNIKKO, P. P. ZABREĬKO, Ja. B. RUTICKIĬ & V. Ja. STECENKO, Approximate Solution of Operator Equations, "Nauka", Moscow, 1969; English transl., Wolters-Noordhoff, Groningen, 1972. MR 41 #4271. MR 0259635 (41:4271)
  • [4] S. G. MIHLIN & H. L. SMOLICKIĬ, Approximate Methods for Solution of Differential and Integral Equations, "Nauka", Moscow, 1965; English transl, by Scripta Technica, Modern Analytic and Computational Methods in Science and Mathematics, vol. 5, American Elsevier, New York, 1967, Chapter 4. MR 33 #855; 36 #1108. MR 0192630 (33:855)
  • [5] F. RIESZ & B. SZ.-NAGY, Functional Analysis, Akad. Kiadó, Budapest, 1952, 2nd ed., 1953; English transl. of 2nd ed., Ungar, New York, 1955. MR 17, 175. MR 0071727 (17:175i)
  • [6] I. H. SLOAN, "Error analysis for a class of degenerate-kernel methods," Numer. Math., v. 25, 1976, pp. 231-238. MR 0443389 (56:1759)
  • [7] I. H. SLOAN, B. J. BURN & N. DATYNER, "A new approach to the numerical solution of integral equations," J. Computational Phys., v. 18, 1975, pp. 92-105. MR 0398137 (53:1992)

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Keywords: Integral equation, compact kernel, Galerkin method, Galerkin-Petrov method, eigenvalue problem
Article copyright: © Copyright 1976 American Mathematical Society

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