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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
DOI: https://doi.org/10.1090/S0025-5718-1976-0474802-4
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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DOI: https://doi.org/10.1090/S0025-5718-1976-0474802-4
Keywords: Integral equation, compact kernel, Galerkin method, Galerkin-Petrov method, eigenvalue problem
Article copyright: © Copyright 1976 American Mathematical Society

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