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 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 (where {} is a given complete set in *H*, and the , are arbitrary numbers) is less accurate than the best approximation of the form , if *n* is sufficiently large. Specifically it is shown that if is chosen optimally (i.e. if the coefficients are chosen to minimize ), and if is chosen to be the first iterate of , i.e. , then , with . A similar result is also obtained, provided the homogeneous equation has no nontrivial solution, if instead is chosen to be the approximate solution by the Galerkin or Galerkin-Petrov method. A generalization of the first result to the approximate forms obtained by further iteration is also shown to be valid, if the range of *K* is dense in *H*.

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Additional Information

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