Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Improvement by iteration for compact operator equations

Author: Ian H. Sloan
Journal: Math. Comp. 30 (1976), 758-764
MSC: Primary 65J05
MathSciNet review: 0474802
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65J05

Retrieve articles in all journals with MSC: 65J05

Additional Information

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