Singular values and condition numbers of galerkin matrices arising from linear integral equations of the first kind
Journal Article
·
· J. Math. Anal. Appl.; (United States)
In (1) the problem of the conditioning of matrices arising in the numerical solution of integral equations of the first kind by Galerkin and collocation schemes was investigated. A lower bound on the matrix condition number was found and its behavior as a function of the smoothness of the kernel of the original equation was studied. Some numerical results presented in (1) and much more extensive studies reported in (2) served to demonstrate the validity and usefulness of the theory. In these studies all basis functions were required to be orthonormal. Current computational methods frequently use non-orthonormal bases (for example, splines), and the question naturally arises as to whether such bases yield better conditions. Actually, in both (1) and (2) piecewise constant splines were examined as a special orthonormal set. They, of course, conformed to the general analytical theory. Computationally they usually proved more satisfactory than more classical orthonormal bases, providing both better conditioning and more satisfactory numerical results. In this paper the authors shall examine much more general bases including some that are spline-like. They show that in general little change is made in the analytical results found in (1). They also provide several corroborating numerical examples.
- Research Organization:
- Univ. of New Mexico, Albuquerque
- OSTI ID:
- 6743653
- Journal Information:
- J. Math. Anal. Appl.; (United States), Journal Name: J. Math. Anal. Appl.; (United States) Vol. 109:2; ISSN JMANA
- Country of Publication:
- United States
- Language:
- English
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