Reliable Computing 3: 287–295, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Attacking a Conjecture in Mathematical Physics
by Combining Methods of Computational
Analysis and Scientiﬁc Computing
URGEN DOBNER and STEFAN RITTER
Mathematisches Institut II, Universit
at Karlsruhe (TH), D-76128 Karlsruhe, Germany
(Received: 12 November 1996; accepted: 18 February 1997)
Abstract. We consider a conjecture on the sum of eigenvalues of two integral operators arising in
potential and scattering theory for the case that the underlying surface is a triaxial ellipsoid. This
concerns computation of Lam
e functions which are anyway of great interest in electromagnetics and
mechanics. We provide a new effective scheme for the numerical treatment of these special functions.
It involves computing the Lam
e functions with high accuracy combined with safe error estimates.
1. Preliminary Remarks
In contrast to many other authors (e.g., Golub , v.d. Velde ) we understand by
scientiﬁc computing, numerical algorithms merging interval analysis and computer
science to develop reliable solution methods and effective implementation. One
of the major point of criticism of such a scientiﬁc computing is the lack of real
applications. In this article we overcome this shortcoming by treating an impor-
tant practical problem arising from mathematical physics, with interval analytic
2. A Conjecture on the Eigenvalues of the Electrostatic Integral Operator
Formulating boundary value problems for harmonic functions as Fredholm integral
equations in the case of ellipsoidal bodies with boundary surface S we meet the
double-layer potential integral operator K which is given for u
and the electrostatic integral operator K
|x − y|
Here n denotes the outward unit normal on S,and|x−y|is the euclidean
distance. The operators K, K
are compact in the Banach space C(S) equipped with