Verified computation of Lamé functions with high accuracy
It is not common knowledge that a sufficiently accurate and efficient computation of higher transcendental functions creates serious problems, even nowadays. For instance, although Lamé (ellipsoidal) functions appear widely in mathematical physics involving ellipsoid geometry, a numerical method for their verified computation is a non trivial task. It is presented here. The necessary numerical tools applied here set high standards. They are e.~g. interval arithmetics (together with the use of algorithms due to \textit{S.~M.~Rump}, A new approach to scientific computation, Proc. Symp., Yorktown Heights, N. Y 1982, Notes Rep. Comput. Sci. Appl. Math. 7, 51-120 (1983; Zbl 0597.65018)), further precise computer arithmetics, so that no floating point number lies between the exact and rounded result of floating point numbers, and automatic differentiation. The high accuracy is illustrated by interval inclusions of various results for Lamé functions of first and second kind.
scientific article; zbMATH DE number 1112323
scientific article