A RECURSIVE TECHNIQUE TO AVOID ARITHMETIC OVERFLOW AND UNDERFLOW WHEN COMPUTING SLOWLY CONVERGENT EIGENFUNCTION TYPE EXPANSION.

Abstract

Eigenfunction expansions for fields scattered by large structures are generally very slowly convergent. The summation often consists of two factors where one factor approaches zero and the other factor grows in magnitude without bound as the summation index increases. Each term of the expansion is bounded; however, due to the extreme magnitude of the individual factors, computational overflow and underflow errors can limit the number of terms that can be computed in the summation thereby forcing the summation to be terminated before it has converged. In this paper an exact technique that circumvents these problems is presented. An auxiliary function is introduced which is proportional to the original factor with its asymptotic behavior factored out. When these auxiliary functions are introduced into the summation, we are left with the task of numerically summing products of well behaved factors. A recursion relationship is developed for computing this auxiliary function. [Vol. 7, No. 1, pp. 67-77 (1992)]