Direct and inverse interpolation for Jacobian elliptic functions, zeta function of Jacobi and complete elliptic integrals of the second kind

Abstract

Computing the value of the Jacobian elliptic functions, given the argument u and the parameter m, is a problem, whose solution can be found either tabulated in tables of elliptic functions [1] or by use of existing software, such as Mathematica, etc. The inverse problem, finding the argument, given the Jacobian elliptic function and the parameter m, is a problem whose solution is found only in tables of elliptic functions. Standard polynomial inverse interpolation procedures fail, due to ill conditioning of the system of linear equations of the unknowns. In this paper, we describe a numerical procedure for inverse interpolation which gives good results in the computation of the argument of the Jacobian elliptic function given the Jacobian elliptic function and the parameter. Also, a direct interpolation is described which gives the Zeta function of Jacobi and the complete elliptic integral of the second kind given the argument and the parameter. These new interpolation procedures are important in problems involving cavities or inclusions of ellipsoidal shape encountered in the mechanical design of bearings, filters and composite materials. They are also important in the modelling of porosity of bones. This porosity may lead to osteoporosis, a disease which affects bone mineral density in humans with bad consequences.