DIRECT AND INVERSE INTERPOLATION FOR JACOBIAN ELLIPTIC FUNCTIONS, ZETA-FUNCTION OF JACOBI AND COMPLETE ELLIPTIC INTEGRALS OF THE 2ND KIND

Computing the value of the Jacobian elliptic functions, given the argu ment mu and the parameter m, is a problem, whose solution can be found either tabulated in tables of elliptic functions [1] or by use of exi sting 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 fun ctions. 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 interpo lation which gives good results in the computation of the argument of the Jacobian elliptic function given the Jacobian elliptic function an d 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 interpola tion procedures are important in problems involving cavities or inclus ions of ellipsoidal shape encountered in the mechanical design of bear ings, 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 cons equences.