An inverse convergence approach for arguments of Jacobian elliptic functions
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 tn, 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 the unknowns. In this paper, we describe a numerical procedure based on the convergence of the unknowns of the solution, by the use of arithmetical method, as an alternative way of solving the problem. The method gives very good results with no significant error, in the computation of the argument of the Jacobian elliptic function given the Jacobian elliptic function and the parameter. This new procedure is 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 modeling of porosity of bones. This porosity may lead to osteoporosis, a disease which affects bone mineral density in humans with bad consequences. Also these procedures are of importance in problems encountered in the physics discipline such as in the analysis of the dependence of the maximum tunneling current on external magnetic field for large area Josephson junctions with overlap boundary conditions. (C) 1999 Elsevier Science Ltd. All rights reserved.
inverse jacobian elliptic functions