Zeros of Nonlinear Functions

The methods described in this note are actually applicable to a wider class of problems than that indicated in the title. However, it was this problem which originally motivated the development of the methods, and it remains the application of most interest. During the past several years powerful methods have been developed for the numerical integration of simultaneous ordinary differential equations on digital computers. The striking feature of these methods is the inclusion of algorithms for automatic modification of the interval of integration in order to preserve a specified degree of accuracy in the integrated results with nearly optimum efficiency of the entire process. This has motivated the reformulation of problems in terms of simultaneous ordinary differential equations whenever possible. For example, the computer time savings involved in automatic and continuous selection of the optimum interval in the evaluation of multiple definite integrals is apparent. This note describes a method of reformulating the problem of finding zeros of nonlinear functions in terms of the solution of simultaneous ordinary differential equations.