Convergence Investigation of the Newton Method Used for Hyperbolic Equations Systems Solving

Abstract

The coordinates determination of the radio emission source by the multi position passive radio monitoring systems is based on the time difference of arrival method. The main measured information to solve this problem is the delay time of signal at the receiving stations. The coordinates determination of the observed object is based on the solution of hyperbolic equations at the central receiving point. The system of nonlinear hyperbolic equations is constructed for the receiving stations, where one of them is the central receiving station, which expresses the structure of distances and the geometry of the mutual arrangement of stations and radio emission source. The most common method for solving such systems is the Newton method. Subject to certain requirements, the Newton method is also the most effective. However, its practical application is complicated by the difficulties connected with the poor conditionality of partial derivative matrices. It reduces the coordinate accuracy finding, the convergence rate, and may eventually create fundamental difficulties in solving problems. The article discusses the conditions of such situations and proposes some approaches to avoid these problems or reduce their negative impact. Consideration must also be given to the significant constraints on calculation time required to complete a computational procedure with a given accuracy, because these requirements are dictated by the real-time principle of the passive radio monitoring systems.