Finding All Solutions of Nonlinear Equations Using Linear Combinations of Functions

As a computational method to find all solutions of nonlinear equations, interval analysis is well-known. In order to improve the computational efficiency of interval analysis, it is necessary to develop a powerful test for nonexistence of a solution in a given region. In this paper, a new nonexistence test is proposed which is more powerful than the conventional test. The basic idea proposed here is to apply the conventional test to linear combinations of functions. Effective linear combinations are proposed which make the nonexistence test very powerful. Using the proposed techniques, all solutions of nonlinear equations (including a system of 100 nonlinear equations and a system with strong nonlinearity which describes a transistor circuit) could be found very efficiently.

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