On the Convergence of Two Methods for the Simultaneous Finding of All Roots of Exponential Equations

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Abstract
Recently, considerable attention has been given to general polynomials and particularly to algebraic, trigonometrical and exponential equations. These polynomials have interesting applications in the theory of approximations, in numerical analysis and in many physical problems. In several previous works the authors developed methods for simultaneous finding of all roots of algebraic and trigonometrical equations and they established their larger domain of convergence in comparison with the methods for individual search of the roots. Two iterative methods of this kind are considered in this paper for the case of exponential equations. It is proved that one of them has a quadratic order of convergence, and the other a cubic one. The advantage of the new methods over the classical methods of Newton and Chebyshev is shown both by numerical experiments and theoretical considerations based on continuous analogues.

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