Computationally improved algorithm to find higher roots of integer order bessel function in gyrotron application

The calculation of higher order roots of Bessel function is computationally intensive, time consuming and not easily available in literature. In real life the solution of many problems comes in the form of Bessel function. Design of fast wave devices, like Gyrotron, requires zeros of the first derivatives of Bessel function of first kind. The Gyrotron is a high frequency, high power microwave device. It operates at higher transverse electric modes. The synthesis of higher operating mode Gyrotron requires zeros of Bessel function of first kind and its first derivative. In this paper an optimized algorithm to efficiently calculate the higher roots of any integer order Bessel function is discussed. Algorithm uses bisection method & property of Bessel function. To find roots, series form of Bessel function is used. Few optimization to this form was done while implemented in code & by applying Bessel function property algorithm proficiently calculate even higher roots of Bessel function of integer order within very less time. Algorithm is implemented in JAVA language.