The polynomial approximation of zeros of the Bessel functions

Two methods of the polynomial approximation of zeros of the Bessel function $J_\nu(x)$ of the first genus and its derivative $J_\nu'(x)$ are considered, where they are assumed to be implicit functions on $\nu$ defined by equations $J_\nu(x)=0$ and $J_\nu'(x)=0$. The first method is based on the approximation of zeros by Taylor polynomials and uses the common algorithm for calculating the high derivative of an implicit function. The asymptotic expressions for zeros of $J_\nu'(x)$ are obtained as well as the numeric values of some first coefficients of decomposition. The region of applications of the formulas is investigated. The second method uses the approximation of the Bessel function by a polynomial of 4th degree and reduces to the solution of a certain system of algebraic equations. An analysis of the accuracy of these methods is carried out.