ELF and GNOME: two tiny codes to evaluate the real zeros of the Bessel functions of the first kind for real orders. Two codes to evaluate the real zeros (j ν,s ) of the Bessel functions of the first kind J ν (x) for real orders are presented. The codes are based on a Newton-Raphson iteration over the monotonic function f ν (x)=x 2ν-1 J ν (x)/J ν-1 (x). The code ELF is a remarkably short program for finding, given any starting value x 0 >0 and any real order, the zero of J ν (x) in the neighborhood of 0 0 (x 0 and the zero in the same branch of f ν (x)). GNOME is a modification of ELF for finding the zeros of J ν (x) inside a given interval [x min ,x max ]; for simplicity, we restrict the code GNOME to work for ν>-1, which is the region of greatest practical use, where all the zeros of J ν (x) are real. The method is especially efficient for moderate values of and for small zeros, where asymptotic expansions tend to fail and, besides, contrary to existing algorithms, enables the search of the real zeros for real orders, including negative orders.
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References in zbMATH (referenced in 10 articles , 1 standard article )
Showing results 1 to 10 of 10.
- Hsu, Cheng-Hsiung; Yang, Chi-Ru: Zeros’ distribution of the first kind Bessel functions (2016)
- Gil, Amparo; Segura, Javier: Computing the real zeros of cylinder functions and the roots of the equation $x\cal C_\nu'(x)+\gamma\cal C_\nu(x)=0$. (2012)
- Pérez-Arancibia, Carlos; Durán, Mario: On the Green’s function for the Helmholtz operator in an impedance circular cylindrical waveguide (2010)
- Gil, Amparo; Segura, Javier: A combined symbolic and numerical algorithm for the computation of zeros of orthogonal polynomials and special functions (2003)
- Gil, Amparo; Segura, Javier: Computing the zeros and turning points of solutions of second order homogeneous linear ODEs (2003)
- MacLeod, Allan J.: Rational expressions for the zeros of several special functions (2002)
- Segura, Javier: The zeros of special functions from a fixed point method (2002)
- Plagianakos, V.P.; Nousis, N.K.; Vrahatis, M.N.: Locating and computing in parallel all the simple roots of special functions using PVM (2001)
- Segura, Javier: Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros (2001)
- Segura, J.; Gil, A.: ELF and GNOME: two tiny codes to evaluate the real zeros of the Bessel functions of the first kind for real orders (1999)