arXiv subjects

Antony Vijesh V

Publications and source records attributed to Antony Vijesh V.

3 recordsLinked to original sources

arXiv2026

Efficient third-order iterative algorithms for computing zeros of special functions

This manuscript presents a novel and reliable third-order iterative procedure for computing the zeros of solutions to second-order ordinary differential equations. By approximating the solution of the related Riccati differential equation using the trapezoidal rule, this study has derived the proposed third-order method. This work establishes sufficient conditions to ensure the theoretical non-local convergence of the proposed method. This study provides suitable initial guesses for the proposed third-order iterative procedure to compute all zeros in a given interval of the solutions to second-order ordinary differential equations. The orthogonal polynomials like Legendre and Hermite, as well as the special functions like Bessel, Coulomb wave, confluent hypergeometric, and cylinder functions, satisfy the proposed conditions for convergence. Numerical simulations demonstrate the effectiveness of the proposed theory. This work also presents a comparative analysis with recent studies.

math.NA↗

arXiv2026

Modified Halley's method for computation of zeros of solution of second order ODEs

This paper develops an efficient iterative method for computing all zeros of solutions of second order ordinary differential equations. A third order Halleys method is first derived by approximating the solution of an associated Riccati differential equation. To improve computational efficiency, a modified Halleys method is proposed by fixing one of the functions in Halleys scheme as a constant. The modified Halleys method also retains third order convergence. Based on the behavior of the coefficients of the second order ODE, nonlocal convergence results are established for both Halleys and modified Halleys methods. Suitable initial guesses for computing all zeros of solutions of second order ODEs in a given interval are also presented for both methods. Furthermore, algorithms based on the modified Halleys method are developed for to compute all nodes and weights for Gauss Legendre and Gauss Hermite quadratures. A comparative numerical study with recent methods demonstrates the efficiency of the proposed algorithms.

math.NA↗

arXiv2024

Infinitely divisible modified Bessel distributions

In this paper we focus on continuous univariate probability distributions, like McKay distributions, $K$-distribution, generalized inverse Gaussian distribution and generalised McKay distributions, with support $[0,\infty),$ which are related to modified Bessel functions of the first and second kinds and in most cases we show that they belong to the class infinitely divisible distributions, self-decomposable distributions, generalized gamma convolutions and hyperbolically completely monotone densities. Some of the results are known, however the proofs are new and we use special functions technique. Integral representations of quotients of Tricomi hypergeometric functions as well as of quotients of Gaussian hypergeometric functions, or modified Bessel functions of the second kind play an important role in our study. In addition, by using a different approach we rediscover a Stieltjes transform representation due to Hermann Hankel for the product of modified Bessel functions of the first and second kinds and we also deduce a series of new Stieltjes transform representations for products, quotients and their reciprocals concerning modified Bessel functions of the first and second kinds. By using these results we obtain new infinitely divisible modified Bessel distributions with Laplace transforms related to modified Bessel functions of the first and second kind. Moreover, we show that the new Stieltjes transform representations have some interesting applications and we list some open problems which may be of interest for further research. In addition, we present a new proof via the Pick function characterization theorem for the infinite divisibility of the ratio of two gamma random variables and we present some new Stieltjes transform representations of quotients of Tricomi hypergeometric functions.

math.PR↗