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

Dhivya Prabhu K, Sanjeev Singh, Antony Vijesh

TL;DR

This paper develops a new third-order iterative method to compute all zeros of solutions to second-order linear ODEs by transforming the ratio of interlaced solution components into a Riccati equation and applying a trapezoidal-rule approximation. The resulting update exhibits cubic convergence and a built-in bracketing property, with global convergence guaranteed between successive singular points under explicit assumptions. The authors validate the method on Legendre and Hermite polynomials as well as Bessel, Cylinder, confluent hypergeometric, and Coulomb wave functions, providing thorough convergence analysis, initial-guess strategies, and numerical comparisons. The approach delivers high accuracy and favorable computational times in large-scale zero-finding problems, offering a unified, implementable framework for zeros of a wide class of special functions. Overall, the work advances robust zero-finding in spectral methods and quantum/wave-related applications by delivering a practical, globally convergent third-order algorithm.

Abstract

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.

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

TL;DR

This paper develops a new third-order iterative method to compute all zeros of solutions to second-order linear ODEs by transforming the ratio of interlaced solution components into a Riccati equation and applying a trapezoidal-rule approximation. The resulting update exhibits cubic convergence and a built-in bracketing property, with global convergence guaranteed between successive singular points under explicit assumptions. The authors validate the method on Legendre and Hermite polynomials as well as Bessel, Cylinder, confluent hypergeometric, and Coulomb wave functions, providing thorough convergence analysis, initial-guess strategies, and numerical comparisons. The approach delivers high accuracy and favorable computational times in large-scale zero-finding problems, offering a unified, implementable framework for zeros of a wide class of special functions. Overall, the work advances robust zero-finding in spectral methods and quantum/wave-related applications by delivering a practical, globally convergent third-order algorithm.

Abstract

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.
Paper Structure (17 sections, 11 theorems, 63 equations, 8 figures, 7 tables)

This paper contains 17 sections, 11 theorems, 63 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of third-order iterative method
  4. Convergence analysis of the proposed scheme
  5. Application
  6. Legendre polynomial
  7. Hermite polynomial
  8. Bessel function
  9. Cylinder function
  10. Confluent hypergeometric function
  11. Coulomb wave function
  12. Numerical results
  13. Legendre polynomial
  14. Hermite polynomial
  15. Bessel function
  16. ...and 2 more sections

Key Result

Theorem 1

20032002 Let $A_{y}(x),$$B_{y}(x),$$A_w(x),$$B_w(x),$$c_1(x),$$c_2(x),$$c_3(x),$ and $c_4(x)$ are real valued continuous functions on $I.$ Let $\{y_1(x), y_2(x)\}$ and $\{w_1(x), w_2(x)\}$, $x \in I$ be fundamental sets of solutions of the second-order linear ODEs and respectively. Further assume that $(y_1(x), w_1(x))$ and $(y_2(x), w_2(x))$ are solutions of coupled differential equation (CDE)

Figures (8)

  • Figure 1: Illustration of Table \ref{["on I' nonzero r"]} for the case $r(z)>0$
  • Figure 2: Illustration of Table \ref{["on I' nonzero r"]} for the case $r(z)<0$
  • Figure 3: Illustration of Table \ref{["on I' when r has zero"]} for the case $z_{w_j}<z_{y_j}<z_r<z_{w_{j+1}}$
  • Figure 4: Illustration of Table \ref{["on I' when r has zero"]} for the case $z_{w_j}<z_r<z_{y_j}<z_{w_{j+1}}$
  • Figure 5: FOM-L seg11 vs SOM-L 2002 vs TOM \ref{['iter']} in finding all zeros of $P_n(x)$.
  • ...and 3 more figures

Theorems & Definitions (24)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Remark 2
  • Remark 3
  • Theorem 3
  • proof
  • Lemma 3
  • ...and 14 more