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A third-order iterative algorithm for inversion of cumulative central beta distribution

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Abstract

An efficient third-order iterative method for inverting the cumulative central beta distribution numerically is proposed. First, a third-order iterative method for finding zeros of the solution of second-order homogeneous linear ODEs is designed. This method is derived by approximating the integration obtained from the second-order ODE. The method is exact for any function f with a constant logarithmic derivative of \(f'\). Sufficient conditions are obtained to ensure the nonlocal convergence of the proposed method. As an application, an interesting numerical algorithm is obtained for inverting the cumulative central beta distribution. To demonstrate the proposed theory, numerical simulation results were presented and compared with the existing algorithms.

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The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

Notes

  1. We observed that the present version of the software package R (Version 4.2.2) is able to invert all the values in Table 4.

References

  1. Abernathy, R.W., Smith, R.P.: Algorithm 724: program to calculate f-percentile. ACM Trans. Math. Soft. 19(4), 481–183 (1993)

    Article  MATH  Google Scholar 

  2. Alshomrani, A.S., Behl, R., Kanwar, V.: An optimal reconstruction of Chebyshev-Halley type methods for nonlinear equations having multiple zeros. J. Comput. Appl. Math. 354, 651–662 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berry, K.J., Mielke, P.W., Cran, G.W.: Algorithm as r83: a remark on algorithm as 109: Inverse of the incomplete beta function ratio. J. R. Stat. Soc. Ser. C. Appl. Stat. 39(2), 309–310 (1990)

    Google Scholar 

  4. Berry, K.J., Mielke, P.W., Cran, G.W.: Correction to algorithm as r83-a remark on algorithm as 109: inverse of the incomplete beta function ratio. J. R. Stat. Soc. Ser. C. Appl. Stat. 40(1), 236 (1991)

    Google Scholar 

  5. Chen, J.Y., Kincaid, D.R., Lin, B.R.: A variant of Newton’s method based on Simpson’s three-eights rule for nonlinear equations. Appl. Math. Lett. 79, 1–5 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cran, G.W., Martin, K.J., Thomas, G.E.: Remark as r19 and algorithm as 109:a remark on algorithms: As 63: The incomplete beta integral as 64: Inverse of the incomplete beta function ratio. J. R. Stat. Soc. Ser. C. Appl. Stat. 26(1), 111–114 (1977)

    Google Scholar 

  7. Devroye, L.: Nonuniform random variate generation. Springer-Verlag, New york (1986)

    Book  MATH  Google Scholar 

  8. Frontini, M., Sormani, E.: Some variant of Newton’s method with third order convergence. Appl. Math. Comput. 140(2–3), 419–426 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Gil, A., Segura, J., Temme, N.M.: A new asymptotic representation and inversion method for the Student’s \(t\) distribution. Integral Transforms Spec. Funct. 33(8), 597–608 (2021)

  10. Gil, A., Segura, J., Temme, N.M.: Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41(3), 827–855 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gil, A., Segura, J., Temme, N.M.: Efficient algorithms for the inversion of the cumulative central beta distribution. Numer. Algorithms. 74(1), 77–91 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heckert, N.A., Filliben, J.J.: NIST/SEMATECH e-Handbook of Statistical Methods; chapter 1: Exploratory data analysis, Gaithersburg , Maryland: NIST Interagency/Internal Report (NISTIR), National Institute of Standards and Technology, Gaithersburg, MD. https://doi.org/10.18434/M32189 (2003)

  13. Ivanov, S.I.: A general approach to the study of the convergence of Picard iteration with an application to Halley’s method for multiple zeros of analytic functions. J. Math. Anal. Appl. 513(2), 1–12 (2022)

    Article  MathSciNet  Google Scholar 

  14. Majumder, K.L., Bhattacharjee, G.P.: Algorithm as 64: inverse of the incomplete beta function ratio. J. R. Stat. Soc. Ser. C. Appl. Stat. 2(3), 411–414 (1973)

    Google Scholar 

  15. Melman, A.: Geometry and convergence of Euler’s and Halley’s methods. SIAM Rev. 39(4), 758–735 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press. Cambridge, UK (2010)

    MATH  Google Scholar 

  17. Petković, M.S., Neta, B.,Petković, L.D., Džunić, D.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Amsterdam (2013)

  18. Petković, M.S., Petković, L.D., Neta, B.: On generalized Halley-like methods for solving nonlinear equations. Appl. Anal. Discrete Math. 13(2), 399–422 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Petković, M.S., Yun, B.I.: Sigmoid-like functions and root finding methods. Appl. Math. Comput. 204(2), 784–793 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Sugiura, H., Hasegawa, T.: On the global convergence of Schröder’s iteration formula for real zeros of entire functions. J. Comput. Appl. Math. 358, 136–145 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Segura, J.: Reliable computation of the zeros of solutions of second order linear ODEs using a fourth order method. SIAM J. Numer. Anal. 48, 452–469 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Segura, J.: The Schwarzian-Newton method for solving nonlinear equations, with applications. Math. Comput. 86(304), 856–879 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Temme, N.M.: Asymptotic inversion of the incomplete beta function. J. Comput. Appl. Math. 41(1–2), 145–157 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yun, B.I.: A non-iterative method for solving non-linear equations. Appl. Math. Comput. 198(2), 691–699 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Yun, B.I., Petković, M.S.: Iterative methods based on the signum function approach for solving nonlinear equations Numer. Algorithms 52(4), 649–662 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yun, B.I.: Transformation methods for finding multiple roots of nonlinear equations. Appl. Math. Comput. 217(2), 599–606 (2010)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the referee for carefully evaluating the manuscript and for their suggestions and comments, enhancing the readability and quality of the paper. The first author is thankful to the Council of Scientific and Industrial Research India (Grant No. 09/1022(11054)/2021-EMR-I) for the financial support.

Funding

Council of Scientific and Industrial Research India (CSIR) provided financial support for Dhivya Prabhu K for this research.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Indore, Indore, 453552, Madhya Pradesh, India

    Dhivya Prabhu K, Sanjeev Singh & Antony Vijesh V

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  1. Dhivya Prabhu K
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  2. Sanjeev Singh
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  3. Antony Vijesh V
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All figures and tables were prepared by K. Dhivya Prabhu. All authors wrote and reviewed the manuscript.

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Correspondence to Antony Vijesh V.

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K, D.P., Singh, S. & V, A.V. A third-order iterative algorithm for inversion of cumulative central beta distribution. Numer Algor 94, 1331–1353 (2023). https://doi.org/10.1007/s11075-023-01537-6

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