Abstract and Applied Analysis

Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques

M. Khalid Mahmood and Farooq Ahmad

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Abstract

In this paper we put forward a family of algorithms for lifting solutions of a polynomial congruence m o d p to polynomial congruence m o d p k . For this purpose, root-finding iterative methods are employed for solving polynomial congruences of the form a x n b ( m o d p k ) , k 1, where a , b, and n > 0 are integers which are not divisible by an odd prime p . It is shown that the algorithms suggested in this paper drastically reduce the complexity for such computations to a logarithmic scale. The efficacy of the proposed technique for solving negative exponent equations of the form a x - n b ( m o d p k ) has also been addressed.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 575064, 9 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425049592

Digital Object Identifier
doi:10.1155/2014/575064

Mathematical Reviews number (MathSciNet)
MR3216060

Zentralblatt MATH identifier
07022640

Citation

Mahmood, M. Khalid; Ahmad, Farooq. Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 575064, 9 pages. doi:10.1155/2014/575064. https://projecteuclid.org/euclid.aaa/1425049592


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References

  • E. V. Krishnamurthy and V. K. Murthy, “Fast iterative division of $p$-adic numbers,” IEEE Transactions on Computers, vol. 32, no. 4, pp. 396–398, 1983.
  • E. V. Krishnamurthy, “On optimal iterative schemes for high-speed division,” IEEE Transactions on Computers, vol. 20, pp. 227–231, 1970.
  • M. P. Knapp and C. Xenophontos, “Numerical analysis meets number theory: using rootfinding methods to calculate inverses mod ${p}^{n}$,” Applicable Analysis and Discrete Mathematics, vol. 4, no. 1, pp. 23–31, 2010.
  • E. Bach, “Iterative root approximation in $p$-adic numerical analysis,” Journal of Complexity, vol. 25, no. 6, pp. 511–529, 2009.
  • B. Kalantari, I. Kalantari, and R. Zaare-Nahandi, “A basic family of iteration functions for polynomial root finding and its characterizations,” Journal of Computational and Applied Mathematics, vol. 80, no. 2, pp. 209–226, 1997.
  • A. S. Householder, The Numerical Treatment of a Single Nonlinear Equation, McGraw-Hill, 1970.
  • J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, 1970.
  • I. Nivan and H. S. Zuckerman, An Introduction to the Theory of Numbers, John Wiley & Sons, 2005.
  • A. Adler and J. E. Coury, The Theory of Numbers, Jones and Bartlett Publishers, Boston, Mass, USA, 1995.
  • S. Abbasbandy, “Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 887–893, 2003.
  • S. Weerakoon and T. G. I. Fernando, “A variant of Newton's method with accelerated third-order convergence,” Applied Mathematics Letters, vol. 13, no. 8, pp. 87–93, 2000.
  • A. Y. Özban, “Some new variants of Newton's method,” Applied Mathematics Letters, vol. 17, no. 6, pp. 677–682, 2004.
  • J. F. Traub, Iterative Methods for Solutions of Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964. \endinput