Partial Fraction Expansions for Newton's and Halley's Iterations for Square Roots

Kouba, Omran

  • Received : 2011.04.16
  • Accepted : 2011.09.28
  • Published : 2012.09.23


When Newton's method, or Halley's method is used to approximate the pth root of 1-z, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).


Newton's method;Halley's method;Series expansion;Square roots;Chebyshev's Polynomials


  1. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, (1972).
  2. G. B. Arfken, and H. J.Weber, Mathematical Methods for Physicists, 6th ed., Elsevier Academic Press, (2005).
  3. Ch-H. Guo, On Newton's method and Halley's method for principal pth root of a matrix, Linear algebra and its applications, 432(8)(2010), 1905-1922.
  4. O. Kouba, A Note on The Positivity of the Coefficients of Some Power Series Expansions, Preprint, (2011), [Online :].
  5. J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, (2003).
  6. G. Polya, and G. Szego, Problems and Theorems in Analysis II, Springer Verlag, New York, Heidelberg, Berlin (1976).

Cited by

  1. The dual Padé families of iterations for the matrix pth root and the matrix p-sector function vol.272, 2014,
  2. A study of Schröder’s method for the matrix pth root using power series expansions pp.1572-9265, 2019,