Partial Fraction Expansions for Newton’s and Halley’s Iterations for Square Roots
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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).
Abstract
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).
참고문헌 (6)
- M. Abramowitz, "Handbook of Mathematical Functions with For- mulas, Graphs, and Mathematical Tables" (1972), Dover Publications
- G. B. Arfken, "Mathematical Methods for Physicists, 6th ed." (2005), Elsevier Academic Press
- 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), pp.1905~1922 (2010)
- "A Note on The Positivity of the Coefficients of Some Power Series Ex- pansions" (2011)[URL]
- J. C. Mason, "Chebyshev Polynomials", pp.2003, Chapman & Hall
- G. Polya, "Problems and Theorems in Analysis II" (1976), Springer Verlag