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Tsuchikura, T. (2011) Calculation Methods of p-th Root by the Ideas That the Mathematicians of Wasan Think about. The Bulletin of Wasan Institute, 3, 10-16. (In Japanese)

has been cited by the following article:

• TITLE: The Formulas to Compare the Convergences of Newton’s Method and the Extended Newton’s Method (Tsuchikura-Horiguchi Method) and the Numerical Calculations

  AUTHORS: Shunji Horiguchi

  KEYWORDS: Recurrence Formula, Newton-Raphson’s Method (Newton’s Method), Extension of Newton’s Method

  JOURNAL NAME: Applied Mathematics, Vol.7 No.1, January 20, 2016

  ABSTRACT: This paper gives the extension of Newton’s method, and a variety of formulas to compare the convergences for the extension of Newton’s method (Section 4). Section 5 gives the numerical calculations. Section 1 introduces the three formulas obtained from the cubic equation of a hearth by Murase (Ref. [1]). We find that Murase’s three formulas lead to a Horner’s method (Ref. [2]) and extension of a Newton’s method (2009) at the same time. This shows originality of Wasan (mathematics developed in Japan) in the Edo era (1603-1868). Suzuki (Ref. [3]) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 2 gives the relations between Newton’s method, Horner’s method and Murase’s three formulas. Section 3 gives a new function defined such as .