On Relations between the General Recurrence Formula of the Extension of Murase-Newton’s Method (the Extension of Tsuchikura*-Horiguchi’s Method) and Horner’s Method
Author(s) Shunji Horiguchi
ABSTRACT

In 1673, Yoshimasu Murase made a cubic equation to obtain the thickness of a hearth. He introduced two kinds of recurrence formulas of square and the deformation (Ref.). We find that the three formulas lead to the extension of Newton-Raphson’s method and Horner’s method at the same time. This shows originality of Japanese native mathematics (Wasan) in the Edo era (1600- 1867). Suzuki (Ref.) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 1 introduces Murase’s three solutions of the cubic equation of the hearth. Section 2 explains the Horner’s method. We give the generalization of three formulas and the relation between these formulas and Horner’s method. Section 3 gives definitions of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), general recurrence formula of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), and general recurrence formula of the extension of Murase-Newton’s method (the extension of Tsuchikura-Horiguchi’s method) concerning n-degree polynomial equation. Section 4 is contents of the title of this paper.

Cite this paper
Horiguchi, S. (2014) On Relations between the General Recurrence Formula of the Extension of Murase-Newton’s Method (the Extension of Tsuchikura*-Horiguchi’s Method) and Horner’s Method. Applied Mathematics, 5, 777-783. doi: 10.4236/am.2014.54074.
References
   Murase, Y. (1673) Sanpoufutsudankai. Nishida, T., Ed., Kenseisha Co., Ltd., Tokyo. (in Japanese)

   Suzuki, T. (2004) Wasan no Seiritsu. Kouseisha Kouseikaku Co., Ltd., Tokyo. (in Japanese)

   Nagasaka, H. (1980) Computer and Numerical Calculations. Asakura Publishing Co., Ltd., Tokyo. (in Japanese)

   Horiguchi, S., Kaneko, T. and Fujii, Y. (2013) On Relation between the Yoshimasu Murase’s Three Solutions of a Cubic Equation of Hearth and Horner’s Method. The Bulletin of Wasan Institute, 13, 3-8. (in Japanese)

   Horiguchi, S. (2011) General Recurrence Formula Obtained from the Murase Yoshimasu’s Recurrence Formulas and Newton’s Method. RIMS, 1739, 234-244. (in Japanese)

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