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

Shunji Horiguchi

Department of Economics, Niigata Sangyo University, Niigata, Japan.

DOI: 10.4236/am.2014.54074                

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.[1]). 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.[2]) 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.

Keywords

Recurrence Formula; Newton-Raphson’s Method (Newton’s Method); Extensions of Murase-Newton’s Method; Horner’s Method

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Murase, Y. (1673) Sanpoufutsudankai. Nishida, T., Ed., Kenseisha Co., Ltd., Tokyo. (in Japanese)
[2]	Suzuki, T. (2004) Wasan no Seiritsu. Kouseisha Kouseikaku Co., Ltd., Tokyo. (in Japanese)
[3]	Nagasaka, H. (1980) Computer and Numerical Calculations. Asakura Publishing Co., Ltd., Tokyo. (in Japanese)
[4]	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)
[5]	Horiguchi, S. (2011) General Recurrence Formula Obtained from the Murase Yoshimasu’s Recurrence Formulas and Newton’s Method. RIMS, 1739, 234-244. (in Japanese)