Share This Article:

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

Abstract Full-Text HTML Download Download as PDF (Size:337KB) PP. 777-783
DOI: 10.4236/am.2014.54074    3,537 Downloads   5,033 Views   Citations

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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

[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)

  
comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.