TITLE:
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
AUTHORS:
Shunji Horiguchi
KEYWORDS:
Recurrence Formula; Newton-Raphson’s Method (Newton’s Method); Extensions of Murase-Newton’s Method; Horner’s Method
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.4,
March
18,
2014
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.