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

In 1673, Yoshimasu Murase made a cubic equation to obtain the thickness of a hearth. He introduced two kinds of recurrence formulas of square 2 k x 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 (16001867). 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.


Introduction
All the references are written in Japanese.We wrote this paper from two kinds of recurrence formulas of the square 2 k x and the deformation of a cubic equation written in Ref. [1], and a hint of Tsuchikura.Therefore, it is enough for readers to know these three formulas.But it is very difficult even for Japanese people to read the Murase's book written in the Japanese ancient writing.Therefore, the readers do not need to read the book.Furthermore, the readers do not need to mind Japanese references.From now on, we explain the Murase's three formulas as introduction.The readers can know the origin of this paper.
Murase made the cubic equation for the next problem in 1673.
There is a rectangular solid (base is a square).We put it together four and make the hearth as Figure 1.We claim one side of length of the square that one side is 14, and a volume becomes 192 of the hearth.Let one side of length of the square be x then the next cubic equation is obtained.
The second method ( ) here he calculates to x 0 = 0, x 1 = 1.85, x 2 = 1.976, x 3 = 1.9989, x 4 = 1.9999907, and decides a solution with 2. An expression (1.4) has better precision than that (1.3), and convergence becomes fast.The third method was nonrecurring in spite of a short sentence for many years.However, Yasuo Fujii (Takakazu Seki Mathematics Research Institute of Yokkaichi University) succeeds in decoding in May, 2009.It is the next equation.
The third method The studies of three formulas of Murase progress by the third method (already decoded).
If we apply Horner's method to an expression (1.2) in case of solution   2 of the calculation of Horner's method in Example 2.1 (Ref.[4]).

Expansions Recurrence Formula of Murase-Newton
In 2009, we found the extension of Newton-Raphson's method from the Murase's three formulas and a hint of Tamotsu Tsuchikura, and called it the Murase-Newton's method or the Tsuchikura-Horiguchi's method.We obtained the extension of Newton-Raphson's method as follows.
Let q x t = where q is a real number that is not 0. We define the function ( ) Applying the Newton-Raphson's method to ( ) g t and express it again in ( ) k f x , we have the next defini- tion.Definition 3.1.For equation ( ) 0 f x = , we call the next recurrence formula the Murase-Newton's method or the Tsuchikura-Horiguchi's method (2009) where q is a real number that is not 0.

( ) ( )
( ) here, if q = 1, then the Formula (3.2) becomes Newton-Raphson's method.Furthermore, we call the next formula general recurrence formula of the Murase-Newton's method or general recurrence formula of the Tsuchikura-Horiguchi's method.Here q and λ are real numbers that are not 0.
The Formula (3.3) switches in various recurrence formula by q, λ, r, i.In particular, if , 1, 1 q r q i λ = = − = , then (3.3) becomes Tsuchikura-Horiguchi's method (Ref.[5]).Let ( ) be a real number.The j-th term of polynomial of n-th degree ( ) with a formula including a real variable m or constant, and denote such ( ) ( ) x .We can associate ( ) ( ) This is equal to Formula (2.9) in Section 2. Definition 3.3.The formula is called general recurrence formula of the extension of Murase-Newton's method or general recurrence formula of the extension of Tsuchikura-Horiguchi's method concerning of n-th degree polynomial equation.

On Relations between General Recurrence Formula of the Extension of Tsuchikura-Horiguchi's Method (the Extension of Murase-Newton's Method) and Horner's Method
We easily explain by the next fifth-degree equation.Here

Horner's Method for an Expression (4.1)
Let α be a real number.If we apply the Horner's method to polynomial ( ) f x , then we obtain the calculation in Table 3. Table 3. Synthetic division for an expression (4.1).( ) ( ) corresponds to the ( ) Proof.We should define the formulas From Theorem 4.1, we obtain the next theorem.Theorem 4.2.There exists the rational recurrence formula 1 k x + obtained from Formula (4.1) so that the denominator equals to (4.2).Similarly, there exists the rational recurrence formula 2 , , so that the denominator equals to x of (4.5), respectively.Proof.We should choose the formulas , , Furthermore, we obtain the next theorem by a simple calculation.Theorem 4.3.The recurrence Formula (4.6) obtained from Formula (4.1) is equal to general recurrence Formula (4.10) of the extension of Tsuchikura-Horiguchi's method of x of Formula (4.1).
Because it is a simple matter, we give only theorems without proof in the following.( ) ( ) ( ) ( ) ( ) ( ) ( )  In this case, Formula (4.13) becomes the next formula.

2 )
This has three solutions of real number 2, 6 2 15 ± .Murase derived two following recurrence formulas and deformed equation from(1 Therefore, these changes correspond to the second line × −1 of the Table the indication of calculating formula of the Horner's method, and it becomes ( ) ( ) real number in m.For an understanding of the notation ( ) ( ) if we take m in 33/4, 34/4, 35/4 and x in 2, then ( ) m f x ′ becomes 10, 12, 14, respectively.These correspond to the second line × −1 of the

Theorem 4 . 5 .
15) corresponds to the second line of the calculation of Horner's method.Similarly, if x k = α, , then the denominator of recurrence Formula (4.15) corresponds to the third, forth, and fifth line of the calculation of Horner's method, respectively.The recurrence Formula (4.15) ( ) 16) of the extension of Tsuchikura-Horiguchi's method of (4.14).

Table 2 .
(1)thetic division for an expression (1.2).We obtain the next theorem from the Murase's three formulas and Table2of Horner's method.Theorem 2.2.(1)We expand the first, second, third method of Murase, and obtain the next recurrence formulas where m is a real number.

Table 2
of the calculation of Horner's method in Example 2.1.

In the Case of Special Fifth-Degree Equations (4.14) of Murase's Type
We transform the fifth-degree Equations (4.14), and obtain the next four recurrence formulas.