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

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.

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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.

1. Introduction

All the references are written in Japanese. We wrote this paper from two kinds of recurrence formulas of the square 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.

Figure 1. Hearth.

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.

(1.1)

that is

(1.2)

This has three solutions of real number 2,.

Murase derived two following recurrence formulas and deformed equation from (1.2).

The first method

(1.3)

Using on an abacus, Murase calculates to x_{0}_{ }= 0 (initial value), x_{1}_{ }= 1.85, x_{2}_{ }= 1.97, x_{3} = 1.9936, and decides a solution with 2.

The second method

(1.4)

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

(1.5)

The studies of three formulas of Murase progress by the third method (already decoded).

2. Horner’s Method

Let

(2.1)

be a n-th degree polynomial where _{ }are real numbers.

The Horner’s method is an algorithm to calculate. Dividing by, we obtain the next formula.

(2.2)

Therefore, becomes R_{1}. Next, dividing by, we obtain the next formula.

(2.3)

Here, R_{2 }is the differential coefficient.

Furthermore, dividing by, we obtain the next formula.

(2.4)

Comparing the coefficients of formula (2.1) and (2.2), (2.2) and (2.3), (2.3) and (2.4) respectively, we obtain the next calculating formula of b_{i}, R_{1}, c_{i}, R_{2 }, d_{i}, R_{3}.

(2.5)

(2.6)

(2.7)

(2.8)

Similarly, we can continue calculating.

The indication of calculating formula by synthetic division is next Table 1.

Table 1. Synthetic division for an expression (2.1) (Ref. [3] ).

Example 2.1. If we apply Horner’s method to an expression (1.2) in case of solution, then it is calculated in Table 2.

Table 2. Synthetic division for an expression (1.2).

We obtain the next theorem from the Murase’s three formulas and Table 2 of 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.

(2.9)

(2) 14, of denominator of formula (1.3), (1.4) respectively, and of formula (1.5) change 14→12→10 if. Therefore, these changes correspond to the second line × −1 of the Table 2 of the calculation of Horner’s method in Example 2.1 (Ref. [4] ).

3. 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 where q is a real number that is not 0. We define the function such as

(3.1)

Applying the Newton-Raphson’s method to and express it again in, we have the next definition.

Definition 3.1. For equation, 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.

(3.2)

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.

(3.3)

The Formula (3.3) switches in various recurrence formula by q, λ, r, i. In particular, if, 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

(3.4)

is. The j-th term of i-th derived function of is this.

(3.5)

We replace the coefficient with a formula including a real variable m or constant, and denote such in. We can associate with the line of the indication of calculating formula of the Horner’s method, and it becomes when we substitute a certain real number in m. For an understanding of the notation, see in the next example and Formulas (4.2)-(4.5) in Section 4.

Example 3.2. In Murase’s formula, let If we take m in 3/2, then becomes. Furthermore if we take m in 33/4, 34/4, 35/4 and x in 2, then becomes 10, 12, 14, respectively. These correspond to the second line × −1 of the Table 2 of the calculation of Horner’s method in Example 2.1.

We make next recurrence formula.

(3.6)

This is equal to Formula (2.9) in Section 2.

Definition 3.3. The formula

(3.7)

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.

4. 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 _{ }are real numbers.

(4.1)

4.1. Horner’s Method for an Expression (4.1)

Let α be a real number. If we apply the Horner’s method to polynomial, then we obtain the calculation in Table 3.

Table 3. Synthetic division for an expression (4.1).

4.2. In the Case of General Fifth-Degree Equation (4.1)

Theorem 4.1. There exists so that it equals to if. Furthermore, if and, then corresponds to the -th line × i! of Table 3 of Horner’s method, respectively.

Proof. We should define the formulas, , , as follows.

(4.2)

(4.3)

(4.4)

(4.5)

From Theorem 4.1, we obtain the next theorem.

Theorem 4.2. There exists the rational recurrence formula obtained from Formula (4.1) so that the denominator equals to (4.2). Similarly, there exists the rational recurrence formula so that the denominator equals to of (4.3), of (4.4), of (4.5), respectively.

Proof. We should choose the formulas, as follows.

(4.6)

(4.7)

(4.8)

(4.9)

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).

(4.10)

Similarly recurrence Formulas (4.7)-(4.9) is equal to general recurrence Formulas (4.11)-(4.13) of the extension of Tsuchikura-Horiguchi’s method of x^{2}, x^{3}, x^{4} of Formula (4.1), respectively.

(4.11)

(4.12)

(4.13)

4.3. In the Case of Special Fifth-Degree Equations (4.14) of Murase’s Type

(4.14)

We transform the fifth-degree Equations (4.14), and obtain the next four recurrence formulas.

(4.15)

Because it is a simple matter, we give only theorems without proof in the following.

Theorem 4.4. If and, then the denominator of the recurrence Formula (4.15) corresponds to the second line of the calculation of Horner’s method. Similarly, if x_{k}_{ }= α, , , and, then the denominator of recurrence Formula (4.15) corresponds to the third, forth, and fifth line of the calculation of Horner’s method, respectively.

Theorem 4.5. The recurrence Formula (4.15) is equal to the next general recurrence formula of (4.16) of the extension of Tsuchikura-Horiguchi’s method of (4.14).

(4.16)

Corollary 4.6. If i = 2, then (4.14) becomes the next formula.

(4.14)

In this case, Formula (4.13) becomes the next formula.

(4.17)

Formula (4.17) is equal to (4.16) if i = 2. In the case of i = 3, 4, 5, a similar thing holds, respectively.

Acknowledgements

Dr. Tamotsu Tsuchikura gave a hint to me. I am deeply grateful to him.

NOTES

^{*}Tsuchikura is Tamotsu Tsuchikura, the professor emeritus of Tohoku University.

Conflicts of Interest

The authors declare no conflicts of interest.

[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) |

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