The Formulas to Compare the Convergences of Newton ’ s Method and the Extended Newton ’ s Method ( Tsuchikura-Horiguchi Method ) and the Numerical Calculations

This paper gives the extension of Newton’s method, and a variety of formulas to compare the convergences for the extension of Newton’s method (Section 4). Section 5 gives the numerical calculations. Section 1 introduces the three formulas obtained from the cubic equation of a hearth by Murase (Ref. [1]). We find that Murase’s three formulas lead to a Horner’s method (Ref. [2]) and extension of a Newton’s method (2009) at the same time. This shows originality of Wasan (mathematics developed in Japan) in the Edo era (1603-1868). Suzuki (Ref. [3]) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 2 gives the relations between Newton’s method, Horner’s method and Murase’s three formulas. Section 3 gives a new function defined such as ( ) ( ) ( ) q y g t f t f x 1 : = = = .


Murase's Three Formulas from the Cubic Equation of a Hearth
We write this paper from two kinds of recurrence formulas of the square 2 k x and the deformation of a cubic equation written in Murase's book (Ref.[1]), and a hint of Tsuchikura (Ref.[4]).It is enough for readers to know these three formulas.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 such 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.This has three solutions of real number 2, 6 2 15 ± .Murase derived two following recurrence formulas (1.3), (1.4) and deformed equation (1.5) from (1.2).The first method: ( ) 48 0,1, 2, .14 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 third method was nonrecurring in spite of a short sentence for many years.However, Yasuo Fujii (Seki Kowa Institute Mathematics 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 have been decoded.Furthermore we obtain the next recurrence formula from (1.5).

Relations between Newton's Method, Horner's Method and the Murase's Three Formulas
Throughout this paper, function f(x) be i ( 1 ≥ ) times differentiable if necessary, and f (i) (x) continuous.We start with the definition of Newton's method.Next Newton's method is explained in a book of the standard numerical computation (Ref.[5]).The recurrence formula to approximate a root of the equation f(x) = 0 ( ) ( ) ( ) is called Newton's method or Newton-Raphson's method.Newton's method is a method of giving the initial value x 0 , calculating 1 2 , , x x  one after another, and to determine for a root.
The quadratic convergence and the linearly convergence of the Newton's method are known as followings.
Let α be a simple root for f(x) = 0, i.e., ( ) 0 f α ′ ≠ .Then Newton's method to the quadratic convergence of the following formula.
( ) ( ) ( ) If α is m ( 2 ≥ ) multiple root, then it will become the linearly convergence of the following formula.
( ) Remark.Concerning choosing the initial value x 0 , the number of iterations until it converges on a root changes.Moreover, it may not be converged on a root.
Example 2.2.Applying the Horner's method to Murase's equation f(x) = x 3 − 14x 2 + 48 = 0 for root 2, we get Table 1.Here, number −14, −12, −10 of the second column corresponds to the denominator 14,14 , 3), (1.4), (1.6), respectively.Therefore, from the Table 1, we find that the Murase's formulas (1.3), (1.4), and (1.6) lead to a Horner's method.Furthermore, please read Ref. [2] if you want to know this deeply.x t = where q is a real number that is not 0. We define the function g(t) such as Because g(x q ) = f(x), the graph of g(x) is extended and contracted by x q = t in the x-axis, without changing the height of f(x).Expansion and contraction come to object in 1 x < and 1 x > .Lemma 3.2.

( )
( ) ( ) The curvature of the curve y = g(x) at the point x q is this.

Extension of Newton's Method and the Convergences
In 2009, we found the extension of Newton's method from the Murase's three formulas as follows.Applying the Newton's method to g(t), we have , 1 This means the intersection 1 with the t(x)-axis of the tangent in the point Returning to the variable x by x q = t, we get an extension of Newton's method below.
( ) ( ) ( ) If α is m ( 2 ≥ ) multiple root, then it will become linearly convergence of the following formula.
( ) α is a simple root for g(t) = 0, then Newton's method for g(t) becomes the quadratic convergence of the following formula.

Varieties of Formulas to Compare the Convergences for the Extension of Newton's Method (Tsuchikura-Horiguchi's Method)
We deform the equation f(x) = 0 to h(x) = 0.That is, two equations have the same root.r-th power of TH-method for h(x) is ( ) ( ) and if α (≠0) is a simple root, then it becomes quadratic convergence ( ) ( ) ( ) We get the following proposition by comparing the coefficients of ( ) of formula (4.5) and (4.12).
Proposition 4.6.Let the equation h(x) = 0 be deformed from f(x) = 0. Let ( ) ( ) 0 f h α α = = , and α(≠0) a simple root.Then the necessary and sufficient condition for the convergence to α of q-th power of TH-method of f(x) to be equal to or faster than that r-th power of TH-method of h(x) is that the real numbers q and r satisfy the following condition.

( ) ( ) ( ) ( )
Theorem 4.7.Let α (≠0) be a simple root of f(x) = 0, and Then a necessary and sufficient condition for the convergence to α of q-th power of TH-method is equal to or faster than that Newton's method is that q satisfies the following conditions.

( ) ( )
Equal signs are the case of q = 1 and ( ) ( ) Proof.Compare the coefficient of ( ) of the quadratic convergence (4.5) of q-th power of TH-method and that (2.2) of Newton's method.Then the necessary and sufficient condition is equivalent to The formula (4.14) is obtained from (4.16).Theorem 4.8.Let α (≠0) be a simple root of f(x) = 0, and e. the graph of f(x) is nearly the straight line in the neighborhood of the point α.).In this case (4.17) holds.
( ) ( ) ( ) This is equivalent to the convergence to α of Newton's method equals to or faster than that q-th power of THmethod.
Proof.By deforming the formula to ( ) We get the conclusion by this.
The following are the results related to the convex-concave of curve and the formulas for comparing convergences of TH-method.
Lemma 4.9.Let 0 x ≠ and Then a necessary and sufficient condition for ( ) ( ) f x ′′ become the same sign (opposite sign resp.).We get the next theorem from Lemma 4.9, directly.Theorem 4.10.Let α(≠0) be a simple root of f(x) = 0, and We divide the Formula (4.14) of Theorem 4.7 into positive and negative range as follows.
( ) ( ) If q satisfies the condition (4.23) ((4.22) resp.)), then the convex-concave of curve of g(x) in the neighborhood of ( )( ) and the f(x) in the neighborhood of ( )( ) 0 f α = are the same (opposite resp.).Theorem 4.11.Let the conditions be the same as the above theorem.We give the following inequality.
Then the convergence to α of q-th power of TH-method is equal to or faster than Newton's method equivalent to the formula (4.24).Proof.By the formula The following are the results related to the curvature and the formulas for comparing the convergences of THmethod.
Theorem 4.13.Let α (≠0) be a simple root of f(x) = 0, and Then the convergence to α of q-th power of TH-method is equal to or faster than that Newton's method is equivalent to that (4.27) holds.
Proof.The formula ( ) hold, then the convergence to α of q-th power of TH-method is equal to or faster than that Newton's method.

Convergence Comparisons of the Numerical Calculations of Newton's Method and Expansion of Newton's Method (Tsuchikura-Horiguchi's Method)
We use formula (4.2') for the numerical calculations of q-th power of TH-method for various equations such as n-th order equations ( 2 n ≥ ), equations of trigonometric, exponential, logarithmic function.We perform numerical calculations in the standard format in Excel of Microsoft.
Example 5.1.Numerical calculation of the p-th root.
Let A be a real number, and p a natural number.The equation for p-th root is this.
( ) ( ) (1) The application of the formula (4.15) 1 p A is p-th root of (5.1.1),and we get In this case, formula (4.15) becomes ( ) Especially p-th power of TH-method for ( ) Therefore, it converges to the root once for any initial value.Hence the number of iterations of formula (5.1.4)is less than that of the recurrence formula other.
(2) Speeds of convergences.The roots of ( ) 2 4 0 The interval of q of (5.1.3)is 1 3 q ≤ ≤ .In the following, we examine the speed of convergence of q-th power of TH-method in case of α = 2.The results of the calculations are Table 2.We explain how to read this.
The first column represents the initial value x 0 and the absolute error ( ) − , and the first row represents the real number q of q k x .Two numbers 3 and 1.36646E−11 of intersection of two rows and two columns mean the following.Number 3 indicates the number of iterations that 0.5-th power of TH-method ( ) ( ) ( ) 2 0.5 0.5 0.5 1 0.5 0.5 1 0 4 0.5 0.5 1.9 2 Table 2. Calculations of q-th power of TH-method for f(x) = x 2 − 4 = 0. 2-th power of TH-methods converges to 2 in number of iterations 1; other TH-methods converge to that in three times.In case of x 0 = 1.9, 1.95, absolute errors of q = 1.2, 1.5, 2.5, 3 are smaller than that q = 1 (Newton's method).Therefore degree of approximations of q = 1.2, 1.5, 2.5, 3 is better than that q = 1.Furthermore, absolute errors of q = 0.5, 3.5 are larger than that q = 1.Thus, these numerical calculations are compatible with the theory of Theorem 4.7.
(3) The application of the formula (4.27) of Theorem 4.13 for Indeed, by calculating the left and right sides of (5.1.6)for q in the Table 3 we get the numbers there.g(x) becomes a straight line x − 4 in case of q = 2, and the curvature is 0. Therefore, the square of TH-method converges to root 2 in the number of iterations 1.For each q, the second and third columns are calculations of formula (5.1.6).The fourth column is the calculations of ( ) 2 µ .Columns 5 and 6 are the calculation of the left-hand side and the right-hand side of the inequality (4.30), respectively.For each q in 1 3 q ≤ ≤ , the numbers of the second column and third column satisfy the condition (5.1.6).

Table 4. Calculations of TH-method for
Right-hand side of (5.2.4) ( ) µ α , (4.30) for α = 2. (2B) We examine the speed of convergence of q-th power of TH-method in 2 9.3 q − ≤ ≤ .The results of the calculations are Table 6.In case of x 0 = 2.1, numerical calculations of q-th power of TH-method are compatible with the theory of Theorem 4.7.
In case of x 0 = 1.9, numerical calculations of q-th power of TH-method are compatible with Theorem 4.7.
(3) Formula (4.27) of Theorem 4.13 becomes ( ) By calculating the left and right sides of (5.3.3) for q in Table 9 formula (5.3.3)holds except for 0.5 and 2.5.
Right-hand side of ( , , x x x , and a root is α = 1.Graph of f(x) is Figure 3. Graph is the convex downward and monotonic decreases in 1 0 x − ≤ < , the convex upward and monotonic decreases in 0 1 x < ≤ , and point (0,3) is a point of inflection.
Right-hand side of (  , , x x x , and a root of (5.5.1) is α ≒ 2.055967397.Graph of f(x) is  becomes minimum at x = 0, which is parallel to the x-axis in the neighborhood.Next it increases and becomes maximum at x = 1.6.Further, it decreases monotonically from here, and intersects with root α.The graph changes intensely in this way in _ 1 < x < 2.5.
( ) ( ) (The value of formula (5.5.2) for q = 16.018 is 1.999993923.) (2) For q = _ 1, 1, 3, 6, 9, 12, 15, 17, we calculate q-th power of TH-method.The results are Table 12.① For x 0 = 1.85, number of iterations of Newton's method and 3-th power of TH-method are the same 5.But absolute error of Newton's method is slightly smaller than that 3-th power of TH-method.The theory compatible with all other cases.② In particular for the initial value is x 0 = 1.5, the number of iterations of the 9-th power of TH-method is 4, which is extremely small than 54 times of the Newton's method.Therefore, we examine the state of convergence of the 9-th power of TH-method.
Converting f(x) by 1 9 x t = , following formula is obtained.
( ) ( ) For the initial value is 1.5 9 , we give in Table 13 the calculations of 9-th power of TH-method to converge to 656.3659005 (=2.055967397 9 ) and the tangents.Then we give the graphs of Figure 5 g(x) and the changes of the tangents.
Straight line 1, 2 and 3 in Figure 5 indicates the tangent to the number of iterations k = 1, 2, 3, respectively.Point ( ) ( ) It becomes convex downward in x < 1.2, minimum at x = 0, and parallel to the x-axis in the neighborhood of x = 0.It becomes convex upward in 1.2 < x, maximum at x = 1.6.Therefore, choosing to 1.5 initial value for Newton's method, x k vibrate, and the number of iterations increase.Graph g(t) (g(x)) becomes minimum at t(x) = 0, but parallel parts to the t(x)-axis do not exist in the neighborhood of this point.Further it becomes convex upward in 9 2 t < , convex downward in 9 2 t < , inter- sects at 656.3659005 t = with t-axis, and close to the shape of a straight line in the neighborhood.Therefore, x become a monotonically increasing sequence, and the number of iterations is reduced.
) sin 0 ( ) of Theorem 4.8 holds, convergence of Newton's method of q = 1 is the fastest in other q-th power of TH-method.For α = π, q = ±1, ±2, ±3, we calculate q-th power of TH-method.The results are Table 15.
For x 0 = 0.73, 0.76, q-th power of TH-method has better approximate degree than Newton's method in the range of (5.8.2).

Figure 4 .
The graph

Table 1 .
Horner's method for Murase's equation.We expand the first, second, third method of Murase, and obtain the next recurrence formula where m is a real number.

Table 13 .
Calculations of 9-th power of TH-method and tangents.