The Convergences Comparison between the Halley ’ s Method and Its Extended One Based on Formulas Derivation and Numerical Calculations

The purpose of this paper is that we give an extension of Halley’s method (Section 2), and the formulas to compare the convergences of the Halley’s method and extended one (Section 3). For extension of Halley’s method we give definition of function by variable transformation in Section 1. In Section 4 we do the numerical calculations of Halley’s method and extended one for elementary functions, compare these convergences, and confirm the theory. Under certain conditions we can confirm that the extended Halley’s method has better convergence or better approximation than Halley’s method.


Introduction
In 1673, Yoshimasu Murase [1] 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 defor- mation.We find that the three formulas lead to a Horner's method (Horiguchi,[2]) and an extension of Newton's method (Horiguchi,[3]).This shows originality of Wasan (mathematics developed in Japan) in the Edo era (1603-1868).We do research similar to Horiguchi, [3] against the Halley's method.We give function ( ) from ( ) f x for extension of Halley's method.
From now on, let x be a real number, and a function ( ) ( ) times differentiable if necessary, and ( ) ( ) i f x continuous.Definition 1.1.Let q x t = where q is a real number that is not 0. We define the function g(t) such as ( ) ( ) ( ) Because ( ) ( ) q g x f x = , the graph of ( ) g x extends or contracts by q x t = in the x -axis, without changing the height of ( ) y f x = .Expansion and contraction come to object in 1 x < and 1 x > .Theorem 1.2.The formulas 1 , give the convex upward (the convex downward resp.) at the point q x of graph of ( ) Proof.It is proved by the next calculations. ( ) ( )

Halley's Method and Extension of Halley's Method
Definition 2.1.The recurrence formula to approximate a root of the equation is called Halley's method 1 .
Halley's method is obtained by improving the Newton's method (11) (Ref.[5]). ( They are methods of giving the initial value 0 x , calculating 1 2 , , x x  one after another, and to determine for a root. From now on we omit the notation ( )  in recurrence formulas.Applying the Halley's method to ( ) If we express this by formula (1) in ( ) k f x , then we get the next definition.
Definition 2.2.Let α be a root of the equation ( ) 0 13) is the recurrence formula to approximate q α .We call this the q -th power of the extension of Halley's method (EH-method).
Calculation formula of q -th power of EH-method is this.( ) ( ) ( ) m ≥ ) multiple root, then it becomes the following linearly convergence.
( ) Proof.There is a brief proof of (15) in wikipedia [4].Therefore we go to the proof of (16).
We merely sketch From these formulas, we obtain the following linearly convergence.x α →∞ = , and q , r an arbitrary real constant number that is not 0, respectively.In this case following formula holds for large enough integer n .
( ) Proof.Applying L'Hospital's rule to ( ) ( ) x sufficiently close to ( ) 0 α ≠ ，then q -th power of EH-method (Extended Halley's method) becomes the third-order convergence of the following formula.
m ≥ ) multiple root, then it will be linearly convergence of the fol- lowing formula.
( ) Proof.If α is a simple root for ( ) In this case Halley's method for ( ) g t becomes the third-order convergence of the following formula.
Therefore we obtain By lemma 3.2, we get ( ) ( ) Therefore, formula (25) becomes By changing the independent variable x of the functions , , f f f ′ ′′ ′′′ and x in numerator to α , we obtain (21).
In case that q α is m multiple root, by ( 16), ( 1) and ( 20) we obtain ( ) be a simple root of ( ) 0 f x = .Then a necessary and suf- ficient condition for the convergence to α of q -th power of EH-method is equal to or faster than Halley's method is that q satisfies the following conditions.
Proof.Compare the coefficient of ( ) of the third-order convergence of q -th power of EH-method and that in the case of Halley's method.Then the necessary and sufficient condition is equivalent to the next formula.
The formula (29) is obtained from this.□ We transform the equation ( ) That is, two equations have the same root.r -th power of EH-method for ( ) and if ( ) We get the following by comparing the coefficient of ( ) sary and sufficient condition for the convergence to α of q -th power of EH-method (Extended Halley's method) (13) of ( ) f x to be equal to or faster than that r -th power of EH-method (34) of ( ) h x is that the real numbers q and r satisfy the following condition (36).
Furthermore, we transform the inequality.

Convergence Comparisons by the Numerical Calculations of Halley's Method and Extensions of Halley's Method
We perform numerical calculations by the calculation formula (14) in the standard format in Excel 2013 of Microsoft.We perform numerical calculations for various equations such as n -th order equations ( 2 n ≥ ), equations of trigonometric, exponential, logarithmic function, respectively.
In the examples of the followings, there are cases where some numerical calculations do not fit in with the inequality (30) a little.Those are probably due to the formula (21) the approximate formula, choosing the initial value 0 x , and the accuracy of using the standard format in Excel is insufficient.However, the results to fit the theories generally have been obtained.
Example 4.1.A quadratic equation The roots of (49 We choose real numbers q and initial values 0 x such as Table 1, Table 2, and do numerical computations.We explain how to read Table 1.The first column represents the initial value 0 x and the absolute error, and the first row represents the real num- ber q of q k x .
Two numbers 1 and 1.11022E−16 of intersection of two row and two column mean the following.
Number 1 indicates the number of iterations that Halley's method ( ) x , then we evaluate the convergences by the absolute errors.In the Table 1, Table 2, all q -th power of EH-method (Extension of Halley's method) converge in root 1 at iteration number 1 k = .But, for the same initial value 0 x , each column of EH-method ( ) has the absolute errors (at least one) that are equal to or smaller than Halley's method ( ) in the ranges of (50).
The results are Table 3.The range of q which satisfies (51) becomes 0.041 Because the root of (52) is 2, the condition (30) becomes ( ) ( ) We choose real numbers q and initial values 0 x such as   x , each column of EH-method ( ) has the absolute errors (at least one) that are equal to or smaller than Halley's method ( ) in the ranges of (53).
Example 4.3.A cubic equation ( ) ( )( )( ) In case of the root 1, the condition (30) becomes ( )( ) We choose real numbers q and initial values 0 x such as Table 6, Table 7, and do numerical computations.Each initial value 0 x , iteration number of EH-method ( ) 1 q ≠ and Halley's method ( ) are the same.But, for the same initial value 0 x , each column of EH-method ( ) has the absolute errors (at least one) that are smaller than Halley's method ( ) in the ranges of (55).
( ) sin 0 The roots of (56) are If we take the root in ( ) We do numerical computations for the real numbers q and initial values 0 x in Ta- ble 8, Table 9.All iteration numbers in Table 8 are 1.But, for the same initial value 0 x , each column of EH-method ( ) has the absolute errors (at least one) that are smaller than Halley's method ( ) number of iterations of EH-methods ( ) are small than Halley's method.( ) tan 0 The roots of (59) are If we take the root in ( ) , then (60) becomes ( ) Table 10 gives numerical computations.In case of 0 3.142 x = , EH-methods ( ) have better approximate degrees than Halley's method ( )  The root of (62) is 1.The condition (30) becomes ( )( ) 0 1 13 2 13.14142 13, 1 1.14142.
Table 11 and Table 12 give the numerical values to almost adapt to Theorem 3.4.
Theorem 1.4.A necessary and sufficient condition for

Theorem 3 . 3 .
20) is obtained.□ Let the condition be the same as Theorem 3.1.If k root.Then a neces-

- verge to the root 1 . 1 .
11022E−16 indicates the absolute error |the value 1 1 x = of the convergence of the numerical calculation-root 1|.If two iteration numbers are the same for the same initial value 0 Example 4.6.

Table 4
, Table5, and do numerical computations.All iteration numbers are 2 or 3. But, for the same initial value