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.
Recurrence Formula Newton’s Method Halley’s Method Extension of Halley’s Method Third-Order Convergence1. 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 and the deformation. 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 defined from for extension of Halley’s method.
From now on, let be a real number, and a function times differentiable if necessary, and continuous.
Definition 1.1. Let where is a real number that is not 0. We define the function g(t) such as
Because, the graph of extends or contracts by in the -axis, without changing the height of. Expansion and contraction come to object in and.
Theorem 1.2. The formulas
give the convex upward (the convex downward resp.) at the point of graph of.
Proof. It is proved by the next calculations.
□
From the formulas (4), (5), we obtain the next theorem.
Theorem 1.3. The curvature of the cure at the point is formulas (6) and (7).
These become the curvature of if in particular.
Proof. Formula (6) is obtained by substituting the formulas (4) and (5) for in the curvature. □
Theorem 1.4. A necessary and sufficient condition for
is that formula (9) holds.
Proof. Formula (9) is obtained from (8). □
Proposition 1.5. If is a simple root (multiple root resp.) of, then becomes the simple root (multiple root resp.) of.
2. 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 method1.
Halley’s method is obtained by improving the Newton’s method (11) (Ref. [5] ).
They are methods of giving the initial value, calculating 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, we get
If we express this by formula (1) in, then we get the next definition.
Definition 2.2. Let α be a root of the equation. (13) is the recurrence formula to approximate. We call this the -th power of the extension of Halley’s method (EH-method).
Here, if then the formula (13) becomes Halley’s method.
Calculation formula of -th power of EH-method is this.
3. Formulas to Compare the Convergences for Extensions of Halley’s Method
Theorem 3.1. Let be a simple root for, i.e.,. Then Halley’s method becomes the following third-order convergence.
If is () 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 with. Since is represented as
is as follows, respectively.
From these formulas, we obtain the following linearly convergence.
Lemma 3.2. In the sequence, let, and, an arbitrary real constant number that is not 0, respectively. In this case following formula holds for large enough integer.
Proof. Applying L’Hospital’s rule to, formula (20) is obtained.
□
Theorem 3.3. Let the condition be the same as Theorem 3.1. If sufficiently close to，then -th power of EH-method (Extended Halley’s method) becomes the third-order convergence of the following formula.
If is () multiple root, then it will be linearly convergence of the following formula.
Proof. If is a simple root for, then also becomes a simple root for. In this case Halley’s method for becomes the third-order convergence of the following formula.
Here become the followings.
Therefore we obtain
By lemma 3.2, we get
Therefore, formula (25) becomes
By changing the independent variable of the functions and in numerator to, we obtain (21).
In case that is multiple root, by (16), (1) and (20) we obtain
□
Theorem 3.4. Let be a simple root of. Then a necessary and sufficient condition for the convergence to of -th power of EH-method is equal to or faster than Halley’s method is that satisfies the following conditions.
That is
Proof. Compare the coefficient of of the third-order convergence of -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. □
Corollary 3.5. (1) If then (30) becomes
(2) If then (30) becomes
We transform the equation into. That is, two equations have the same root. -th power of EH-method for is
and if is a simple root, then it becomes the third-order convergence (35).
We get the following by comparing the coefficient of of formula (21) and (35).
Proposition 3.6. Let, and a simple root. Then a necessary and sufficient condition for the convergence to of -th power of EH-method (Extended Halley’s method) (13) of to be equal to or faster than that -th power of EH-method (34) of is that the real numbers and satisfy the following condition (36).
Theorem 3.7. Let be a simple root for, i.e.,. Inequality (29) is represented by the second derivative
which distinguishes the convex-concave of the curve. It shows the next complicated inequalities (38), (39).
Proof. We lead inequalities (38), (39) from (29). Let
Then inequality (29) becomes
We transform the formula B.
Therefore inequality (29) becomes (44).
Furthermore, we transform the inequality.
From (45), we get (38), (39) according to plus, minus number of respectively. □
Theorem 3.8. Let the condition be the same as the above Theorem. Inequality (29) is represented by the curvature
Those are the next complicated inequalities (47) and (48).
Proof. We get (47), (48) by dividing formula (38), (39) in, respectively.
4. 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 -th order equations (), 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, 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) are. Because, in case of, condition (33) becomes
We choose real numbers and initial values such as Table 1, Table 2, and do numerical computations. We explain how to read Table 1. The first column represents the initial value and the absolute error, and the first row represents the real number of.
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 to converge to the root 1. 1.11022E−16 indicates the absolute error |the value of the convergence of the numerical calculation―root 1|. If two iteration numbers are the same for the same initial value, then we evaluate the convergences by the absolute errors. In the Table 1, Table 2, all -th power of EH-method (Extension of Halley’s method) converge in root 1 at iteration number. But, for the same initial value, 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).
We confirm Theorem 3.7. Because in, inequality (39) is applied. In this case (39) becomes the following inequality.
The results are Table 3. The range of which satisfies (51) becomes.
Example 4.2. A cubic equation
Because the root of (52) is 2, the condition (30) becomes
We choose real numbers and initial values such as Table 4, Table 5, and do numerical computations. All iteration numbers are 2 or 3. But, for the same initial value
Calculations of (14) for root 1, −0.041 ≤ q ≤ 1
q x0
1
−0.041
−0.025
−0.01
0.1
0.2
0.4
0.6
0.8
0.9
0.999999993
k = 1
1
1
1
1
1
1
1
1
1
Absolute errors
1.11022E−16
0
0
0
0
0
0
2.22045E−16
1.11022E−16
2.22045E−16
1.000000003
1
1
1
1
1
1
1
1
1
1
Absolute errors
2.22045E−16
0
0
0
0
0
4.44089E−16
0
2.22045E−16
2.22045E−16
Calculations of (14) for root 1, 23 ≤ q ≤ 24.042
q x0
1
23.1
23.3
23.5
23.7
23.9
24.042
24.1
0.999999992
1
1
1
1
1
1
1
1
Absolute errors
2.22045E−16
0
0
0
0
0
0
0
1.000000015
1
1
1
1
1
1
1
1
Absolute errors
2.22045E−16
0
0
0
0
0
0
0
Calculations of (51) for root 1, −0.041 ≤ q ≤ 1
q
(1 − q)f'(α)/α = q − 1
Left-hand side of g"(t)
g"(t) = q + 1
Right-hand side of g"(t)
−0.4
−1.4
−1.35
0.6
0.078571429
−0.3
−1.3
−1.241666667
0.7
0.296794872
−0.2
−1.2
−1.133333333
0.8
0.533333333
−0.1
−1.1
−1.025
0.9
0.793181818
−0.042
−1.042
−0.962166667
0.958
0.95721913
−0.041
−1.041
−0.961083333
0.959
0.960146254
0
−1
−0.916666667
1
1.083333333
0.1
−0.9
−0.808333333
1.1
1.413888889
0.2
−0.8
−0.7
1.2
1.8
0.3
−0.7
−0.591666667
1.3
2.26547619
0.4
−0.6
−0.483333333
1.4
2.85
0.5
−0.5
−0.375
1.5
3.625
0.6
−0.4
−0.266666667
1.6
4.733333333
0.7
−0.3
−0.158333333
1.7
6.508333333
0.8
−0.2
−0.05
1.8
9.95
0.9
−0.1
0.058333333
1.9
20.05833333
0.9999
−1E−04
0.166558333
1.9999
20000.16656
1
0
#DIV/0!
2
#DIV/0!
1.001
0.001
0.16775
2.001
−1999.83225
1.1
0.1
0.275
2.1
−19.725
1.2
0.2
0.383333333
2.2
−9.616666667
Calculations of (14) for root 2, −1.8875 ≤ q ≤ −1.462
q x0
−1.8875
−1.7
−1.6
−1.5
−1.462
1
1.999
2
2
2
2
2
2
Absolute errors
5.28466E−14
5.28466E−14
5.24025E−14
5.28466E−14
5.28466E−14
5.28466E−14
2.05
3
3
3
3
3
3
Absolute errors
4.26326E−14
4.26326E−14
4.35207E−14
4.30767E−14
4.35207E−14
4.39648E−14
2.1
3
3
3
3
3
3
Absolute errors
1.12594E−11
1.14033E−11
1.14744E−11
1.15419E−11
1.15663E−11
1.18154E−11
2.15
3
3
3
3
3
3
Absolute errors
2.92275E−10
2.97491E−10
3.00076E−10
3.02517E−10
3.03407E−10
3.12511E−10
Calculations of (14) for root 2, 1 ≤ q ≤ 1.4262
q x0
1
1.001
1.1
1.2
1.3
1.4262
1.999
2
2
2
2
2
2
Absolute errors
5.28466E−14
5.24025E−14
5.24025E−14
5.28466E−14
5.24025E−14
5.24025E−14
2.05
3
3
3
3
3
3
Absolute errors
4.39648E−14
4.39648E−14
4.35207E−14
4.35207E−14
4.35207E−14
4.30767E−14
2.1
3
3
3
3
3
3
Absolute errors
1.18154E−11
1.18154E−11
1.17693E−11
1.17186E−11
1.1664E−11
1.15885E−11
2.15
3
3
3
3
3
3
Absolute errors
3.12511E−10
3.12495E−10
3.10816E−10
3.08966E−10
3.06965E−10
3.04226E−10
, 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 and initial values such as Table 6, Table 7, and do numerical computations. Each initial value, iteration number of EH-method and Halley’s method are the same. But, for the same initial value, each column of EH-method has the absolute errors (at least one) that are smaller than Halley’s method in the ranges of (55).
Example 4.4.
The roots of (56) are The condition (30) becomes
If we take the root in, then (57) becomes
We do numerical computations for the real numbers and initial values in Table 8, Table 9. All iteration numbers in Table 8 are 1. But, for the same initial value, each column of EH-method has the absolute errors (at least one) that are smaller than Halley’s method in. In case of, number of iterations of EH-methods are small than Halley’s method.
Calculations of (14) for root 1, −0.1934 ≤ q ≤ 1
q x0
−0.1934
0.1
0.3
0.5
0.7
0.99
1
0.875
3
3
3
3
3
3
3
Absolute error
0
0
0
4.44089E−16
2.22045E−16
2.22045E−16
4.44089E−16
0.99999741
1
1
1
1
1
1
1
Absolute error
0
0
0
2.22045E−16
1.11022E−16
2.22045E−16
2.22045E−16
1.000001
1
1
1
1
1
1
1
Absolute error
4.44089E−16
0
0
4.44089E−16
2.22045E−16
2.22045E−16
4.44089E−16
1.1
3
3
3
3
3
3
3
Absolute error
1.11022E−15
0
3.33067E−16
0
1.11022E−16
2.22045E−16
3.33067E−16
Calculations of (14) for root 1, 35 ≤ q ≤ 36
q x0
1
34
35
35.2
35.4
35.6
35.8
36.1934
0.999999256
1
1
1
1
1
1
1
1
Absolute error
5.55112E−16
4.44089E−16
4.44089E−16
4.44089E−16
4.44089E−16
4.44089E−16
4.44089E−16
4.44089E−16
1.000001
1
1
1
1
1
1
1
1
Absolute error
4.44089E−16
2.22045E−16
2.22045E−16
2.22045E−16
2.22045E−16
2.22045E−16
2.22045E−16
2.22045E−16
Calculations of (14) for root π, −0.3455 ≤ q ≤ 0.458
q x0
−0.3455
−0.3
−0.1
0.1
0.3
0.458
1
3.1415
1
1
1
1
1
1
1
Absolute errors
4.04578E−10
4.04076E−10
3.95939E−10
4.21089E−10
4.12902E−10
4.11561E−10
4.10339E−10
3.141585
1
1
1
1
1
1
1
Absolute errors
4.10203E−10
4.10203E−10
4.10198E−10
4.10209E−10
4.10208E−10
4.10208E−10
4.10207E−10
3.1417
1
1
1
1
1
1
1
Absolute errors
4.18962E−10
4.19741E−10
4.32396E−10
4.06015E−10
3.93276E−10
4.08099E−10
4.1E−10
Example 4.5.
The roots of (59) are The condition (30) becomes
If we take the root in, then (60) becomes
Table 10 gives numerical computations. In case of, EH-methods have better approximate degrees than Halley’s method in.
Example 4.6.
The root of (62) is 1. The condition (30) becomes
Table 11 and Table 12 give the numerical values to almost adapt to Theorem 3.4.
Example 4.7.
The root of (64) is 1. The condition (30) becomes
Table 13 and Table 14 give the numerical values to almost adapt to Theorem 3.4.
Calculations of (14) for root π, 1 ≤ q ≤ 1.8043
q x0
0.8
1
1.2
1.4
1.6
1.8043
2
3.14005
2
2
1
1
1
1
2
4.5
6
5
4
4
3
4
5
Calculations of (14) for root π, 0.46 ≤ q ≤ 1
q x0
0.4
0.46
0.6
0.7
0.8
0.9
1
3.14
2
2
2
2
2
2
2
Absolute errors
4.10207E−10
4.10207E−10
4.10207E−10
4.10207E−10
4.10206E−10
4.10206E−10
4.10207E−10
3.142
1
1
1
1
1
1
1
Absolute errors
2.8809E−11
9.20344E−11
1.86697E−10
2.28724E−10
2.58712E−10
2.80826E−10
2.97554E−10
Calculations of (14) for root 1, −13.14142 ≤ q ≤ −13
q x0
1
−13.2
−13.14142
−13.1
−13.08
−13.05
−13.02
−13
0.999999998
1
1
1
1
1
1
1
1
Absolute errors
1.11022E−16
0
0
0
0
0
0
0
1.000000009
1
1
1
1
1
1
1
1
Absolute errors
1.11022E−16
0
0
0
0
0
0
2.22045E−16
Calculations of (14) for root 1, 1 ≤ q ≤ 1.14142
q x0
0.9
1
1.02
1.04
1.06
1.08
1.14142
1.15
0.99999996
1
1
1
1
1
1
1
1
Absolute errors
2.22045E−16
1.11022E−16
0
1.11022E−16
1.11022E−16
0
0
0
1.00000004
1
1
1
1
1
1
1
1
Absolute errors
2.22045E−16
1.11022E−16
0
0
1.11022E−16
1.11022E−16
0
1.11022E−16
Calculations of (14) for root 1, 1 ≤ q ≤ 1.2041684
q x0
0.9
1
1.04
1.08
1.12
1.16
1.2
1.2041684
0.96
3
3
3
3
3
3
3
3
Absolute errors
6.73195E−12
4.75153E−12
4.19242E−12
3.72524E−12
3.33122E−12
2.9966E−12
2.70994E−12
2.67797E−12
1.01
3
3
3
3
3
3
3
3
Absolute errors
2.22045E−16
1.11022E−16
1.11022E−16
1.11022E−16
0
1.11022E−16
0
1.11022E−16
Calculations of (14) for root 1, 10.795834 ≤ q ≤ 11
q x0
1
10
10.795834
10.9
10.94
10.98
11
12
0.9
4
3
3
3
3
3
3
4
1.15
4
4
4
4
4
4
4
5
Absolute errors
1.10389E−12
1.66533E−15
2.20934E−14
8.53762E−14
1.36779E−13
2.14606E−13
2.67009E−13
Acknowledgements
Dr. Hideko Nagasaka (Nihon University former professor) taught me numerical computations. I am deeply grateful to her.
Cite this paper
Horiguchi, S. (2016) The Convergences Comparison between the Halley’s Method and Its Extended One Based on Formulas Derivation and Numerical Cal- culations. Applied Mathematics, 7, 2394- 2410. http://dx.doi.org/10.4236/am.2016.718188
NOTESReferencesMurase, Y. (1673) Sanpoufutsudankai. In: Nishida, T., Ed., Kenseisha Co., Ltd., Tokyo. (In Japanese)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. https://doi.org/10.4236/am.2014.54074Horiguchi, S (2016) The Formulas to Compare the Convergences of Newton’s Method and the Extended Newton’s Method (Tsuchikura-Horiguchi Method) and the Numerical Calculations. Applied Mathematics, 7, 40-60. https://doi.org/10.4236/am.2016.71004http://en.wikipedia.org/wiki/Halley’s_methodNagasaka, H. (1980) Computer and Numerical Analysis (Japanese Title Keisanki to suuchikaiseki). Asakura Publishing Co., Ltd., Tokyo. (In Japanese)