_{1}

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.

In 1673, Yoshimasu Murase [

From now on, let

Definition 1.1. Let

Because

Theorem 1.2. The formulas

give the convex upward (the convex downward resp.) at the point

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

These become the curvature

Proof. Formula (6) is obtained by substituting the formulas (4) and (5) for

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

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. [

They are methods of giving the initial value

From now on we omit the notation

If we express this by formula (1) in

Definition 2.2. Let α be a root of the equation

Here, if

Calculation formula of

Theorem 3.1. Let

If

Proof. There is a brief proof of (15) in wikipedia [

We merely sketch

From these formulas, we obtain the following linearly convergence.

Lemma 3.2. In the sequence

Proof. Applying L’Hospital’s rule to

□

Theorem 3.3. Let the condition be the same as Theorem 3.1. If

If

Proof. If

Here

Therefore we obtain

By lemma 3.2, we get

Therefore, formula (25) becomes

By changing the independent variable

In case that

□

Theorem 3.4. Let

That is

Proof. Compare the coefficient of

The formula (29) is obtained from this. □

Corollary 3.5. (1) If

(2) If

We transform the equation

and if

We get the following by comparing the coefficient of

Proposition 3.6. Let

Theorem 3.7. Let

which distinguishes the convex-concave of the curve

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

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

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

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

Example 4.1. A quadratic equation

The roots of (49) are

We choose real numbers

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

We confirm Theorem 3.7. Because

The results are

Example 4.2. A cubic equation

Because the root of (52) is 2, the condition (30) becomes

We choose real numbers

q x_{0} | 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 |

q x_{0} | 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 |

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 |

q x_{0} | −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 |

q x_{0} | 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 |

Example 4.3. A cubic equation

In case of the root 1, the condition (30) becomes

We choose real numbers

Example 4.4.

The roots of (56) are

If we take the root in

We do numerical computations for the real numbers

q x_{0} | −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 |

q x_{0} | 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 |

q x_{0} | −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

If we take the root in

Example 4.6.

The root of (62) is 1. The condition (30) becomes

Example 4.7.

The root of (64) is 1. The condition (30) becomes

q x_{0} | 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 |

q x_{0} | 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 |

q x_{0} | 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 |

q x_{0} | 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 |

q x_{0} | 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 |

q x_{0} | 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 |

Dr. Hideko Nagasaka (Nihon University former professor) taught me numerical computations. I am deeply grateful to her.

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