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 .
We write this paper from two kinds of recurrence formulas of the square
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
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.
that is
This has three solutions of real number 2,
Murase derived two following recurrence formulas (1.3), (1.4) and deformed equation (1.5) from (1.2).
The first method:
Using on an abacus, Murase calculates to x0 = 0 (initial value), x1 = 1.85, x2 = 1.97, x3 = 1.9936, and decides a solution with 2.
The second method:
Here he calculates to x0 = 0, x1 = 1.85, x2 = 1.976, x3 = 1.9989, x4 = 1.9999907, and decides a solution with 2. Formula (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 (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).
Throughout this paper, function f(x) be i (
Next Newton’s method is explained in a book of the standard numerical computation (Ref. [
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 x0, calculating
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.,
If α is m (
Remark. Concerning choosing the initial value x0, the number of iterations until it converges on a root changes. Moreover, it may not be converged on a root.
Example 2.1. By the transformation of variable
It becomes the following formula if Newton’s method is applied to
This becomes the following formula by t = x2.
This is a middle formula of (1.4) and (1.6) exactly. That is, Murase’s formulas (1.3), (1.4), and (1.5) lead to extension of a Newton’s method (2009).
Example 2.2. Applying the Horner’s method to Murase’s equation f(x) = x3 − 14x2 + 48 = 0 for root 2, we get
2) | 1 | −14 | 0 | 48 |
---|---|---|---|---|
+) | 2 | −24 | −48 | |
1 | −12 | −24 | 0 | |
+) | 2 | −20 | ||
1 | −10 | −44 | ||
+) | 2 | |||
1 | −8 |
Proposition 2.3. We expand the first, second, third method of Murase, and obtain the next recurrence formula where m is a real number.
Definition 3.1. Let
Because g(xq) = f(x), the graph of g(x) is extended and contracted by xq = t in the x-axis, without changing the height of f(x). Expansion and contraction come to object in
Lemma 3.2.
Proof. It is proved by the next calculations.
Theorem 3.3. The curvature of the curve y = g(x) at the point xq is this.
These become
Proof. Formula (3.5) is obtained by substituting the formulas (3.2), (3.3) for
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
This means the intersection
Definition 4.1. For equation f(x) = 0, we call the next recurrence formulas the extension of Newton’s method or Murase-Newton’s method, Tsuchikura-Horiguchi’s method.
Here, if q = 1, then the formulas (4.2), (4.2′) become Newton’s method.
Example 4.2. In the case of q = 2, applying the formula (4.2) to the Murase’s equation (1.2) of the hearth, we get
The formula (4.3) equals to (2.6).
Lemma 4.3. In the sequence {xn}, let
Proof. Applying L’Hospital’s rule to
Proposition 4.4. If α is a simple root (m (>1) multiple root resp.) of f(x) = 0, then
Theorem 4.5. Let α (≠0) be a simple root for f(x) = 0, i.e.,
If α is m (
Proof. If
Since
(4.7) becomes
Here by the formula (4.4),
is obtained. Similarly formula (4.6) is obtained from (2.3).
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
Proposition 4.6. Let the equation h(x) = 0 be deformed from f(x) = 0. Let
Theorem 4.7. Let α (≠0) be a simple root of f(x) = 0, and
i.e.,
Equal signs are the case of q = 1 and
Proof. Compare the coefficient of
The formula (4.14) is obtained from (4.16).
Theorem 4.8. Let α (≠0) be a simple root of f(x) = 0, and
This is equivalent to the convergence to α of Newton’s method equals to or faster than that q-th power of TH- method.
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
Proof. Because
according to
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
If q satisfies the condition (4.23) ((4.22) resp.)), then the convex-concave of curve of g(x) in the neighborhood of
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
and (4.14) of Theorem 4.7, (4.24) is obtained.
Corollary 4.12. If
The following are the results related to the curvature and the formulas for comparing the convergences of TH- method.
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
and (4.14) of Theorem 4.7, (4.27) is obtained.
Theorem 4.14. Let the conditions be same as the above theorem. Then formulas (4.29) and (4.30) are the equivalent.
Proof. (4.30) is obtained from (4.29).
Theorem 4.15. Let the conditions be same as Theorem 4.13. If
and
hold, then the convergence to α of q-th power of TH-method is equal to or faster than that Newton’s method.
Proof. Assertion is obtained from (4.14) of Theorem 4.7 and (4.30), (4.31).
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 (
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)
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
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
The first column represents the initial value x0 and the absolute error
q x0 | 0.5 | 1 (N-method) | 1.2 | 1.5 | 2 | 2.5 | 3 | 3.5 |
---|---|---|---|---|---|---|---|---|
1.9 | 3 (x3 = 2) | 3 | 3 | 3 | 1 | 3 | 3 | 3 |
Absolute error | 1.36646E−11 | 7.47402E−13 | 1.52323E−13 | 5.77316E−15 | 4.66294E−15 | 5.73097E−13 | 9.17288E−12 | |
1.95 | 3 | 3 | 3 | 3 | 1 | 3 | 3 | 3 |
Absolute error | 4.66294E−14 | 2.66454E−15 | 4.44089E−16 | 0 | 0 | 2.22045E−15 | 3.79696E−14 | |
2.05 | 3 | 3 | 3 | 3 | 1 | 3 | 3 | 3 |
Absolute error | 3.59712E−14 | 2.22045E−15 | 4.44089E−16 | 0 | 0 | 2.44249E−15 | 4.37428E−14 | |
2.1 | 3 | 3 | 3 | 3 | 1 | 3 | 3 | 3 |
Absolute error | 8.01581E−12 | 5.00933E−13 | 1.0747E−13 | 4.44089E−15 | 4.66294E−15 | 6.5481E−13 | 1.19822E−11 |
to converge to a root 2. 1.36646E−11 indicates the absolute error |the value 2 of the convergence of the numerical calculation xk+1 − root 2|.
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 x0 = 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
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
(4) Formulas (4.29), (4.30) and (4.31). In case of q = 1.2, the formulas (4.29), (4.30) of Theorem 4.14 do not hold, respectively. Formulas (4.29), (4.31) of Theorem 4.15 hold in
Example 5.2. A quadratic equation
(1) The roots of (5.2.1) are α = 1, 2. Because
q | Right-hand side of (5.1.6) | Right-hand side of (4.30) | |||
---|---|---|---|---|---|
0.6 | 0.01951429 | 0.013938779 | 0.028533603 | 1.4 | 2.047066242 |
0.8 | 0.024972421 | 0.020810351 | 0.028533603 | 1.2 | 1.371125523 |
1 | 0.028533603 | 0.028533603 | 0.028533603 | 1 | 1 |
1.2 | 0.029122212 | 0.036402765 | 0.028533603 | 0.8 | 0.783830657 |
1.4 | 0.02591774 | 0.043196233 | 0.028533603 | 0.6 | 0.660557671 |
1.6 | 0.018954714 | 0.047386784 | 0.028533603 | 0.4 | 0.602142634 |
1.8 | 0.009553789 | 0.047768943 | 0.028533603 | 0.2 | 0.597325396 |
2 | 0 | 0.044194174 | 0.028533603 | 0 | 0.645641732 |
2.2 | 0.007549705 | 0.037748524 | 0.028533603 | 0.2 | 0.755886593 |
2.4 | 0.012053832 | 0.030134579 | 0.028533603 | 0.4 | 0.946872458 |
2.6 | 0.013697643 | 0.022829405 | 0.028533603 | 0.6 | 1.249861901 |
2.8 | 0.013327903 | 0.016659878 | 0.028533603 | 0.8 | 1.712713749 |
3 | 0.011858541 | 0.011858541 | 0.028533603 | 1 | 2.406164671 |
3.2 | 0.009973947 | 0.008311623 | 0.028533603 | 1.2 | 3.432976253 |
3.4 | 0.008084781 | 0.005774843 | 0.028533603 | 1.4 | 4.941017728 |
A. In case of α = 1
Numerical calculations of TH-method, formulas (4.27),
(2A) We examine the speed of convergence of q-th power of TH-method in
The results of the calculations are
In case of x0 = 1.05, 1.1, q-th (q = _3, _2, _1, 0.5) power of TH-method converges better than Newton’s method, respectively. Therefore, these are compatible with the theory of Theorem 4.7.
(3A) For
Indeed, by calculating the left and right sides of (5.2.4) for q in the
q x0 | −4 | −3 | −2 | −1 | 0.5 | 1 | 2 |
---|---|---|---|---|---|---|---|
0.95 | 4 | 3 | 3 | 2 | 3 | 3 | 3 |
Absolute error | 9.75731E−11 | 9.1771E−13 | 3.2816E−12 | 2.6439E−11 | 4.52769E−10 | ||
1.05 | 3 | 3 | 3 | 2 | 3 | 3 | 4 |
Absolute error | 2.63685E−10 | 1.49318E−11 | 7.9714E−14 | 8.35199E−12 | 5.88803E−11 | ||
1.1 | 4 | 4 | 3 | 3 | 4 | 4 | 4 |
Absolute error | 1.33227E−15 | 2.22045E−16 | 0 | 5.55112E−16 | 2.42584E−13 | ||
1.2 | 4 | 4 | 3 | 3 | 4 | 4 | 5 |
Absolute error | 1.06004E−12 | 4.44089E−16 | 4.31544E−12 | 2.32831E−10 |
q | Right-hand side of (5.2.4) | Right-hand side of (4.30) | |||
---|---|---|---|---|---|
−4 | 0.171201618 | 0.114134412 | 0.707106781 | 1.5 | 6.195386388 |
−3.5 | 0.181419613 | 0.14513569 | 0.707106781 | 1.25 | 4.872039263 |
−3 | 0.18973666 | 0.18973666 | 0.707106781 | 1 | 3.726779962 |
−2.5 | 0.192098626 | 0.256131501 | 0.707106781 | 0.75 | 2.760717751 |
−2 | 0.178885438 | 0.357770876 | 0.707106781 | 0.5 | 1.976423538 |
−1.5 | 0.128007738 | 0.51203095 | 0.707106781 | 0.25 | 1.380984452 |
−0.5 | 0.178885438 | 0.715541753 | 0.707106781 | 0.25 | 0.988211769 |
0.5 | 0.536656315 | 0.715541753 | 0.707106781 | 0.75 | 0.988211769 |
1 | 0.707106781 | 0.707106781 | 0.707106781 | 1 | 1 |
1.5 | 0.640038688 | 0.51203095 | 0.707106781 | 1.25 | 1.380984452 |
(4A) Formulas (4.29), (4.30) of Theorem 4.14 hold. Formula (4.31) of Theorem 4.15 hold for q = _0.5, 0.5, 1.
B. In case of α = 2
Numerical calculations of TH-method, formulas (4.27),
(2B) We examine the speed of convergence of q-th power of TH-method in
The results of the calculations are
In case of x0 = 2.1, numerical calculations of q-th power of TH-method are compatible with the theory of Theorem 4.7.
(3B) For α = 2, formula (4.27) of Theorem 4.13 becomes
Indeed, by calculating the left and right sides of (5.2.6) for q in
(4B) Formulas (4.29), (4.30) of Theorem 4.14 hold except for q = _2. In this case, according to q increases, the value of the right-hand side of (4.30) increases rapidly. Formula (4.31) of Theorem 4.15 holds the equal sign only q = 1.
Example 5.3. Murase’s third degree equation
Graph of f(x) is this (
The graph is drawn in Bear Graph of free software.
(1) For a root 2 of (5.3.1), condition (4.15) becomes
q x0 | −2 | −1 | 1 | 2 | 3 |
---|---|---|---|---|---|
1.9 | 4 | 4 | 4 | 4 | 3 |
Absolute error | 3.79385E−12 | 3.08198E−13 | 0 | 0 | |
2.1 | 4 | 4 | 4 | 4 | 3 |
Absolute error | 6.17284E−14 | 7.10543E−15 | 4.44089E−16 | 0 | |
2.2 | 5 | 4 | 4 | 4 | 4 |
Absolute error | 9.30589E−11 | 3.53939E−13 | 4.44089E−15 | 4.44089E−16 |
4 | 5 | 6 | 7 | 8 | 9 | 9.3 |
---|---|---|---|---|---|---|
3 | 3 | 3 | 3 | 3 | 4 | 4 |
2.22045E−16 | 0 | |||||
3 | 3 | 3 | 3 | 4 | 4 | 4 |
2.22045E−16 | 4.44089E−16 | 2.22045E−15 | ||||
3 | 3 | 4 | 4 | 4 | 5 | 5 |
3.38884E−12 | 0 | 9.99201E−15 | 4.47709E−12 |
q | Right-hand side of (5.2.6) | Right-hand side of (4.30) | |||
---|---|---|---|---|---|
−2 | 0.798940882 | 0.456537647 | 0.707106781 | 1.75 | 1.548846597 |
−1 | 0.684806471 | 0.456537647 | 0.707106781 | 1.5 | 1.548846597 |
1 | 0.707106781 | 0.707106781 | 0.707106781 | 1 | 1 |
2 | 0.085600809 | 0.114134412 | 0.707106781 | 0.75 | 6.195386388 |
3 | 0.006872729 | 0.013745459 | 0.707106781 | 0.5 | 51.44293798 |
4 | 0.000487567 | 0.001950267 | 0.707106781 | 0.25 | 362.5691315 |
5 | 0 | 0.000312427 | 0.707106781 | 0 | 2263.272051 |
6 | 1.35628E−05 | 5.42513E−05 | 0.707106781 | 0.25 | 13033.92252 |
7 | 4.98242E−06 | 9.96485E−06 | 0.707106781 | 0.5 | 70960.11004 |
8 | 1.43051E−06 | 1.90735E−06 | 0.707106781 | 0.75 | 370728.1304 |
9 | 3.7676E−07 | 3.7676E−07 | 0.707106781 | 1 | 1876809.006 |
10 | 9.53674E−08 | 7.62939E−08 | 0.707106781 | 1.25 | 9268190.533 |
11 | 2.36448E−08 | 1.57632E−08 | 0.707106781 | 1.5 | 44858040.14 |
(2) In case of q = 0.5, 1, 1.5, 2, 2.45, 2.5, we calculate q-th power of TH-method. The results are
In case of x0 = 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
(4) Formulas (4.29), (4.30) of Theorem 4.14 hold for q = 0.5, 1, 1.5. Formula (4.31) of Theorem 4.15 holds for q = 1, 1.5.
q x0 | 0.5 | 1 | 1.5 | 2 | 2.45 | 2.5 | |
---|---|---|---|---|---|---|---|
0.6 | 6 | 5 | 4 | 4 | 5 | 5 | |
Absolute error | 1.92246E−12 | 0 | 0 | ||||
1 | 5 | 5 | 4 | 4 | 4 | 4 | |
Absolute error | 3.12452E−11 | 0 | |||||
1.5 | 4 | 4 | 3 | 3 | 4 | 4 | |
Absolute error | 6.38245E−12 | 1.33227E−15 | 0 | 0 | |||
1.9 | 3 | 3 | 3 | 3 | 3 | 3 | |
Absolute error | 3.41949E−12 | 8.39329E−14 | 2.22045E−16 | 0 | 5.83977E−14 | 9.30367E−14 | |
2.1 | 3 | 3 | 3 | 3 | 3 | 3 | |
Absolute error | 1.92957E−12 | 5.15143E−14 | 0 | 0 | 6.79456E−14 | 1.08802E−13 | |
q | Right-hand side of (5.3.3) | Right-hand side of (4.30) | |||
---|---|---|---|---|---|
0.5 | 0.000112052 | 6.6401E−05 | 0.000187683 | 1.6875 | 2.826510815 |
1 | 0.000187683 | 0.000187683 | 0.000187683 | 1 | 1 |
1.5 | 0.000124081 | 0.00039706 | 0.000187683 | 0.3125 | 0.472682782 |
2 | 0.000278286 | 0.000742096 | 0.000187683 | 0.375 | 0.25290959 |
2.45 | 0.001207242 | 0.001214835 | 0.000187683 | 0.99375 | 0.15449283 |
2.5 | 0.001358204 | 0.00127831 | 0.000187683 | 1.0625 | 0.146821424 |
Example 5.4. A fifth degree equation
f(x) has no terms of
Graph is the convex downward and monotonic decreases in
(1) Condition (4.15) becomes
(2) In case of q = _1, 1, 3, 5, 6, 6.81, 7, we calculate q-th power of TH-method. The results are
Notation 5(#DIV/0!) denotes that it is #DIV/0! in 5 iterations. In case of x0 = 1.063, numerical calculations of q-th power of TH-method are compatible with Theorem 4.7. There is a noteworthy thing. In case of x0 = 0.1, number of iterations of Newton’s method is 22 times but that 3-th power of TH-method is 5 times only.
(3) Formula (4.27) of Theorem 4.13 becomes
Indeed, by calculating the left and right sides of (5.4.3) for q = _1, 1, 3, 5, 6, 6.81, 8, 10 we get
q x0 | −1 | 1 | 3 | 5 | 6 | 6.81 | 7 |
---|---|---|---|---|---|---|---|
0.1 | 5(#DIV/0!) | 22 | 5 | 6 | 6 | 7 | 7 |
0.5 | 6(#NUM!) | 8 | 4 | 5 | 5 | 5 | 6 |
0.8 | 6 | 5 | 4 | 4 | 4 | 4 | 4 |
0.9 | 5 | 4 | 3 | 3 | 4 | 4 | 4 |
Absolute error | 8.28226E−14 | 0 | 6.21725E−15 | 1.44329E−14 | |||
1.063 | 4 | 4 | 3 | 3 | 3 | 4 | 4 |
Absolute error | 1.06581E−14 | 0 | 0 | 1.11022E−16 |
q | Right-hand side of (5.4.3) | Right-hand side of (4.30) | |||
---|---|---|---|---|---|
−1 | 0.040073199 | 0.023747081 | 0.023747081 | 1.6875 | 1 |
1 | 0.023747081 | 0.023747081 | 0.023747081 | 1 | 1 |
3 | 0.0202398 | 0.064767361 | 0.023747081 | 0.3125 | 0.366651974 |
5 | 0.034011201 | 0.090696536 | 0.023747081 | 0.375 | 0.261830077 |
6 | 0.070150312 | 0.097600434 | 0.023747081 | 0.71875 | 0.243309173 |
6.81 | 0.100353783 | 0.100636824 | 0.023747081 | 0.9971875 | 0.235968107 |
8 | 0.143068791 | 0.101737807 | 0.023747081 | 1.40625 | 0.233414515 |
(4) Formulas (4.29), (4.30) of Theorem 4.14 hold for q = 1 and 3. Similarly formulas (4.29), (4.31) of Theorem 4.15 also hold for q = 1 and 3.
Example 5.5. Fifth degree equation
f(x) has no terms of
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.
(1) The formula (4.15) of Theorem 4.7 becomes (5.5.2).
(The value of formula (5.5.2) for q = 16.018 is 1.999993923.)
(2) For q = _1, 1, 3, 6, 9, 12, 15, 16, 17, we calculate q-th power of TH-method. The results are
① For x0 = 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 x0 = 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
The formula of the tangent of the curve of g(t) at point
For the initial value is 1.59, we give in
Straight line 1, 2 and 3 in
q x0 | −1 | 1 | 3 | 6 | 9 | 12 |
---|---|---|---|---|---|---|
0.5 | #NUM! | 28 | Oscillation | #NUM! | 13 | #NUM! |
1 | #DIV/0! | 45 | Oscillation | #NUM! | 9 | #NUM! |
1.5 | #DIV/0! | 54 | Oscillation | #NUM! | 4 | #NUM! |
1.65 | #DIV/0! | 10 | 7 | 5 | 5 | 5 |
1.7 | 16 | 8 | 6 | 5 | 4 | 4 |
1.8 | 7 | 6 | 5 | 4 | 4 | 4 |
1.85 | 6 | 5 | 5 | 4 | 3 | 4 |
Absolute error | 2.66674E−10 | 2.87181E−10 | ||||
1.89 | 5 | 5 | 4 | 4 | 3 | 4 |
Absolute error | 2.59915E−10 | 2.87176E−10 |
15 | 16 | 17 | q x0 |
---|---|---|---|
13 | #NUM! | 15 | 0.5 |
9 | #NUM! | 10 | 1 |
6 | #NUM! | 6 | 1.5 |
4 | 4 | 4 | 1.65 |
4 | 5 | 5 | 1.7 |
4 | 5 | 5 | 1.8 |
4 | 4 | 5 | 1.85 |
2.87181E−10 | Absolute error | ||
4 | 4 | 4 | 1.89 |
Absolute error |
k | xk | Gradient of tangent | Intercept | ||
---|---|---|---|---|---|
1 | 1.5 | 38.44335938 | −444.234375 | 0.007315958 | 3.25 |
2 | −1.968698131 | −444.234375 | 459.9298761 | −0.067041182 | 30.83424256 |
3 | 1.976307982 | 459.9298761 | 656.2643436 | −0.006934223 | 4.550683281 |
4 | 2.055932049 | 656.2643436 | 656.3659 | −0.006895787 | 4.526159387 |
5 | 2.055967397 | 656.3659 | 656.3659005 | −0.006895729 | 4.526121193 |
6 | 2.055967397 | 656.3659005 | 656.3659005 | −0.006895729 | 4.526121193 |
vibration is only once,
(3) Formula (4.27) of Theorem 4.13 becomes
Indeed, by calculating the left and right sides of (5.5.5) for q = _1, 1, 3, 6, 9, 12, 15, 16, 17 we get
(4) Formulas (4.29), (4.30) of Theorem 4.14 do not hold for q = 3. Theorem 4.15 holds as equality for q = 1.
Equation (5.4.1),(5.5.1) has only one term which degree is smaller than highest degree, respectively. These equations have the trend that the convergences of TH-methods are extremely fast than that Newton’s method.
Example 5.6.
Roots of equation (5.6) are α = mπ (m is an integer, π ? 3.141592654), and
Example 5.7.
A root of equation (5.7.1) is α. Because
For q = _2, _1, 1, 2, 3, we calculate q-th power of TH-method, and get
Example 5.8.
(1) The root of (5.8.1) is ln2 ? 0.693147181, and
(2) For q = 0.5, 1, 1.5, 2, 2.386294361, 2.5, we calculate q-th power of TH-method.
However, we calculate the absolute error as ln2 ? 0.693147181 root. The results are
For x0 = 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).
(3) Formula (4.27) of Theorem 4.13 for (5.8.1) becomes
q | Right-hand side of (5.5.5) | Right-hand side of (4.30) | |||
---|---|---|---|---|---|
−1 | 0.002786695 | 0.002200579 | 0.009268403 | 1.266346241 | 4.211802053 |
1 | 0.009268403 | 0.009268403 | 0.009268403 | 1 | 1 |
3 | 0.05171836 | 0.070494235 | 0.009268403 | 0.733653759 | 0.13147746 |
6 | 0.000491737 | 0.001471676 | 0.009268403 | 0.334134398 | 6.297857366 |
9 | 5.73089E−07 | 8.76484E−06 | 0.009268403 | 0.065384963 | 1057.452359 |
12 | 3.03503E−08 | 6.5283E−08 | 0.009268403 | 0.464904324 | 141972.7121 |
15 | 4.782E−10 | 5.53201E−10 | 0.009268403 | 0.864423686 | 16754135.22 |
16 | 1.14749E−10 | 1.15025E−10 | 0.009268403 | 0.997596806 | 80577151.16 |
17 | 2.72569E−11 | 2.41048E−11 | 0.009268403 | 1.130769926 | 384505213.3 |
q x0 | −3 | −2 | −1 | 1 | 2 | 3 | ||
---|---|---|---|---|---|---|---|---|
2.7 | 5 | 5 | 4 | 3 | 3 | 4 | ||
Absolute error | 4.10207E−10 | 4.11536E−10 | ||||||
2.9 | 4 | 4 | 4 | 3 | 3 | 4 | ||
Absolute error | 4.10207E−10 | 4.12394E−10 | ||||||
3.1 | 3 | 3 | 3 | 2 | 3 | 3 | ||
3.3 | 4 | 3 | 3 | 2 | 3 | 4 | ||
q x0 | −2 | −1 | 1 | 2 | 3 |
---|---|---|---|---|---|
1.8 | 4 | 4 | 3 | 4 | 4 |
1.9 | 4 | 3 | 3 | 3 | 3 |
Absolute error | 4.98299E−11 | 4.44089E−16 | 2.24532E−12 | 1.38187E−10 | |
2.1 | 4 | 3 | 3 | 3 | 3 |
Absolute error | 1.10953E−10 | 4.44089E−16 | 7.34968E−14 | 3.54139E−11 | |
2.2 | 4 | 4 | 3 | 3 | 4 |
Absolute error | 4.44089E−16 | 1.3034E−13 |
q x0 | 0.5 | 1 | 1.5 | 2 | 2.386294361 | 2.5 |
---|---|---|---|---|---|---|
0.68 | 3 | 3 | 2 | 3 | 3 | 3 |
Absolute error | 3.33067E−16 | 0 | 0 | 0 | 0 | |
0.7 | 3 | 2 | 2 | 2 | 3 | 3 |
Absolute error | 1.11022E−16 | 2.74411E−10 | 6.18072E−12 | 2.31052E−11 | 0 | 0 |
0.73 | 3 | 3 | 3 | 3 | 3 | 3 |
Absolute error | 9.97646E−13 | 2.53131E−14 | 1.11022E−16 | 0 | 2.32037E−14 | 6.9722E−14 |
0.76 | 3 | 3 | 3 | 3 | 3 | 3 |
Absolute error | 1.01667E−10 | 2.8485E−12 | 5.55112E−16 | 5.10703E−15 | 2.43627E−12 | 7.55507E−12 |
By calculating the left and right sides of (5.8.3) for q in
(4) In
Example 5.9.
(1) The root of (5.9.1) is α = 1, and
(2) The calculations for TH-method are
For x0 = 1.05, 1.1, 1.2, q-th (q = _1, _0.5, 0.5) power of TH-method converges better than Newton’s method, respectively.
(3) Formula (4.27) of Theorem 4.13 for (5.9.1) is this.
By calculating the left and right sides of (5.9.3) for q in
(4) Formulas (4.29), (4.30), (4.31) hold for q = _1, _0.5, 0.5, 1.
q | Right-hand side of (5.8.3) | Right-hand side of (4.30) | |||
---|---|---|---|---|---|
0.8 | 0.214901835 | 0.166779456 | 0.178885438 | 1.288539008 | 1.072586771 |
1 | 0.178885438 | 0.178885438 | 0.178885438 | 1 | 1 |
1.2 | 0.132153552 | 0.18574954 | 0.178885438 | 0.711460992 | 0.963046469 |
1.4 | 0.0801186 | 0.189440613 | 0.178885438 | 0.422921984 | 0.944282406 |
1.6 | 0.025705605 | 0.191286171 | 0.178885438 | 0.134382975 | 0.935171827 |
1.8 | 0.029614548 | 0.192107617 | 0.178885438 | 0.154156033 | 0.931173064 |
2 | 0.085174223 | 0.192399316 | 0.178885438 | 0.442695041 | 0.929761302 |
2.2 | 0.140725657 | 0.192449541 | 0.178885438 | 0.731234049 | 0.929518655 |
2.386294361 | 0.192420845 | 0.192420845 | 0.178885438 | 1 | 0.929657275 |
2.4 | 0.196222738 | 0.192418045 | 0.178885438 | 1.019773057 | 0.9296708 |
q x0 | −1.5 | −1 | −0.5 | 0.5 | 1 | 1.5 |
---|---|---|---|---|---|---|
0.9 | 4 | 3 | 3 | 3 | 3 | 3 |
Absolute error | 1.57945E−10 | 8.54383E−12 | 6.45084E−12 | 8.99983E−11 | 3.98149E−10 | |
0.95 | 3 | 3 | 3 | 3 | 3 | 3 |
Absolute error | 2.33324E−12 | 4.29878E−13 | 2.4869E−14 | 2.17604E−14 | 3.2685E−13 | 1.54754E−12 |
1.05 | 3 | 3 | 3 | 3 | 3 | 3 |
Absolute error | 1.04716E−12 | 2.2049E−13 | 1.46549E−14 | 1.68754E−14 | 2.85993E−13 | 1.54754E−12 |
1.1 | 3 | 3 | 3 | 3 | 3 | 3 |
Absolute error | 1.85532E−10 | 4.14198E−11 | 2.93099E−12 | 3.77964E−12 | 6.88853E−11 | 3.98203E−10 |
1.2 | 4 | 4 | 3 | 4 | 4 | 4 |
Absolute error | 2.22045E−16 | 0 | 0 | 1.11022E−16 | 5.44009E−15 |
q | Right-hand side of (5.9.3) | Right-hand side of (4.30) | |||
---|---|---|---|---|---|
−1.5 | 0.384023213 | 0.256015475 | 0.353553391 | 1.5 | 1.380984452 |
−1 | 0.353553391 | 0.353553391 | 0.353553391 | 1 | 1 |
−0.5 | 0.178885438 | 0.357770876 | 0.353553391 | 0.5 | 0.988211769 |
0.5 | 0.178885438 | 0.357770876 | 0.353553391 | 0.5 | 0.988211769 |
1 | 0.353553391 | 0.353553391 | 0.353553391 | 1 | 1 |
1.5 | 0.384023213 | 0.256015475 | 0.353553391 | 1.5 | 1.380984452 |
Dr. Tamotsu Tsuchikura (1923-2015, professor emeritus of Tohoku University) and Dr. Mitsuo Morimoto (professor emeritus of Sophia University) gave hints to me. I am deeply grateful to them.
ShunjiHoriguchi, (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,07,40-60. doi: 10.4236/am.2016.71004