Applied Mathematics, 06, 7, 40-60 Published Online January 06 in SciRes. http://www.scirp.org/journal/am http://d.doi.org/0.436/am.06.7004 The Formulas to Compare the Convergences of Newton s Method and the Etended Newton s Method (Tsuchiura-Horiguchi Method and the Numerical Calculations Shunji Horiguchi Department of Economics, Niigata Sangyo University, Niigata, Japan Received 4 November 05; accepted 7 January 06; published 0 January 06 Copyright 06 by author and Scientific Research Publishing Inc. This wor is licensed under the Creative Commons Attribution International License (CC BY. http://creativecommons.org/licenses/by/4.0/ Abstract This paper gives the etension of Newton s method, and a variety of formulas to compare the convergences for the etension of Newton s method (Section 4. Section 5 gives the numerical calculations. Section introduces the three formulas obtained from the cubic euation of a hearth by Murase (Ref. []. We find that Murase s three formulas lead to a Horner s method (Ref. [] and etension of a Newton s method (009 at the same time. This shows originality of Wasan (mathematics developed in Japan in the Edo era (603-868. Suzui (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 gives the relations between Newton s method, Horner s method and Murase s y = g t : = f t = f. three formulas. Section 3 gives a new function defined such as Keywords Recurrence Formula, Newton-Raphson s Method (Newton s Method, Etension of Newton s Method. Murase s Three Formulas from the Cubic Euation of a Hearth We write this paper from two inds of recurrence formulas of the suare and the deformation of a cubic euation written in Murase s boo (Ref. [], and a hint of Tsuchiura (Ref. [4]. It is enough for readers to now these three formulas. It is very difficult even for Japanese people to read the Murase s boo written in the How to cite this paper: Horiguchi, S. (06 The Formulas to Compare the Convergences of Newton s Method and the Etended Newton s Method (Tsuchiura-Horiguchi Method and the Numerical Calculations. Applied Mathematics, 7, 40-60. http://d.doi.org/0.436/am.06.7004
Japanese ancient writing. Therefore, the readers do not need to read the boo. Furthermore, the readers do not need to mind Japanese references. From now on, we eplain the Murase s three formulas as introduction. The readers can now the origin of this paper. Murase made the cubic euation for the net problem in 673. There is a rectangular solid (base is a suare. We put it together four and mae the hearth such as Figure. We claim one side of length of the suare that one side is 4, and a volume becomes 9 of the hearth. Let one side of length of the suare be, then the net cubic euation is obtained. that is ( 4 4 9 = (. 3 f = 4 + 48 = 0. (. This has three solutions of real number, 6 ± 5. Murase derived two following recurrence formulas (.3, (.4 and deformed euation (.5 from (.. The first method: 3 48 + + = ( = 0,,,. (.3 4 Using on an abacus, Murase calculates to 0 = 0 (initial value, =.85, =.97, 3 =.9936, and decides a solution with. The second method: 48 = + ( 0,,,. 4 = (.4 Here he calculates to 0 = 0, =.85, =.976, 3 =.9989, 4 =.9999907, and decides a solution with. Formula (.4 has better precision than that (.3, and convergence becomes fast. The third method was nonrecurring in spite of a short sentence for many years. However, Yasuo Fujii (Sei Kowa Institute Mathematics of Yoaichi University succeeds in decoding in May 009. It is the net euation. The third method: 3 ( 4 = 48. (.5 The studies of three formulas of Murase progress by the third method have been decoded. Furthermore we obtain the net recurrence formula from (.5. + 3 48 = ( = 0,,,. (.6 4. Relations between Newton s Method, Horner s Method and the Murase s Three Formulas Throughout this paper, function f( be i ( times differentiable if necessary, and f (i ( continuous. We start with the definition of Newton s method. Figure. Hearth. 4
Net Newton s method is eplained in a boo of the standard numerical computation (Ref. [5]. The recurrence formula to approimate a root of the euation f( = 0 ( f = 0,,, + f = (. is called Newton s method or Newton-Raphson s method. Newton s method is a method of giving the initial value 0, calculating,, one after another, and to determine for a root. The uadratic convergence and the linearly convergence of the Newton s method are nown as followings. Let α be a simple root for f( = 0, i.e., f ( α 0. Then Newton s method to the uadratic convergence of the following formula. + ( α ( α α f α (. f If α is m ( multiple root, then it will become the linearly convergence of the following formula. + α ( α (.3 m Remar. Concerning choosing the initial value 0, the number of iterations until it converges on a root changes. Moreover, it may not be converged on a root. Eample.. By the transformation of variable = t, Murase s euation f( = 3 4 + 48 = 0 becomes 3 It becomes the following formula if Newton s method is applied to g( t. t+ = t = t = This becomes the following formula by t =. g t = t 4t+ 48 = 0. (.4 g t t 4t + 48 0.5t 48 3 4 3 3 g t t 4 t + 48 0.5 = 4.5 This is a middle formula of (.4 and (.6 eactly. That is, Murase s formulas (.3, (.4, and (.5 lead to etension of a Newton s method (009. Eample.. Applying the Horner s method to Murase s euation f( = 3 4 + 48 = 0 for root, we get Table. Here, number 4,, 0 of the second column corresponds to the denominator 4,4, 4 for = of (.3, (.4, (.6, respectively. Therefore, from the Table, we find that the Murase s formulas (.3, (.4, and (.6 lead to a Horner s method. Furthermore, please read Ref. [] if you want to now this deeply. Table. Horner s method for Murase s euation. 4 0 48 + 4 48 3 4 0 + 0 0 44 + 8 (.5 (.6 4
Proposition.3. We epand the first, second, third method of Murase, and obtain the net recurrence formula where m is a real number. 3. Function y = g(t Defined by Definition 3.. Let + ( m 3 48 = 4 m = of y = f( t = t where is a real number that is not 0. We define the function g(t such as : (.7 g t = f t = f. (3. Because g( = f(, the graph of g( is etended and contracted by = t in the -ais, without changing the height of f(. Epansion and contraction come to object in < and >. g g f, f as follows. Lemma 3.., are represented by f ( ( Proof. It is proved by the net calculations. g = (3. ( f + f (3.3 g ( = f ( f ( + f (, f ( 0 (3.4 ( dg t df df d g ( t = = = = f dt dt d dt dg t d f d f d g ( t = = = dt dt d dt ( f f = Theorem 3.3. The curvature of the curve y = g( at the point is this. ( ( f + f (3.5 3 f ( + g ( t µ ( t = = µ 3 ( = f ( ( + g ( t f ( + f (, f ( 0 (3.6 3 f ( + f ( These become µ ( = of f( if = in particular. ( + f ( 3 Proof. Formula (3.5 is obtained by substituting the formulas (3., (3.3 for g ( t, g ( t ( t µ. in the curvature 43
4. Etension of Newton s Method (Tsuchiura-Horiguchi s Method 4.. Etension of Newton s Method and the Convergences In 009, we found the etension of Newton s method from the Murase s three formulas as follows. Applying the Newton s method to g(t, we have ( ( t f ( t ( g t t+ = t, t+ = t g f t t. (4. This means the intersection t+ = +, t g t =, g of the graph of y = g( t ( g(. Returning to the variable by = t, we get an etension of Newton s method below. Definition 4.. For euation f( = 0, we call the net recurrence formulas the etension of Newton s method or Murase-Newton s method, Tsuchiura-Horiguchi s method. with the t(-ais of the tangent in the point ( ( ( f = + ( 0, f R (4. ( ( f + = (4. f Here, if =, then the formulas (4., (4. become Newton s method. Eample 4.. In the case of =, applying the formula (4. to the Murase s euation (. of the hearth, we get 3 3 4 + 48 48 0.5 + = =. (4.3 3 8 4.5 The formula (4.3 euals to (.6. Lemma 4.3. In the seuence { n }, let lim n = α, and, r anarbitrary real constant that is not 0, respectively. n In this case, following formula holds for large enough integer n. r r r n α α ( n α r Proof. Applying L Hospital s rule to ( α ( r α r (4.4, (4.4 is obtained. Proposition 4.4. If α is a simple root (m (> multiple root resp. of f( = 0, then α becomes the simple root (m multiple root resp. of g(. Theorem 4.5. Let α ( 0 be a simple root for f( = 0, i.e., f ( α 0. For sufficiently close to α, -th power of TH-method (Tsuchiura-Horiguchi s method becomes the uadratic convergence of the following formula. + ( α f α + ( α (4.5 f α If α is m ( multiple root, then it will become linearly convergence of the following formula. + α ( α (4.6 m Proof. If α is a simple root for g(t = 0, then Newton s method for g(t becomes the uadratic convergence of the following formula. Since t + ( α ( t α α g α (4.7 g 44
(4.7 becomes f g ( α = ( α α g, ( α + ( f α f α α =, (4.8 α + ( α f α f α α α α f ( α α +. (4.9 Here by the formula (4.4, α f α + f α α ( + α α α α f α is obtained. Similarly formula (4.6 is obtained from (.3. (4.0 4.. Varieties of Formulas to Compare the Convergences for the Etension of Newton s Method (Tsuchiura-Horiguchi s Method We deform the euation f( = 0 to h( = 0. That is, two euations have the same root. r-th power of TH-method for h( is ( ( r r r h = + r h and if α ( 0 is a simple root, then it becomes uadratic convergence + ( α, (4. h r α + ( α. (4. h α of formula (4.5 and (4.. = =, and α( 0 a simple root. Then the necessary and sufficient condition for the convergence to α of -th power of TH-method of f( to be eual to or faster than that r-th power of TH-method of h( is that the real numbers and r satisfy the following condition. We get the following proposition by comparing the coefficients of ( α Proposition 4.6. Let the euation h( = 0 be deformed from f( = 0. Let f h 0 ( α ( α f + f α h r + h α (4.3 Theorem 4.7. Let α ( 0 be a simple root of f( = 0, and f ( α 0. Then a necessary and sufficient condition for the convergence to α of -th power of TH-method is eual to or faster than that Newton s method is that satisfies the following conditions. i.e., ( α f + f α ( α f 0. f α Eual signs are the case of = and α f f Proof. Compare the coefficient of (4.4 (4.5 = +. α of the uadratic convergence (4.5 of -th power of TH-method and that (. of Newton s method. Then the necessary and sufficient condition is euivalent to f f +. (4.6 f α α f α 45
The formula (4.4 is obtained from (4.6. Theorem 4.8. Let α ( 0 be a simple root of f( = 0, and f ( α = 0 (i.e. the graph of f( is nearly the straight line in the neighborhood of the point α.. In this case (4.7 holds. 0 ( ( µ α = µ α (4.7 This is euivalent to the convergence to α of Newton s method euals to or faster than that -th power of THmethod. µ α. Proof. By deforming the formula to ( α µ α, we compare it with ( f ( f 0 + α α µ α = 0 µ α = = 0 3 3 f f + + ( α α α f f f = 0 f = 0 0 + α α α (4.8 (4.9 We get the conclusion by this. The following are the results related to the conve-concave of curve and the formulas for comparing convergences of TH-method. Lemma 4.9. Let 0 and f ( 0. Then a necessary and sufficient condition for g ( and f ( are the same sign (opposite sign resp. is Proof. Because according to ( f ( f ( f + > 0( < 0 resp.. (4.0 f ( f ( f ( f ( + f ( + ( f ( g ( = =, (4. + > 0 < 0 resp. and f (, g ( resp.. We get the net theorem from Lemma 4.9, directly. ( Theorem 4.0. Let α( 0 be a simple root of f( = 0, and f ( α 0 Theorem 4.7 into positive and negative range as follows. ( α f + < 0 f α ( α f 0< + f α become the same sign (opposite sign. We divide the Formula (4.4 of (4. (4.3 If satisfies the condition (4.3 ((4. resp., then the conve-concave of curve of g( in the neighborhood g α = 0 and the f( in the neighborhood of f ( α ( = 0 are the same (opposite resp.. Theorem 4.. Let the conditions be the same as the above theorem. We give the following ineuality. of ( α ( α f f g (4.4 ( α Then the convergence to α of -th power of TH-method is eual to or faster than Newton s method euivalent 46
to the formula (4.4. Proof. By the formula ( + = + f f f f f and (4.4 of Theorem 4.7, (4.4 is obtained. Corollary 4.. If α then ineuality (4.4 becomes ( α (4.5 f g f. (4.6 The following are the results related to the curvature and the formulas for comparing the convergences of THmethod. Theorem 4.3. Let α ( 0 be a simple root of f( = 0, and f ( α 0. Suppose that the curvature μ ( of g( satisfies the condition µ α ( α f ( α ( α f + α 3. (4.7 Then the convergence to α of -th power of TH-method is eual to or faster than that Newton s method is euivalent to that (4.7 holds. Proof. The formula f ( α ( α ( f ( α 3 + α f f ( α + 3 = + α f f α α f α + α (4.8 and (4.4 of Theorem 4.7, (4.7 is obtained. Theorem 4.4. Let the conditions be same as the above theorem. Then formulas (4.9 and (4.30 are the euivalent. and ( α ( f ( α f + α f ( α µ ( α = = µ 3 3 ( f f α ( α + α + α ( α ( α α 3 f + f α f + ( + f ( α Proof. (4.30 is obtained from (4.9. Theorem 4.5. Let the conditions be same as Theorem 4.3. If ( α 3 (4.9 (4.30 µ α µ α (4.9 3 f 3 ( α ( f + + α hold, then the convergence to α of -th power of TH-method is eual to or faster than that Newton s method. (4.3 47
Proof. Assertion is obtained from (4.4 of Theorem 4.7 and (4.30, (4.3. 5. Convergence Comparisons of the Numerical Calculations of Newton s Method and Epansion of Newton s Method (Tsuchiura-Horiguchi s Method We use formula (4.' for the numerical calculations of -th power of TH-method for various euations such as n-th order euations ( n, euations of trigonometric, eponential, logarithmic function. We perform numerical calculations in the standard format in Ecel of Microsoft. Eample 5.. Numerical calculation of the p-th root. Let A be a real number, and p a natural number. The euation for p-th root is this. ( The application of the formula (4.5 In this case, formula (4.5 becomes p 0 f = A= p (5.. p A is p-th root of (5.., and we get p p p p p f A = p A = pa p p p f A = p p A = p p A. p pa 0 p p p p A A ( f = A= is p Especially p-th power of TH-method for 0 p p p p + p p (5.. i.e. p. (5..3 A = p = A. (5..4 Therefore, it converges to the root once for any initial value. Hence the number of iterations of formula (5..4 is less than that of the recurrence formula other. ( Speeds of convergences. The roots of f ( = 4= 0 are α = ±. The interval of of (5..3 is 3. In the following, we eamine the speed of convergence of -th power of TH-method in case of α =. The results of the calculations are Table. We eplain how to read this. The first column represents the initial value 0 and the absolute error + ( =, and the first row represents the real number of. Two numbers 3 and.36646e 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 ( f 4 = 0.5 = 0.5 =.9 0.5 0.5 0.5 0.5 0.5 + 0 f Table. Calculations of -th power of TH-method for f( = 4 = 0. (5..5 0 0.5 (N-method..5.5 3 3.5.9 3 ( 3 = 3 3 3 3 3 3 Absolute error.36646e 7.4740E 3.533E 3 5.7736E 5 4.6694E 5 5.73097E 3 9.788E.95 3 3 3 3 3 3 3 Absolute error 4.6694E 4.66454E 5 4.44089E 6 0 0.045E 5 3.79696E 4.05 3 3 3 3 3 3 3 Absolute error 3.597E 4.045E 5 4.44089E 6 0 0.4449E 5 4.3748E 4. 3 3 3 3 3 3 3 Absolute error 8.058E 5.00933E 3.0747E 3 4.44089E 5 4.6694E 5 6.548E 3.98E 48
to converge to a root..36646e indicates the absolute error the value of the convergence of the numerical calculation + root. -th power of TH-methods converges to in number of iterations ; other TH-methods converge to that in three times. In case of 0 =.9,.95, absolute errors of =.,.5,.5, 3 are smaller than that = (Newton s method. Therefore degree of approimations of =.,.5,.5, 3 is better than that =. Furthermore, absolute errors of = 0.5, 3.5 are larger than that =. Thus, these numerical calculations are compatible with the theory of Theorem 4.7. f = 4 is this. (3 The application of the formula (4.7 of Theorem 4.3 for µ α = 3 3 4 4 + + ( ( (5..6 Indeed, by calculating the left and right sides of (5..6 for in the Table 3 we get the numbers there. g( becomes a straight line 4 in case of =, and the curvature is 0. Therefore, the suare of TH-method converges to root in the number of iterations. For each, the second and third columns are calculations of formula (5..6. The fourth column is the calculations of µ. Columns 5 and 6 are the calculation of the left-hand side and the right-hand side of the ineuality (4.30, respectively. For each in 3, the numbers of the second column and third column satisfy the condition (5..6. (4 Formulas (4.9, (4.30 and (4.3. In case of =., the formulas (4.9, (4.30 of Theorem 4.4 do not hold, respectively. Formulas (4.9, (4.3 of Theorem 4.5 hold in.4 ecept for =.. However, in.6, formula (4.9 holds, but (4.3 does not hold. Eample 5.. A uadratic euation ( The roots of (5.. are α =,. Because f ( f ( Table 3. Calculations of (5..6, µ, (4.30 for f( = 4 = 0. f = = 3+ = 0 (5.. = 3, =, condition (4.5 becomes µ ( Right-hand side of (5..6 µ f ( f + Right-hand side of (4.30 0.6 0.09549 0.03938779 0.08533603.4.0470664 0.8 0.04974 0.008035 0.08533603..37553 0.08533603 0.08533603 0.08533603. 0.09 0.03640765 0.08533603 0.8 0.783830657.4 0.059774 0.0439633 0.08533603 0.6 0.66055767.6 0.0895474 0.047386784 0.08533603 0.4 0.604634.8 0.009553789 0.047768943 0.08533603 0. 0.59735396 0 0.0449474 0.08533603 0 0.6456473. 0.007549705 0.03774854 0.08533603 0. 0.755886593.4 0.005383 0.03034579 0.08533603 0.4 0.94687458.6 0.03697643 0.089405 0.08533603 0.6.498690.8 0.0337903 0.06659878 0.08533603 0.8.773749 3 0.085854 0.085854 0.08533603.4066467 3. 0.009973947 0.008363 0.08533603. 3.4397653 3.4 0.00808478 0.005774843 0.08533603.4 4.940778 49
A. In case of α = Numerical calculations of TH-method, formulas (4.7, α 3 0. (5.. α 3. (5..3 µ α, (4.30 for α =. (A We eamine the speed of convergence of -th power of TH-method in 4. The results of the calculations are Table 4. In case of 0 =.05,., -th ( = _ 3, _, _, 0.5 power of TH-method converges better than Newton s method, respectively. Therefore, these are compatible with the theory of Theorem 4.7. f = 3+ and α=, formula (4.7 of Theorem 4.3 becomes (3A For + µ ( =..5.5 + + (5..4 Indeed, by calculating the left and right sides of (5..4 for in the Table 5 we get the numbers there. For each in 3, the numbers of the second column and third column satisfy the condition (5..4. Table 4. Calculations of TH-method for f( = 3 + = 0, α =. 0 4 3 0.5 0.95 4 3 3 3 3 3 Absolute error 9.7573E 9.77E 3 3.86E.6439E 4.5769E 0.05 3 3 3 3 3 4 Absolute error.63685e 0.4938E 7.974E 4 8.3599E 5.88803E. 4 4 3 3 4 4 4 Absolute error.337e 5.045E 6 0 5.55E 6.4584E 3. 4 4 3 3 4 4 5 Absolute error.06004e 4.44089E 6 4.3544E.383E 0 Table 5. Calculations of (5..4, µ (, (4.30 for f( = 3 + = 0. µ ( Right-hand side of (5..4 µ ( + f ( ( f ( Right-hand side of (4.30 4 0.7068 0.4344 0.7070678.5 6.95386388 3.5 0.84963 0.453569 0.7070678.5 4.8703963 3 0.8973666 0.8973666 0.7070678 3.7677996.5 0.909866 0.56350 0.7070678 0.75.7607775 0.78885438 0.357770876 0.7070678 0.5.97643538.5 0.8007738 0.503095 0.7070678 0.5.38098445 0.5 0.78885438 0.7554753 0.7070678 0.5 0.988769 0.5 0.53665635 0.7554753 0.7070678 0.75 0.988769 0.7070678 0.7070678 0.7070678.5 0.640038688 0.503095 0.7070678.5.38098445 50
(4A Formulas (4.9, (4.30 of Theorem 4.4 hold. Formula (4.3 of Theorem 4.5 hold for = _ 0.5, 0.5,. B. In case of α = Numerical calculations of TH-method, formulas (4.7, 9. (5..5 µ α, (4.30 for α =. (B We eamine the speed of convergence of -th power of TH-method in 9.3. The results of the calculations are Table 6. In case of 0 =., numerical calculations of -th power of TH-method are compatible with the theory of Theorem 4.7. (3B For α =, formula (4.7 of Theorem 4.3 becomes + µ =. 3 3 + + ( ( (5..6 Indeed, by calculating the left and right sides of (5..6 for in Table 7 formula (5..6 holds in 9. (4B Formulas (4.9, (4.30 of Theorem 4.4 hold ecept for = _. In this case, according to increases, the value of the right-hand side of (4.30 increases rapidly. Formula (4.3 of Theorem 4.5 holds the eual sign only =. 3 Eample 5.3. Murase s third degree euation f ( = 4 + 48 = 0 (5.3. = (.. Graph of f( is this (Figure. The graph is drawn in Bear Graph of free software. ( For a root of (5.3., condition (4.5 becomes 7 =.4545. (5.3. Table 6. Calculations of TH-method for f( = 3 + = 0, α =. 0 3.9 4 4 4 4 3 Absolute error 3.79385E 3.0898E 3 0 0. 4 4 4 4 3 Absolute error 6.784E 4 7.0543E 5 4.44089E 6 0. 5 4 4 4 4 Absolute error 9.30589E 3.53939E 3 4.44089E 5 4.44089E 6 4 5 6 7 8 9 9.3 3 3 3 3 3 4 4.045E 6 0 3 3 3 3 4 4 4.045E 6 4.44089E 6.045E 5 3 3 4 4 4 5 5 3.38884E 0 9.990E 5 4.47709E 5
Table 7. Calculations of (5..6, µ, (4.30 for f( = 3 + = 0. µ ( Right-hand side of (5..6 µ + f ( f Right-hand side of (4.30 0.79894088 0.456537647 0.7070678.75.548846597 0.68480647 0.456537647 0.7070678.5.548846597 0.7070678 0.7070678 0.7070678 0.085600809 0.4344 0.7070678 0.75 6.95386388 3 0.0068779 0.03745459 0.7070678 0.5 5.4493798 4 0.000487567 0.0095067 0.7070678 0.5 36.56935 5 0 0.000347 0.7070678 0 63.705 6.3568E 05 5.453E 05 0.7070678 0.5 3033.95 7 4.984E 06 9.96485E 06 0.7070678 0.5 70960.004 8.4305E 06.90735E 06 0.7070678 0.75 37078.304 9 3.7676E 07 3.7676E 07 0.7070678 876809.006 0 9.53674E 08 7.6939E 08 0.7070678.5 96890.533.36448E 08.5763E 08 0.7070678.5 44858040.4 Figure. Graph of 3 f = 4 + 48. ( In case of = 0.5,,.5,,.45,.5, we calculate -th power of TH-method. The results are Table 8. In case of 0 =.9, numerical calculations of -th power of TH-method are compatible with Theorem 4.7. (3 Formula (4.7 of Theorem 4.3 becomes 6 6 µ =. 3 3 44 44 + + ( ( (5.3.3 By calculating the left and right sides of (5.3.3 for in Table 9 formula (5.3.3 holds ecept for 0.5 and.5. (4 Formulas (4.9, (4.30 of Theorem 4.4 hold for = 0.5,,.5. Formula (4.3 of Theorem 4.5 holds for =,.5. 5
Table 8. Calculations of TH-method for (5.3.. 0 0.5.5.45.5 0.6 6 5 4 4 5 5 Absolute error.946e 0 0 5 5 4 4 4 4 Absolute error 3.45E 0.5 4 4 3 3 4 4 Absolute error 6.3845E.337E 5 0 0.9 3 3 3 3 3 3 Absolute error 3.4949E 8.3939E 4.045E 6 0 5.83977E 4 9.30367E 4. 3 3 3 3 3 3 Absolute error.9957e 5.543E 4 0 0 6.79456E 4.0880E 3 Table 9. Calculations of (5.3.3, µ, (4.30 for (5.3.. µ ( Right-hand side of (5.3.3 µ + f ( f Right-hand side of (4.30 0.5 0.00005 6.640E 05 0.00087683.6875.865085 0.00087683 0.00087683 0.00087683.5 0.000408 0.00039706 0.00087683 0.35 0.476878 0.0007886 0.00074096 0.00087683 0.375 0.590959.45 0.00074 0.004835 0.00087683 0.99375 0.544983.5 0.0035804 0.00783 0.00087683.065 0.46844 Eample 5.4. A fifth degree euation 5 3 f = + 3= 0 (5.4. 4 f( has no terms of,,, and a root is α =. Graph of f( is Figure 3. Graph is the conve downward and monotonic decreases in < 0, the conve upward and monotonic decreases in 0<, and point (0,3 is a point of inflection. ( Condition (4.5 becomes 6.88. (5.4. ( In case of = _,, 3, 5, 6, 6.8, 7, we calculate -th power of TH-method. The results are Table 0. Notation 5(#DIV/0! denotes that it is #DIV/0! in 5 iterations. In case of 0 =.063, numerical calculations of -th power of TH-method are compatible with Theorem 4.7. There is a noteworthy thing. In case of 0 = 0., number of iterations of Newton s method is times but that 3-th power of TH-method is 5 times only. (3 Formula (4.7 of Theorem 4.3 becomes 3 3 µ =. 3 3 + + (5.4.3 Indeed, by calculating the left and right sides of (5.4.3 for = _,, 3, 5, 6, 6.8, 8, 0 we get Table. Formula (5.4.3 holds for =, 3, 5, 6, 6.8, respectively. 53
Figure 3. Graph of 5 3 f = + 3. Table 0. Calculations of TH-method for f( = 5 3 + 3 = 0. 0 3 5 6 6.8 7 0. 5(#DIV/0! 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 8.86E 4 0 6.75E 5.4439E 4.063 4 4 3 3 3 4 4 Absolute error.0658e 4 0 0.0E 6 Table. Calculations of (5.4.3, µ (, (4.30 for f( = 5 3 + 3 = 0. µ ( Right-hand side of (5.4.3 µ ( + f ( ( f ( Right-hand side of (4.30 0.04007399 0.0374708 0.0374708.6875 0.0374708 0.0374708 0.0374708 3 0.00398 0.06476736 0.0374708 0.35 0.36665974 5 0.03400 0.090696536 0.0374708 0.375 0.6830077 6 0.070503 0.097600434 0.0374708 0.7875 0.4330973 6.8 0.00353783 0.0063684 0.0374708 0.997875 0.3596807 8 0.4306879 0.0737807 0.0374708.4065 0.334455 (4 Formulas (4.9, (4.30 of Theorem 4.4 hold for = and 3. Similarly formulas (4.9, (4.3 of Theorem 4.5 also hold for = and 3. Eample 5.5. Fifth degree euation f( has no terms of 5 4 f = + + = 0 (5.5. 3,,, and a root of (5.5. is α.055967397. Graph of f( is Figure 4. The graph 54
Figure 4. Graph of 5 4 f = + +. becomes minimum at = 0, which is parallel to the -ais in the neighborhood. Net it increases and becomes maimum at =.6. Further, it decreases monotonically from here, and intersects with root α. The graph changes intensely in this way in _ < <.5. ( The formula (4.5 of Theorem 4.7 becomes (5.5.. f (.055967397 0 ie.. about 6.08 (5.5. f.055967397.055967397 (The value of formula (5.5. for = 6.08 is.99999393. ( For = _,, 3, 6, 9,, 5, 6, 7, we calculate -th power of TH-method. The results are Table. For 0 =.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. In particular for the initial value is 0 =.5, the number of iterations of the 9-th power of TH-method is 4, which is etremely small than 54 times of the Newton s method. Therefore, we eamine the state of convergence of the 9-th power of TH-method. 9 Converting f( by = t, following formula is obtained. y g t f t t t The formula of the tangent of the curve of g(t at point ( n, ( n 9 59 49 = = = + + (5.5.3 t g t is the following. 5 8 5 8 y = + 49 59t + t 49 59 n + g t 9tn 9tn 9tn 9tn n (5.5.4 For the initial value is.5 9, we give in Table 3 the calculations of 9-th power of TH-method to converge to 656.3659005 (=.055967397 9 and the tangents. Then we give the graphs of Figure 5 g( and the changes of the tangents. Straight line, and 3 in Figure 5 indicates the tangent to the number of iterations =,, 3, respectively. Point (., (. f is a point of inflection of graph f(. It becomes conve downward in <., minimum at = 0, and parallel to the -ais in the neighborhood of = 0. It becomes conve upward in. <, maimum at =.6. Therefore, choosing to.5 initial value for Newton s method, vibrate, and the number of iterations increase. Graph g(t (g( becomes minimum at t( = 0, but parallel parts to the t(-ais do not eist in the 9 9 neighborhood of this point. Further it becomes conve upward in t <, conve downward in < t, intersects at t = 656.3659005 with t-ais, and close to the shape of a straight line in the neighborhood. Therefore, 55
Table. Calculations of TH-method for f( = 5 + 4 + = 0. 0 3 6 9 0.5 #NUM! 8 Oscillation #NUM! 3 #NUM! #DIV/0! 45 Oscillation #NUM! 9 #NUM!.5 #DIV/0! 54 Oscillation #NUM! 4 #NUM!.65 #DIV/0! 0 7 5 5 5.7 6 8 6 5 4 4.8 7 6 5 4 4 4.85 6 5 5 4 3 4 Absolute error.66674e 0.878E 0.89 5 5 4 4 3 4 Absolute error.5995e 0.8776E 0 5 6 7 3 #NUM! 5 0.5 9 #NUM! 0 6 #NUM! 6.5 4 4 4.65 4 5 5.7 4 5 5.8 4 4 5.85.878E 0 Absolute error 4 4 4.89 Absolute error 0 Table 3. Calculations of 9-th power of TH-method and tangents. 9 9 8 f ( f ( 9 Gradient of tangent Intercept.5 38.44335938 444.34375 0.00735958 3.5.9686983 444.34375 459.99876 0.067048 30.834456 3.97630798 459.99876 656.643436 0.0069343 4.5506838 4.05593049 656.643436 656.3659 0.006895787 4.5659387 5.055967397 656.3659 656.3659005 0.00689579 4.5693 6.055967397 656.3659005 656.3659005 0.00689579 4.5693 9 vibration is only once, become a monotonically increasing seuence, and the number of iterations is reduced. (3 Formula (4.7 of Theorem 4.3 becomes µ α 3 3 3 0α + 4α + 5α + 8α 0α + 4α = 3 3 4 3 4 3 5α + 8α 5α + 8α + + ( α ( α α α. (5.5.5 56
Figure 5. Graph g( and tangents (5.5.4 of g(. Indeed, by calculating the left and right sides of (5.5.5 for = _,, 3, 6, 9,, 5, 6, 7 we get Table 4. Formula (5.5.5 holds for =, 3, 6, 9,, 5, 6. (4 Formulas (4.9, (4.30 of Theorem 4.4 do not hold for = 3. Theorem 4.5 holds as euality for =. Euation (5.4.,(5.5. has only one term which degree is smaller than highest degree, respectively. These euations have the trend that the convergences of TH-methods are etremely fast than that Newton s method. Eample 5.6. f ( = sin = 0 (5.6 f mπ = sin mπ = 0. Because µ ( α = 0 µ ( α ( m 0, of Theorem 4.8 holds, convergence of Newton s method of = is the fastest in other -th power of TH-method. For α = π, = ±, ±, ±3, we calculate -th power of TH-method. The results are Table 5. Eample 5.7. Roots of euation (5.6 are α = mπ (m is an integer, π 3.459654, and n α α 0 ( α 0, ( 3 : natural number f = + = n (5.7. A root of euation (5.7. is α. Because n f n α n = + α, f ( = n( n ( α, f ( α 0 this is also an eample of Theorem 4.8. Particular if n 4 then root α of f ( 0 For n = 3, α =, euation (5.7. is the following. 3 3 =, = becomes multiple root. f = + = 6 + 4 = 0 (5.7. For = _, _,,, 3, we calculate -th power of TH-method, and get Table 6. Eample 5.8. ( The root of (5.8. is ln 0.693478, and 0 f = e = (5.8. ( n 0 f = e. Applying (4.5 of Theorem 4.7, we have + ln.3869436. (5.8. ( For = 0.5,,.5,,.3869436,.5, we calculate -th power of TH-method. However, we calculate the absolute error as ln 0.693478 root. The results are Table 7. For 0 = 0.73, 0.76, -th power of TH-method has better approimate degree than Newton s method in the range of (5.8.. (3 Formula (4.7 of Theorem 4.3 for (5.8. becomes 57
Table 4. Calculations of (5.5.5, µα, (4.30 for f( = 5 + 4 + = 0. µ ( α Right-hand side of (5.5.5 µ ( α + f ( α f Right-hand side of (4.30 0.00786695 0.0000579 0.00968403.663464 4.80053 0.00968403 0.00968403 0.00968403 3 0.057836 0.07049435 0.00968403 0.733653759 0.347746 6 0.00049737 0.0047676 0.00968403 0.33434398 6.97857366 9 5.73089E 07 8.76484E 06 0.00968403 0.065384963 057.45359 3.03503E 08 6.583E 08 0.00968403 0.46490434 497.7 5 4.78E 0 5.530E 0 0.00968403 0.86443686 675435. 6.4749E 0.505E 0 0.00968403 0.997596806 805775.6 7.7569E.4048E 0.00968403.3076996 3845053.3 Table 5. Calculations of TH-method for f( = sin = 0. 0 3 3.7 5 5 4 3 3 4 Absolute error 4.007E 0 4.536E 0.9 4 4 4 3 3 4 Absolute error 4.007E 0 4.394E 0 3. 3 3 3 3 3 3.3 4 3 3 3 4 Table 6. Calculations of TH-method for (5.7.. 0 3.8 4 4 3 4 4.9 4 3 3 3 3 Absolute error 4.9899E 4.44089E 6.453E.3887E 0. 4 3 3 3 3 Absolute error.0953e 0 4.44089E 6 7.34968E 4 3.5439E. 4 4 3 3 4 Absolute error 4.44089E 6.3034E 3 Table 7. Calculations of TH-method for f( = e = 0. 0 0.5.5.3869436.5 0.68 3 3 3 3 3 Absolute error 3.33067E 6 0 0 0 0 0.7 3 3 3 Absolute error.0e 6.744E 0 6.807E.305E 0 0 0.73 3 3 3 3 3 3 Absolute error 9.97646E 3.533E 4.0E 6 0.3037E 4 6.97E 4 0.76 3 3 3 3 3 3 Absolute error.0667e 0.8485E 5.55E 6 5.0703E 5.4367E 7.55507E 58
µ (( ln = ( ( ( + ln ln + ( ln ln + ( ln 3 3. (5.8.3 By calculating the left and right sides of (5.8.3 for in Table 8 we get the numbers there. Formula (5.8.3 holds for ecept for 0.8 and.4. (4 In Table 8, formulas (4.9, (4.30 hold in the range of (5.8. ecept for =.3869436. (4.3 holds in (5.8.. Eample 5.9. ( The root of (5.9. is α =, and f ( = 0. Applying (4.5 we have f ( = ln = 0 (5.9.. (5.9. ( The calculations for TH-method are Table 9. For 0 =.05,.,., -th ( = _, _ 0.5, 0.5 power of TH-method converges better than Newton s method, respectively. (3 Formula (4.7 of Theorem 4.3 for (5.9. is this. µ ( ( + = = 3 3 3 + + + (5.9.3 By calculating the left and right sides of (5.9.3 for in Table 0 we get the numbers in its. In euality (5.9.3 holds for ecept for _.5 and.5. (4 Formulas (4.9, (4.30, (4.3 hold for = _, _ 0.5, 0.5,. Table 8. Calculations of (5.8.3, µα, (4.30 for f( = e = 0. µ ( Right-hand side of (5.8.3 µ ( α + f ( α f Right-hand side of (4.30 0.8 0.490835 0.66779456 0.78885438.88539008.0758677 0.78885438 0.78885438 0.78885438. 0.35355 0.8574954 0.78885438 0.746099 0.963046469.4 0.08086 0.8944063 0.78885438 0.49984 0.9448406.6 0.05705605 0.9867 0.78885438 0.3438975 0.935787.8 0.0964548 0.90767 0.78885438 0.5456033 0.9373064 0.085743 0.939936 0.78885438 0.4469504 0.997630. 0.4075657 0.944954 0.78885438 0.7334049 0.9958655.3869436 0.940845 0.940845 0.78885438 0.9965775.4 0.96738 0.948045 0.78885438.09773057 0.996708 59
Table 9. Calculations of TH-method for f( = ln = 0. 0.5 0.5 0.5.5 0.9 4 3 3 3 3 3 Absolute error.57945e 0 8.54383E 6.45084E 8.99983E 3.9849E 0 0.95 3 3 3 3 3 3 Absolute error.3334e 4.9878E 3.4869E 4.7604E 4 3.685E 3.54754E.05 3 3 3 3 3 3 Absolute error.0476e.049e 3.46549E 4.68754E 4.85993E 3.54754E. 3 3 3 3 3 3 Absolute error.8553e 0 4.498E.93099E 3.77964E 6.88853E 3.9803E 0. 4 4 3 4 4 4 Absolute error.045e 6 0 0.0E 6 5.44009E 5 Table 0. Calculations of (5.9.3, µ (, (4.30 for f( = ln = 0. µ ( Right-hand side of (5.9.3 µ ( + f ( ( f ( Right-hand side of (4.30.5 0.384033 0.5605475 0.35355339.5.38098445 0.35355339 0.35355339 0.35355339 0.5 0.78885438 0.357770876 0.35355339 0.5 0.988769 0.5 0.78885438 0.357770876 0.35355339 0.5 0.988769 0.35355339 0.35355339 0.35355339.5 0.384033 0.5605475 0.35355339.5.38098445 Acnowledgements Dr. Tamotsu Tsuchiura (93-05, professor emeritus of Tohou University and Dr. Mitsuo Morimoto (professor emeritus of Sophia University gave hints to me. I am deeply grateful to them. References [] Murase, Y. (673 Sanpoufutsudanai. Nishida, T., Ed., Kenseisha Co., Ltd., Toyo. (In Japanese [] Horiguchi, S. (04 On Relations between the General Recurrence Formula of the Etension of Murase-Newton s Method (the Etension of Tsuchiura-Horiguchi s Method and Horner s Method. Applied Mathematics, 5, 777-783. http://d.doi.org/0.436/am.04.54074 [3] Suzui, T. (004 Wasan no Seiritsu. Kouseisha Kouseiau Co., Ltd., Toyo. (In Japanese [4] Tsuchiura, T. (0 Calculation Methods of p-th Root by the Ideas That the Mathematicians of Wasan Thin about. The Bulletin of Wasan Institute, 3, 0-6. (In Japanese [5] Nagasaa, H. (980 Computer and Numerical Calculations. Asaura Publishing Co., Ltd., Toyo. (In Japanese 60