- Email: [email protected]

Xiao-Diao Chen, Jiaer Shi, Weiyin Ma

PII: DOI: Reference:

S0893-9659(16)30380-9 http://dx.doi.org/10.1016/j.aml.2016.12.013 AML 5150

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Applied Mathematics Letters

Received date : 28 October 2016 Revised date : 22 December 2016 Accepted date : 22 December 2016 Please cite this article as: X. Chen, et al., A fast and robust method for computing real roots of nonlinear equations, Appl. Math. Lett. (2016), http://dx.doi.org/10.1016/j.aml.2016.12.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A fast and robust method for computing real roots of nonlinear equations Xiao-Diao Chen a,b, Jiaer Shi a, Weiyin Ma b a School

of Computer, Hangzhou Dianzi University, Hangzhou, China.

b Department

of MBE, City University of Hong Kong, Hong Kong, China

Abstract The root-finding problem of a univariate nonlinear equation is a fundamental and long-studied problem, and has wide applications in mathematics and engineering computation. This paper presents a fast and robust method for computing the simple root of a nonlinear equation within an interval. It turns the root-finding problem of a nonlinear equation into the solution of a set of linear equations, and explicit formulae are also provided to obtain the solution in a progressive manner. The method avoids the computation of derivatives, and achieves the convergence order 2n−1 by using n evaluations of the function, which is optimal according to Kung and Traub’s conjecture. Comparing with the prevailing Newton’s methods, it can ensure the convergence to the simple root within the given interval. Numerical examples show that the performance of the derived method is better than those of the prevailing methods. Key words: Root finding; Non-linear equations; Linear equations; Progressive formula; Optimal convergence order.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 Introduction Finding the roots of a nonlinear equation f (t) = 0 is a common and important problem in mathematics and engineering computation. Many modified iterative methods have been developed to improve the local order of convergence, including Newton, Halley or Ostrowski’s methods. In an iterative step, the convergence order p can be improved by increasing the number n of functional evaluations (FE). The balance between p and n is measured by using p1/n , which is called an efficiency index [1, 6, 7, 10–13, 16–18]. It is conjectured that the order of convergence of any multi-point method cannot exceed the optimal bound 2n−1 [8]. Some classical methods, e.g., Newton’s method, Newton-Secant method and Ostrowski’s method, have efficiency indices of 21/2 ≈ 1.414, 31/3 ≈ 1.442 and 41/3 ≈ 1.587, respectively, and they usually need evaluations of its derivatives. In [9], Li, Mu, Ma and Wang presented a method of sixteen convergence order, whose efficiency index is 161/6 ≈ 1.587. Email addresses: [email protected] (Xiao-Diao Chen), [email protected] (Weiyin Ma).

Preprint submitted to Elsevier

22 December 2016

35

Some optimal methods with eighth convergence order have also been developed, whose efficiency index is 81/4 ≈ 1.682 [4, 14, 15]. For cases having one root within an interval, the above methods still start from an initial value from one side, and their convergence is sensitive to the selection of the initial value. In some worst cases, the iterative methods cannot converge to the proper simple root, even with a good initial value [3] (see also Example 3 in Section 3 for more details). Chen et. al. provided an efficient method based on progressive interpolation, which starts from an interval bounding the unique simple root, and achieves convergence order 3 · 2n−3 with n FEs and no derivative evaluations [3]. It ensures that the roots of progressive interpolation polynomials are within the given interval, and converge to the simple root. This paper focuses on how to compute the simple root t⋆ of a nonlinear equation within an interval [a, b]. It turns the root-finding problem of the given nonlinear equation f (t) into those of a sequence of linear equations. It ensures that each linear equation li (t) has a root ti ∈ [a, b] with explicit formulae, where ti is well-approximated to t⋆ in a progressive way, i = 3, 4, · · · . It can ensure the convergence to the simple root with convergence order 2n−1 by using n FEs without derivative evaluations, which is higher than that of the method in [3] and is optimal in the conjecture of [8] (see also Table 1). Numerical examples show that the performance of the proposed method is better than those of prevailing methods.

36

2

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

37 38 39 40 41 42 43 44 45 46 47 48

The main result

Note that a root t⋆ of f (t) should be a simple root of f (t)/f ′ (t). Without loss of generality, we suppose that the simple root t⋆ is the unique root of f (t) within [a, b], and h = b − a. In cases with only one given initial value x0 , one can obtain x1 such that |x1 −t⋆ | < |x0 −x⋆ |/2 by using some iterative methods, such as the Newton’s method, and can thus obtain an interval bounding t⋆ . The details are as follows. If f (x0 ) · f (x1 ) < 0, x0 and x1 already bound the simple root t⋆ ; otherwise, we have that 2x1 − x0 and x1 bound t⋆ . In this paper, we discuss how to compute the simple root t⋆ ∈ [a, b] of f (t), where f (a) · f (b) < 0. We want to find a sequence of linear equations which have roots well-approximating to t⋆ . Let t1 = a and t2 = b. Let the first linear equation l3 (t) be the one interpolating f (t) at two points t1 and t2 , which is trivial to compute its root t3 . Let t − ti li (t) = i−1 , i = 3, 4, . . . , n, (1) P αi,j tj−2 j=2

49 50 51 52 53 54 55

where the i − 1 unknowns, i.e., ti and αi,j , j = 2, 3, · · · , i − 1, are determined li (tj ) = f (tj ), j = 1, 2, . . . , i − 1. (2) by

Note that the denominator of li (t) should not be zero within [a, b], and li (t) = 0 is equivalent to a linear equation with root ti . On the other hand, combining the assumption that f (a) · f (b) < 0 and the constraints that li (a) = f (a) and li (b) = f (b), li (t) must have the root ti . We introduce Theorem 3.5.1 in Page 67, Chapter 3.5 of [5] as follows. 2

56 57 58

59

60 61

Theorem 1. Let w0 , w1 , · · · , wr be r+1 distinct points in [a, b], and n0 , · · · , nr be r+1 integers ≥ 0. Let N = n0 + · · · + nr + r. Suppose that g(t) is a polynomial of degree N such that g (i) (wj ) = f (i) (wj ), i = 0, · · · , nj , j = 0, · · · , r. Then there exists ξ1 (t) ∈ [a, b] such that r f (N +1) (ξ1 (t)) Y f (t) − g(t) = (t − wi )ni . (N + 1)! i=0

From Eq. (2), tj is a root of the polynomial Ei (t) = f (t)−li (t), j = 1, 2, · · · , i− 1. Combining with Theorem 1, wei−1 have that Y (3) Ei (t) = Fi (t) · (t − tj ), i = 3, 4, · · · , j=1

62 63

where Fi (t) is a bounded rational polynomial within [a, b]. From Eq. (3), we have that |t⋆ − ti | = O(|li (t⋆ ) − li (ti )|) = O(|li (t⋆ )|) (4) i−1 Q = O(|li (t⋆ ) − f (t⋆ )|) = Ei (t⋆ ) = O(| (t⋆ − tj )|), i = 3, 4, · · · , j=1

64

65 66 67 68

69 70

which leads to

), i = 3, 4, · · · .

(5)

Remark 1. In principle, the selection of t1 and t2 can be optimized by using the methods in [17, 18], which can lead to a better approximation effect to the root t⋆ . Eq. (2) is linear in i − 1 unknowns canbibe Mi and · vi = , turned into a matrix form (6) where Mi is a matrix of size (i − 1) × (i − 1), both vi and bi are vectors of i−3 size i − 1, which 1 f (t1 ) can f (tbe · · · f (tas αi,2 t1 1 ) · tdenoted 1 1 ) · t1 Mi =

71

i−2

|t⋆ − ti | = O(h2

f (t2 ) . ..

f (t2 ) · t2 . ..

···

···

f (t2 ) · ti−3 2 . ..

1 . , vi ..

=

f (ti−1 ) f (ti−1 ) · ti−1 · · · f (ti−1 ) · ti−3 i−1 1

. .. , αi,i−1 ti

Combining Eq. (6) with Cramer’s thatt1 f (t1 ) f (t1rule, ) · t1 we · · · obtain f (t1 ) · ti−3 1 ti =

f (t2 ) · t2 · · · f (t2 ) · ti−3 t2 f (t2 ) 2 . .. .. .. . . . ··· . . f (ti−1 ) f (ti−1 ) · ti−1 · · · f (ti−1 ) · ti−3 ti−1 i−1 . f (t1 ) f (t1 ) · t1 · · · f (t1 ) · ti−3 1 1 f (t2 ) · t2 · · · f (t2 ) · ti−3 1 f (t2 ) 2 . .. .. .. . . . ··· . . f (ti−1 ) f (ti−1 ) · ti−1 · · · f (ti−1 ) · ti−3 1

bi =

t2 . ..

ti−1

.

(7)

i−1

72 73

To expand the two determinants of Eq. (7) in their last columns, it can be verified that i−1 i−1 i−1 Q P P ( Ai,j tj ) · ( f (tk )) Ai,j tj j=1 j=1 k=1 ti = i−1 (8) = i−1 , i ≥ 3, i−1 Q P P ( Ai,j ) · ( f (tk )) Ai,j j=1

j=1

k=1

3

74

where Di,j is the Vandermonde of the matrix such that 1 t1determinant · · · ti−3 1 ¯ i,j ) = Di,j = det(M

. . . 1 det 1 . . .

.. .

···

tj−1 · · · tj+1 · · · .. . ···

i−3 tj−1 ti−3 j−1 .. . .. .

=

1 ti−1 · · · ti−3 i−1

75

and Ai,j = (−1)j ·

76

result as follows.

77 78 79 80 81 82 83

84

85 86 87 88 89 90 91 92 93 94 95 96 97 98

1,

Q

i = 3, (tk − tr ), i > 3,

1 ≤ k < r ≤ i − 1, k, r 6= j,

Di,j . Combining Eq. (5) with Eq. (8), we have the main f (tj )

Theorem 2. There are n functional evaluations for computing tn+1 from Formula (8), and it achieves the convergence rate 2n−1 , which is optimal in the Kung’s conjecture. Proof. From Formula (8), there are n functional evaluations f (tj ), j = 1, 2, · · · n+1−2 , n, for computing tn+1 . From Eq. (5), we have that |t⋆ − tn+1 | = O(h2 )= n−1 O(h2 ), and the corresponding convergence rate is 2n−1 , which is directly from its definition. 3

Numerical results and discussions

Let M1 , M2 , M3 , M4 and M5 denote the methods in [9], [19], [15], [3] and the method in this paper, respectively, where q in [19] is set as 2. We show several examples to illustrate M5 and its comparisons with other four methods. In principle, M1 , M2 and M3 need 6, 6 and 4 FEs in each iterative step, while M4 and M5 can stop at arbitrary n ≥ 3 FEs. We utilize 12 FEs for comparing the above five methods. Firstly, we compare convergence order (CO) and efficiency index (EI) of different methods. Table 1 shows that both M2 and M5 have the best EI among the five methods. Example 1 (Ref. [9]): Given f1 (t) = (t − 1)3 − 1 with exact root t⋆ = 2, and initial value is t1 = 1.9. With the Newton’s method, we obtain that t2 = t1 − f1 (t1 )/f1′ (t1 ) = 2.011, and t⋆ is bounded by t1 and t2 . So the initial interval of both M4 and M5 is set as [t1 , t2 ] = [1.9, 2.011]. The comparison results are shown in Table 2. It shows that M5 achieves a better performance than that of M4 . Table 1. Efficiency indexes and convergence orders between different methods M1 [9]

M2 [19]

M3 [15]

CO(n = 12)

28

211

29

3·

EI

1.587

1.887

1.681

1.843 2

99 100 101 102

M4 [3] 29

M5 211 1.887

Example 2 (Ref. [3]): Given f2 (t) = 10150−5x − 1 with exact root t⋆ ≈ 5.477 and initial value t1 = 5.494. Similarly, we obtain t2 = t1 −f2 (t1 )/f2′ (t1 ) = 5.436, and the initial interval [t2 , t1 ] bounds t⋆ . The corresponding results are shown in Table 2. As shown in Table 2, both M1 and M3 diverge (denoted by “/”), 4

103 104

M2 doesn’t converge within 80 iterative processes, while M4 and M5 converge, and M5 achieves a slightly better result than M4 . Table 2. Comparisons of Examples 1 and 2 between difference methods with 12 FEs

105 106

Example

M1 [9]

M2 [19]

M3 [15]

M4 [3]

M5

Exam. 1

4.2e − 18

2.5e − 2695

5.3e − 540

2.5e − 2607

2e − 3305

Exam. 2

/

2.57

/

1e − 371

8e − 476

Example 3 (extracted from [9]): We have tested the following eight smooth functions 2 f3 (t) = e−t +t+3 − t + 2, f4 (t) = et − 1, f5 (t) = t3 − 10, f (t) = et2 +7t−30 − 1,

t⋆ = 2.490, x0 = 2.3, t⋆ = 0,

6

107

f7 (t) = t5 + t − 104 , √ f8 (t) = t − 1/t − 3, f9 (t) = et + t − 20, f (t) = ln(t) + √t − 5,

109

= 2.154, x0 = 2,

t⋆

= 3,

x0 = 2.97,

t⋆ = 6.308, x0 = 6, t⋆ = 9.633, x0 = 9.4, t⋆ = 2.842, x0 = 2.5, t⋆ = 8.309, x0 = 7.9.

10

108

x0 = 0.14,

t⋆

The corresponding results are shown in Table 3. It shows that M5 can achieve a better performance than those of other four methods. Table 3. Errors of Example 3 between different methods with 12 FEs

110 111 112 113 114 115 116

Method

f3 (t)

f4 (t)

f5 (t)

f6 (t)

f7 (t)

f8 (t)

f9 (t)

f10 (t)

M1

2e-188

1e-314

2e-322

2e-185

2e-274

4e-520

7e-208

3e-428

M2

1e-1481

6e-2485

5e-1339

1e-1246

3e-506

3e-4101

1e-206

1e-3449

M3

5e-364

3e-641

5e-615

1e-403

2e-544

2e-1094

3e-438

3e-909

M4

2e-1638

3e-2749

9e-2851

2e-2044

6e-2622

7e-4301

1e-2195

2e-3622

M5

4e-1962

1e-3452

2e-3745

2e-2793

1e-3495

2e-4198

3e-2966

3e-3585

1 Example 4. We have also tested the oscillating function f11 (t) = cos( ) + t et − 2.1 (with three roots t⋆0 ≈ 0.1506, t⋆1 ≈ 0.1704 and t⋆2 ≈ 0.6863) and 1 f12 (t) = cos( ) + et − 2.5 (with one root t⋆3 ≈ 0.7890) within [0.01,1.1], see t also Fig. 1. The comparison results are shown in Table 4, where t1 (t⋆i ) denotes the initial value t1 and its closest root t⋆i . It shows that M1 , M2 and M3 may converge to a wrong root, M2 converges very slowly for the f12 case, and M5 work much better and more robust than M1 , M2 , M3 and M4 . Table 4. Errors of Example 4 between difference methods with 12 FEs Exam

f11 (t)

f12 (t)

0.13(t⋆0 )

0.1596(t⋆0 )

0.2 (t⋆1 )

0.22 (t⋆1 )

M1

3e-104

7e-1(t⋆2 )

1e-105

5e-85

3e-295

7e-90

8e-113

M2

6e-789(t⋆2 )

3e-528(t⋆1 )

2e-1138

2e-368(t⋆2 )

8e-2079

1.03

6e-1

M3

4e-181

5e-53(t⋆1 )

1e-274

3e-34(t⋆0 )

4e-617

2e-190

5e-74

M4

5e-1194

4e-89

1e-1084

1e-895

6e-2218

5e-903

1e-913

M5

1e-559

1.4e-502

4e-1261

5e-1026

2e-3690

6e-1413

2e-1361

5

0.78(t⋆2 )

0.1065(t⋆3 )

0.15958(t⋆3 )

(a)

(b)

Fig. 1. Plots of oscillating functions (a) f11 (t); and (b) f12 (t). 117

4

Conclusions

124

This paper presents an improved method of the one in [3]. It achieves the optimal convergence order 2n−1 by using n FEs and no derivative evaluations, which is much higher than 3/4 · 2n−1 of the method in [3]. Different from prevailing iterative methods starting from an initial value, it starts from an interval bounding the simple root and ensures the convergence. Numerical examples show that the new method can achieve better performance and better approximation effect than those of M1 , M2 , M3 and M4 .

125

Acknowledgement

118 119 120 121 122 123

127

This research was partially supported by City University of Hong Kong (SRG 7004452), the National Science Foundation of China (61672009, 61370218).

128

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133 134

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135 136 137

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