On some modified families of multipoint iterative ... - Semantic Scholar
Applied Mathematics and Computation 218 (2012) 7382–7394
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On some modified families of multipoint iterative methods for multiple roots of nonlinear equations Sanjeev Kumar a,⇑, V. Kanwar b,⇑, Sukhjit Singh c a
Department of Mathematics, Maharishi Markandeshwar University, Sadopur, Ambala 134 007, Haryana, India University Institute of Engineering and Technology, Panjab University, Chandigarh 160 014, India c Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148 106, Punjab, India b
a r t i c l e
i n f o
Keywords: Newton’s method Modified Newton’s method Schröder’s method Multiple roots Power means Order of convergence
a b s t r a c t In this paper, we propose a new one-parameter family of Schröder’s method for finding the multiple roots of nonlinear equations numerically. Further, we derive many new cubically convergent families of Schröder-type methods. Proposed families are derived from the modified Newton’s method for multiple roots and one-parameter family of Schröder’s method. Furthermore, we introduce new families of third-order multipoint iterative methods for multiple roots free from second-order derivative by semi discrete modifications of the above proposed methods. One of the families requires two evaluations of the function and one evaluation of its first-order derivative and the other family requires one evaluation of the function and two evaluations of its first-order derivative per iteration. Numerical examples are also presented to demonstrate the performance of proposed iterative methods. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Solving nonlinear equations is a common and important problem in science and engineering [1–25]. Analytical methods for solving such equations are almost non-existent and therefore, it is only possible to obtain approximate solutions by relying on numerical methods based on iterative procedures. With the advancement of computers, the problem of solving nonlinear equations by numerical methods gained more importance than before. In this paper, we consider iterative methods for finding a multiple root rm of multiplicity m > 1, i.e. f ðrm Þ ¼ f 0 ðrm Þ ¼ ¼ f ðm1Þ ðrm Þ ¼ 0 and f ðmÞ ðr m Þ – 0, of a nonlinear equation:
f ðxÞ ¼ 0;
ð1:1Þ
where f : I R ! R be a nonlinear smooth and continuous function on an open interval I. A lot of research work [1–9,11,13–22] has been carried out to compute the roots of nonlinear equation (1.1). The design of iterative methods for solving such equations is very important and interesting tasks in computational mathematics. Newton’s method for multiple roots appears in the work of Schröder [4], which is given as:
xnþ1 ¼ xn
f ðxn Þf 0 ðxn Þ : f 02 ðxn Þ f ðxn Þf 00 ðxn Þ
⇑ Corresponding authors. E-mail addresses: [email protected] (S. Kumar), [email protected] (V. Kanwar). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.12.081
ð1:2Þ
S. Kumar et al. / Applied Mathematics and Computation 218 (2012) 7382–7394
7383
This method has second-order convergence, including the case of multiple roots. It may be obtained by applying Newton’s method to the function uðxÞ ¼ ff0ðxðxnnÞÞ, which has simple roots in each multiple root of f(x). Another quadratically convergent method for multiple roots is the modified Newton’s method [8]:
xnþ1 ¼ xn m
f ðxn Þ : f 0 ðxn Þ
ð1:3Þ
More recently, Kumar et al. [10] have developed a family of Newton’s method for finding the simple roots of a nonlinear equation (1.1) and is given by
xnþ1 ¼ xn
f 0 ðxn Þ
2f ðxn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; f 02 ðxn Þ þ 4a2 f 2 ðxn Þ
ð1:4Þ
where a 2 R. In (1.4), the sign in the denominator should be chosen so that the denominator is the largest in magnitude. The beauty of this method is that it converges quadratically and moreover, has the same error equation as Newton’s method. Therefore, this method is an efficient alternative to the classical Newton’s method. In this paper, we further extend our work [10] and construct a modified family of Schröder’s method. We propose new cubically convergent families of Schröder-type methods based on power means, when multiplicity m of the root is known in advance. Thereafter by discretizing second-order derivative, we obtain many new second derivative free cubically convergent families of multipoint iterative methods for multiple roots. It is also shown that some well-known methods like super-Halley method [5], Halley’s method [1–7], Ostrowski’s square-root method [1–7], Newton–Secant method [11], etc. can be regarded as particular cases of the proposed family. 2. Review of definition of various means For a given finite real number c, the cth- power mean Pc [12] of positive scalars a and b, is defined as follows:
c c 1 a þb c Pc ¼ : 2
ð2:1Þ
It is easy to see that:
For c ¼ 1; For c ¼
1 ; 2
2ab ðHarmonic meanÞ; P 1 ¼ aþb (pffiffiffi pffiffiffi)2 aþ b P1 ¼ ; 2 2 aþb ðArithmetic meanÞ; s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
For c ¼ 1;
P1 ¼
For c ¼ 2;
P2 ¼
For c ! 0;
we have P 0 ¼ limc!0 P c ¼
a2 þ b 2
ð2:2Þ ð2:3Þ ð2:4Þ
2
ðRoot mean squareÞ;
ð2:5Þ
and
pffiffiffiffiffiffi ab ðGeometric meanÞ:
ð2:6Þ
For given positive scalars a and b, some other well-known means are defined as:
pffiffiffiffiffiffi ab þ b ðHeronian meanÞ; Ho M ¼ 3 2 2 a þb CoM ¼ ðContra-harmonic meanÞ; aþb 2 2 a2 þ ab þ b ðCentroidal meanÞ; CeM ¼ 3ð a þ b Þ aþ
ð2:7Þ ð2:8Þ ð2:9Þ
and
Lo M ¼
ab logðaÞ logðbÞ
ðLogarithmic meanÞ:
ð2:10Þ
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3. Development of the iterative methods 3.1. Modified family of Schröder’s method The family of Newton’s method (1.4) for finding simple root of a nonlinear equation (1.1) can be written as:
xnþ1 ¼ xn
2f ðxn Þ #: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi f ðxn Þ 2 0 f ðxn Þ 1 1 þ 4a f 0 ðxn Þ "
ð3:1Þ
Now consider the factor:
4a2
2 f ðxn Þ : f 0 ðxn Þ
ð3:2Þ
Since the scaling parameter a appears in the numerator of (3.2), it is clear that there exits some real values of a such that:
2 2 f ðxn Þ 1; 4a f 0 ðxn Þ
ð3:3Þ
holds. With this assumption, the binomial theorem is applicable in Eq. (3.1) and one can get the following formula free from square root term as:
xnþ1 ¼ xn
f ðxn Þf 0 ðxn Þ : 2 2 n Þ þ a f ðxn Þ
ð3:4Þ
f 02 ðx
Further, applying the well known Newton’s method:
xnþ1 ¼ xn
f ðxn Þ ; f 0 ðxn Þ
ð3:5Þ 0
n Þf ðxn Þ to the modified function uf ðxÞ ¼ f 02 ðxf ðxn Þþ a2 f 2 ðxn Þ of formula (3.4), we obtain:
xnþ1 ¼ xn Hðxn Þ
f ðxn Þf 0 ðxn Þ ; f 02 ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:6Þ
where
Hðxn Þ ¼
f 02 ðxn Þ þ a2 f 2 ðxn Þ : f 02 ðxn Þ a2 f 2 ðxn Þ
ð3:7Þ
This is a one-parameter modified family of Schröder’s method [4] for an equation having multiple root of multiplicity m > 1 unknown. It is interesting to note that by ignoring the term a, method (3.6) reduces to classical Schröder’s method. 3.2. Schröder-type cubically convergent iterative methods Using the modified Newton’s formulae (1.3) and (3.6), one can obtain:
xnþ1 ¼ xn
1 f ðxn Þ f ðxn Þf 0 ðxn Þ m 0 þ Hðxn Þ 02 : 2 f ðxn Þ f ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:8Þ
This is a new one-parameter family of super-Halley type methods for the case of multiple roots of multiplicity m > 1 known. It is interesting to note that by ignoring the term a, method (3.8) reduces to super-Halley method [5] for the case of multiple roots. Method (3.8) can further be rewritten as:
xnþ1
n o3 2 f 02 ðxn Þ f ðxn Þ 4m þ Hðxn Þ f 02 ðxn Þf ðxn Þf 00 ðxn Þ 5 : ¼ xn 0 2 f ðxn Þ
Now we shall generalize the formula (3.9) by cth-power mean. For this, we take:
a ¼ m and b ¼
Hðxn Þ
f 02 ðxn Þ : f 02 ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:9Þ
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Approximating the functions in (3.9) as follows, we have:
xnþ1
2 n oc 31c f 02 ðxn Þ c f ðxn Þ 6m þ Hðxn Þ f 02 ðxn Þf ðxn Þf 00 ðxn Þ 7 ¼ xn 0 4 5: f ðxn Þ 2
ð3:10Þ
Suppose that initial guess xn a is reasonably good approximation to the multiple roots of multiplicity m > 1 so that the quantity:
f ðxn Þf 00 ðxn Þ f 02 ðx Þ 1: n
ð3:11Þ
It can be seen that the quantities a and b are already positive in view of (3.3) and (3.11). This family may be called the cthpower mean iterative family of Schröder-type methods for multiple roots. Special cases: It is interesting to note that for different values of c, various well-known methods can be deduced from formula (3.10) as follows: (i) For c = 1 (arithmetic mean), formula (3.10) corresponds to cubically convergent super-Halley type methods [5] for multiple roots:
xnþ1 ¼ xn
1 f ðxn Þ f 02 ðxn Þ m þ Hðxn Þ 02 : 0 00 2 f ðxn Þ f ðxn Þ f ðxn Þf ðxn Þ
ð3:12Þ
(ii) For c = 1 (harmonic mean), formula (3.10) corresponds to the well-known Halley’s type methods [1–7] for multiple roots:
xnþ1 ¼ xn
2mHðxn Þf ðxn Þf 0 ðxn Þ : m½f 02 ðxn Þ f ðxn Þf 00 ðxn Þ þ Hðxn Þf 02 ðxn Þ
ð3:13Þ
(iii) For c ¼ 12, formula (3.10) reduces to a new method for multiple roots given by
xnþ1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ffi#2 1 f ðxn Þ pffiffiffiffiffi f 02 ðxn Þ mþ Hðxn Þ 02 ¼ xn : 4 f 0 ðxn Þ f ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:14Þ
(iv) For c = 2 (root-mean-square), formula (3.10) reduces to a new method for multiple roots given by
xnþ1
1 f ðxn Þ ¼ xn pffiffiffi 0 2 f ðxn Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f 02 ðxn Þ : m2 þ Hðxn Þ 02 f ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:15Þ
(v) For c ! 0 (geometric mean), formula (3.10) reduces to the well-known Ostrowski’s square-root type methods [1–7] given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m : xnþ1 ¼ xn f ðxn Þ Hðxn Þ 02 f ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:16Þ
It is interesting to note that by ignoring the term a, methods (3.12), (3.13) and (3.16) reduce to super-Halley, Halley’s and Ostrowski’s square-root methods for multiple roots. In addition to this, if m = 1 the methods (3.12), (3.13) and (3.16) reduce to super-Halley, Halley’s and Ostrowski’s square-root methods for simple roots [1–7]. Some other new cubically convergent iterative methods based on Heronian mean, contra-harmonic mean, centroidal mean, logarithmic mean, etc. can also be obtained from formula (3.10) respectively. 4. Convergence analysis Now, we shall present the mathematical proof for the order of convergence of the proposed iterative family (3.6) and (3.10). Theorem 4.1. Let f : I # R ! R be a sufficiently differentiable function defined on an open interval I, enclosing a multiple root of f (x), say x ¼ r m 2 I with multiplicity m > 1. Assume that initial guess x ¼ x0 is sufficiently close to rm , then for a 2 R, the iteration scheme (3.6) has quadratic order of convergence and satisfies the following error equation:
enþ1 ¼
C2 2 e þ Oðe3n Þ: m n
ð4:1Þ
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Proof. Since f(x) is sufficiently differentiable function, therefore, expanding f ðxn Þ about x ¼ r m by Taylor’s expansion and using f ðrm Þ ¼ f 0 ðr m Þ ¼ f ðm1Þ ðr m Þ ¼ 0 and f ðmÞ ðrm Þ – 0 (a condition for x ¼ r m to be root of multiplicity m), we have:
f ðxn Þ ¼
f ðmÞ ðrm Þ m en 1 þ C 2 en þ C 3 e2n þ Oðe3n Þ ; m!
ð4:2Þ
f ðmþk1Þ ðr
mÞ where en ¼ xn r m and C k ¼ ðmþ1Þðmþ2Þ...ðmþk1Þf ðmÞ ðr Þ ; k ¼ 2; 3; . . .. m
Similarly for f 0 ðxn Þ and f 00 ðxn Þ, it may be shown that:
f 0 ðxn Þ ¼
f ðmÞ ðrm Þ m1 ðm þ 1Þ ðm þ 2Þ en C 2 en þ C 3 e2n þ Oðe3n Þ ; 1þ ðm 1Þ! m m
ð4:3Þ
f 00 ðxn Þ ¼
f ðmÞ ðr m Þ ðm2Þ mþ1 ðm þ 1Þðm þ 2Þ en C 2 en þ C 3 e2n þ Oðe3n Þ : 1þ ðm 2Þ! m1 mðm 1Þ
ð4:4Þ
and
From (4.2)–(4.4), we obtain:
" # f ðxn Þ en C2 fðm þ 1Þ2 C 2 2mC 3 g 2 3 u¼ 0 en þ Oðen Þ ; ¼ 1 en þ f ðxn Þ m m2 m
ð4:5Þ
and
" # f 00 ðxn Þ ðm 1Þ ðm þ 1Þ f2mðm þ 2ÞC 3 ðm þ 1Þ2 C 22 g 2 3 C 2 en þ 1þ ¼ en þ Oðen Þ : f ðxn Þ en mðm 1Þ m2 ðm 1Þ
ð4:6Þ
Further, Eq. (3.6) can be rewritten as:
xnþ1 ¼ xn
1 1 2 2 f ðxn Þ f ðxn Þf 00 ðxn Þ 2 f ðxn Þ 2 f ðxn Þ : 1 a 1 þ a 1 f 0 ðxn Þ f 02 ðxn Þ f 02 ðxn Þ f 02 ðxn Þ
ð4:7Þ
From (4.5) and (4.6), we get:
1 f ðxn Þf 00 ðxn Þ 1 2 e2n þ Oðe3n Þ; 1 ¼ m þ 2C e þ 6C þ C 3 þ 2 n 3 2 f 02 ðxn Þ m f 2 ðxn Þ a2 1 þ a2 02 ¼ 1 þ 2 e2n þ Oðe3n Þ f ðxn Þ m
ð4:8Þ ð4:9Þ
and
1 f 2 ðxn Þ a2 1 a2 02 ¼ 1 þ 2 e2n þ Oðe3n Þ: f ðxn Þ m
ð4:10Þ
Substituting (4.5), (4.8)–(4.10) in Eq. (4.7) and simplifying, we finally obtain:
enþ1 ¼
C2 2 e þ Oðe3n Þ: m n
ð4:11Þ
This completes the proof of the theorem. h Theorem 4.2. Let f : I # R ! R be a sufficiently differentiable function defined on an open interval I, enclosing a multiple root of f(x), say x ¼ r m 2 I with multiplicity m > 1. Assume that initial guess x ¼ x0 is sufficiently close to rm , then for a; c 2 R, the iteration scheme (3.10) has cubic order of convergence and satisfies the following error equation:
h enþ1 ¼
mðC 22 2C 3 Þ ðcC 22 þ 2a2 Þ 2m2
i e3n þ Oðe4n Þ:
ð4:12Þ
Proof. Further, formula (3.10) can be rewritten as:
1c
xnþ1 ¼ xn 2
" c c c #1c 2 2 f ðxn Þ f ðxn Þf 00 ðxn Þ c 2 f ðxn Þ 2 f ðxn Þ 1 a 02 1 : m þ 1 þ a 02 f 0 ðxn Þ f ðxn Þ f ðxn Þ f 02 ðxn Þ
ð4:13Þ
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Now using (4.5) and (4.6), we obtain:
c f 2 ðxn Þ a2 c 2C 2 a2 c 3 1 þ a2 02 ¼ 1 þ 2 e2n en þ Oðe4n Þ; f ðxn Þ m m3 c f 2 ðxn Þ a2 c 2C 2 a2 c 3 ¼ 1 þ 2 e2n en þ O e4n ; 1 a2 02 f ðxn Þ m m3
ð4:14Þ ð4:15Þ
and
c i f ðxn Þf 00 ðxn Þ 2cC 2 c h 1 ¼ mc 1 þ en þ 2 6mC 3 3ðm þ 1ÞC 22 þ 2ðc þ 1ÞC 22 e2n þ Oðe3n Þ : 02 f ðxn Þ m m
ð4:16Þ
Substituting (4.5), (4.14)–(4.16) in Eq. (4.13) and simplifying, we finally obtain:
h enþ1 ¼
mðC 22 2C 3 Þ ðcC 22 þ 2a2 Þ 2m2
i
e3n þ O e4n :
ð4:17Þ
This completes the proof of the theorem. h
5. Generalized families of multipoint iterative methods The main difficulty associated with recently proposed third-order iterative methods is the evaluation of second-order derivative. In recent years, many modifications of Newton’s method for multiple roots have been proposed by Kim and Lee [11], Victory and Neta [13], Dong [14,15], Neta [17,18], Chun et al. [19], Li et al. [20,21], Neta and Johnson [22] and the references cited there in. All these modifications are targeted at increasing the local order of convergence with the view of increasing their efficiency index [1]. Some of these multipoint iterative methods are of order three [11,13–15,17,19], while others are of order four [20–22]. The fourth order methods proposed by Li et al. in [20] and last two formulae in [21] have optimal order of convergence [25], since they require three evaluations per step, namely one evaluation of function and two evaluations of the first-order derivative. All these modifications of Newton’s method require the prior knowledge of multiplicity m. Here, we also intend to develop and unify general class of multipoint iterative methods for multiple roots. The main idea of the proposed methods lies in the discretization of second-order derivative involved in the cth-power mean family of Schröder-type iterative methods (3.10). In this section, we have derived two families by discretizing second-order derivative involved in the family (3.10). 5.1. First family Expanding the function f ðxn huÞ where h – 0 2 R, but finite, about the point x ¼ xn with f ðxn Þ – 0, we have: 0
f ðxn huÞ ¼ f ðxn Þ huf ðxn Þ þ Let us take u ¼
f ðxn Þ f 0 ðxn Þ
f ðxn Þf 00 ðxn Þ
h2 u2 00 f ðxn Þ þ Oðu3 Þ: 2
ð5:1Þ
and inserting this into equation (5.1), we obtain:
2f 02 ðxn Þ h2 f ðxn Þ
ff ðxn huÞ ð1 hÞf ðxn Þg:
ð5:2Þ
Using this approximate value of f ðxn Þf 00 ðxn Þ into the formula (3.10), we have:
xnþ1 ¼ xn
0c 1=c f ðxn Þ a0c þ b ; f 0 ðxn Þ 2
ð5:3Þ
where
( 0
a0 ¼ m and b ¼
Hðxn Þ
h2 f ðxn Þ
)
ðh2 þ 2 2hÞf ðxn Þ 2f ðxn huÞ
:
The iteration formula (5.3) requires two evaluations of function and one of first-order derivative per iteration. Further, family (5.3) forms the basis of the proposed family of schemes which may be described through the following theorem: Theorem 5.1. Let f : I # R ! R be a sufficiently differentiable function defined on an open interval I, enclosing a multiple root of f(x), say x ¼ rm 2 I with multiplicity m > 1. Assume that initial guess x ¼ x0 is sufficiently close to rm , then for a; c 2 R, (i) The iteration scheme:
xnþ1 ¼ xn
0c 1=c f ðxn Þ k1 a0c þ k2 b ; f 0 ðxn Þ 2
ð5:4Þ
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S. Kumar et al. / Applied Mathematics and Computation 218 (2012) 7382–7394
where h – m, will have third-order convergence in the vicinity of rm , if:
k1 ¼ lm
ðh mÞ þ 2lm ; a0
ð5:5Þ
and ð1þcÞ
k2 ¼ mc ðm hÞlm a0
:
ð5:6Þ
(ii) The proposed iterative scheme (5.4) satisfies the following error equation:
enþ1 ¼
2 m2
(
3 ) 2 m a1 h þ a2 h2 þ a3 h þ a4 2 3 al 2ðh mÞ þ C 3 þ a0 C 2 en þ Oðe4n Þ; a0 ðm hÞh2
ð5:7Þ
where
l¼ 1 a0 ¼
h ; m
ð5:8Þ
h2 2
2 þ h 2h 2lm
;
a1 ¼ ð1 þ 2m þ cÞ; m
ð5:9Þ ð5:10Þ
2
a2 ¼ 2 2ð1 þ cÞl þ m þ 6m þ mc þ 2c;
ð5:11Þ
a3 ¼ 2 þ ð2 þ 4m þ 2cÞlm 2m2 8m 2mc 2c;
ð5:12Þ
a4 ¼ ð4m 2m2 2mcÞlm þ 2m2 þ 4m þ 2mc:
ð5:13Þ
and
Proof. Further, using Eq. (4.5), we obtain:
$ m2 þ 2h þ mð3 þ 2hÞ 2 2 2 f ðmÞ ðr m Þ m m ðm2 mh þ h2 Þ C 2 en þ f ðxn huÞ ¼ l en 1 þ h C 2 en m! mðm hÞ 2m2 ðm hÞ2 % 4 m 2m3 h þ 4m2 h2 4mh3 þ h4 C 3 e2n þ Oðe3n Þ : þ m2 ðm hÞ2
ð5:14Þ
Substituting (4.2)–(4.5) and (5.14) in Eq. (5.4), we obtain:
enþ1 ¼ K 1 en þ K 2 e2n þ K 3 e3n þ Oðe4n Þ;
ð5:15Þ
where
c 1=c 1 mc k1 þ a0 k2 ; m 2 " #
c 1=c ð1þcÞ c 2lm a0 k2 C 2 m k1 þ a0 k2 K2 ¼ 2 ; 1 m 2 ðm hÞ mc k1 þ ac0 k2
K1 ¼ 1
K 3 ¼ M 1 C 3 þ M 2 C 22 ; M1 ¼
c 1=c k2 ac0 ð3m hÞa0 lm a2 ðm hÞ 2 mc k1 þ a0 k2 m þ ; m3 2 ðm hÞðmc k1 þ ac0 k2 Þ
and
c 1=c " ð1þcÞ m k1 þ ac0 k2 2lm a0 k2 a20 M2 ¼ ðm þ 1Þ þ 2 c c 2 4 3 2 ðm hÞ mc k1 þ ac0 k2 m h ðm hÞ m k1 þ a0 k2 ( mc k1 þ ac0 k2 h2 lm ð3mðm þ 1Þ þ 2ð1 þ 2mÞhÞð2 þ hðh 2ÞÞ ac0 k2 2ð1 þ cÞh4 l2m ac0 k2 þ ðm!Þ2 )!# 2a2 ðm hÞ2 h4 ðm!Þ2 : þ 2lm 3m2 þ mð3 4hÞ þ hð2 þ h þ chÞ ðm!Þ2 þ a20 The proposed scheme (5.4) for multiple roots will be cubically convergent if K 1 ¼ 0 and K 2 ¼ 0 simultaneously.
ð5:16Þ ð5:17Þ ð5:18Þ
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Hence after simplification, we obtain:
mc k1 þ ac0 k2 ¼ 2mc
ð5:19Þ
ð1þcÞ ðm hÞ mc k1 þ ac0 k2 2lm a0 k2 ¼ 0:
ð5:20Þ
and
Solving Eqs. (5.19) and (5.20), k1 and k2 are determined as follows:
k1 ¼ lm
ðh mÞ þ 2l m a0
ð5:21Þ
and ð1þcÞ
k2 ¼ mc ðm hÞlm a0
ð5:22Þ
:
Clearly, for finite and non-zero values of k1 and k2 ; h – m. This completes the proof of the first assertion. For the second part of the theorem, we need to compute the term K3. While doing so, substitute the values of k1 and k2 from Eqs. (5.21) and (5.22) into Eq. (5.18), we get:
2 K3 ¼ 2 m
(
2ðh mÞ
a2 lm a0
þ C 3 þ a0
a1 h3 þ a2 h2 þ a3 h þ a4 ðm hÞh2
) C 22
:
ð5:23Þ
Finally from Eq. (5.15), we obtain:
enþ1
2 ¼ 2 m
(
3 ) 2 m a1 h þ a2 h2 þ a3 h þ a4 2 3 al 2ðh mÞ þ C 3 þ a0 C 2 en þ Oðe4n Þ: a0 ðm hÞh2
ð5:24Þ
This completes the proof of the second assertion. For m = 1 (simple roots), k1 and k2 given by Eqs. (5.21) and (5.22) are both unity respectively. h Special cases: For different specific values of parameters c and h, the following various families of multipoint iterative methods for multiple roots can be derived from formula (5.4). (i) For (c, h) = (1, 1), we get the formula:
xnþ1 ¼ xn
1 f ðxn Þ f ðxn Þ m þ k Hðx Þ k ; 1 2 n 2 f 0 ðxn Þ f ðxn Þ 2f ðxn uÞ
ð5:25Þ
where
ðm1Þ 1 k1 ¼ 2m m 1 m
ð5:26Þ
m 2 ðm1Þ 1 1 k2 ¼ m2 1 2 1 1 : m m
ð5:27Þ
and
This is a new cubically convergent Traub–Ostrowski’s type family of iterative methods for multiple roots. This is a modification over the well-known Traub–Ostrowski’s formula for simple roots. (ii) For (c, h) = (1, 1), we get the formula:
xnþ1 ¼ xn
f ðxn Þ 2mHðxn Þf ðxn Þ ; f 0 ðxn Þ k1 Hðxn Þf ðxn Þ þ k2 m½f ðxn Þ 2f ðxn uÞ
ð5:28Þ
where
ðm1Þ 1 k1 ¼ 2m m 1 m
ð5:29Þ
ðm1Þ 1 k2 ¼ 1 : m
ð5:30Þ
and
This is a new cubically convergent Newton–Secant type family for multiple roots. This is a modification over the Newton– Secant formula derived by Kim and Lee [11].
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Note that the family (5.4) can produce many more new cubically convergent families of multipoint iterative methods for multiple roots by choosing different values of the parameters. 5.2. Second family Replacing the second-order derivative in Eq. (3.10) by following definition:
f 0 ðxn Þ f 0 ðxn huÞ ; hu
f 00 ðxn Þ
h – 0 2 R;
ð5:31Þ
we get the following new generalized family as:
xnþ1 ¼ xn
00c 1=c f ðxn Þ a00c þ b ; f 0 ðxn Þ 2
ð5:32Þ
where 00
a00 ¼ m and b ¼
Hðxn Þ
hf 0 ðxn Þ : ðh 1Þf 0 ðxn Þ þ f 0 ðxn huÞ
The iteration formula (5.32), requires one evaluation of function and two of first-order derivatives per iteration. Further, formula (5.32) forms the basis of the proposed family of schemes, which may be described through the following theorem: Theorem 5.2. Let f : I # R ! R be a sufficiently differentiable function defined on an open interval I, enclosing a multiple root of f(x), say x ¼ rm 2 I with multiplicity m > 1. Assume that initial guess x ¼ x0 is sufficiently close to rm , then for a; c 2 R: (i) The iteration scheme:
xnþ1 ¼ xn
00c 1=c f ðxn Þ d1 a00c þ d2 b ; f 0 ðxn Þ 2
where h – m and h –
" d1 ¼ 2m
2m , mþ1
ð5:33Þ
will have third-order convergence in the vicinity of r m , if:
ðm hÞ2 ð1 þ h þ lm Þ mhð1 hÞlm hðm hÞ½mð2 þ hÞ þ h
#
lðm1Þ
ð5:34Þ
and
d2 ¼ 2
ðh 1Þðm hÞ þ mlm ð1þcÞ c ðm1Þ b0 l : m h½mð2 þ hÞ þ h
ð5:35Þ
(ii) The proposed iterative scheme (5.33) satisfies the following error equation:
enþ1
( ) 2ðm hÞ3 d1 þ d2 h þ d3 h2 þ C 22 b0 b1 þ b2 h þ b3 h2 þ b4 h3 þ b5 h4 e3n þ Oðe4n Þ; ¼2 hm2 ðm hÞ2 ½mð2 þ hÞ þ h
ð5:36Þ
where
h ; m h ; b0 ¼ 1 þ h þ lðm1Þ
l¼ 1
ð5:37Þ ð5:38Þ
b1 ¼ 2m3 ðc þ m þ 2m2 Þð1 lm Þ; 2
m
ð5:39Þ 3
m
2
m
b2 ¼ 7m ½1 þ 2m ð1 þ cÞl m ð1 þ cÞð3 l Þ m ð5c 8ml Þ; b3 ¼ 4mð1 þ cÞ þ 18m2 þ 13m3 þ m4 þ 7m2 c þ m3 c lm 5mð1 þ cÞ þ 7m2 þ 5m2 c þ 2m3 ;
ð5:41Þ
b4 ¼ 1 c 7m 5mc 13m2 2m2 c 3m3 þ lm ð1 þ cÞð1 þ mÞ2 ;
ð5:42Þ
2
b5 ¼ 1 þ 3m þ 2m þ mð1 þ cÞ; 2
1
ð5:40Þ
ð5:43Þ
d1 ¼ 2ma ð1 l Þ;
ð5:44Þ
d2 ¼ 2C 3 þ 2a2 ðm þ 1Þlm ;
ð5:45Þ
d3 ¼ 2C 3 C 3 m 2a2 lm :
ð5:46Þ
and
S. Kumar et al. / Applied Mathematics and Computation 218 (2012) 7382–7394
7391
Proof. On the similar lines as in the previous theorem. h Special cases: For different specific values of parameters c and h, the following various families of multipoint iterative methods for multiple roots can be derived from family (5.33). (I) For c = 1 in formula (5.33), we obtain the family based on arithmetic mean given by
xnþ1 ¼ xn
1 f ðxn Þ hf 0 ðxn Þ m þ d Hðx Þ d ; 1 2 n 2 f 0 ðxn Þ ðh 1Þf 0 ðxn Þ þ f 0 ðxn huÞ
ð5:47Þ
where
ðm hÞlm ððm hÞðh 1Þ þ mlm Þ d1 ¼ 2 1 þ hðhðm þ 1Þ 2mÞ
ð5:48Þ
and
d2 ¼
2mlm ððm hÞðh 1Þ þ mlm Þ2 h2 ðhðm þ 1Þ 2mÞ
ð5:49Þ
:
Some interesting particular cases of the family (5.47) are: (i) For h ¼ 1 in (5.47), we get the formula:
xnþ1 ¼ xn
1 f ðxn Þ f 0 ðxn Þ m þ d Hðx Þ d ; 1 2 n 2 f 0 ðxn Þ f 0 ðxn uÞ
ð5:50Þ
where
d1 ¼ 2ðm 1Þ
ð5:51Þ
ðm1Þ 1 d2 ¼ 2m2 1 : m
ð5:52Þ
and
This is a new cubically convergent family of iterative methods for multiple roots. This is a modification over the Özban’s formula (5) in [20] for simple roots. 2m (ii) For h ¼ mþ2 in Eq. (5.47), we get the formula:
xnþ1
2 m f ðxn Þ 4 2f 0 ðxn Þ ¼ xn d1 þ d2 Hðxn Þ 2 f 0 ðxn Þ ðm 2Þf 0 ðxn Þ þ ðm þ 2Þf 0 xn
3
5;
2m u mþ2
ð5:53Þ
where
m3 þ 4m2 8 þ m2 ðm 2Þ d1 ¼
m mþ2
m ð5:54Þ
2ðm þ 2Þ
and
d2 ¼
m mþ2
m h
m i2 m ðm þ 2Þ2 þ mðm 2Þ mþ2 4ðm þ 2Þ
:
ð5:55Þ
This is a new cubically convergent Jarratt-type family of iterative methods for multiple roots. This is a modification over the well known Jarratt’s formula [21] for simple roots. (II) For c = 1 in (5.33), we obtain the family based on harmonic mean given by
xnþ1 ¼ xn
2mhHðxn Þf ðxn Þ ; d1 hHðxn Þf 0 ðxn Þ þ d2 m½ðh 1Þf 0 ðxn Þ þ f 0 ðxn huÞ
ð5:56Þ
where
" # ðm hÞ2 ð1 þ h þ lm Þ mhð1 hÞlm ðm1Þ l hðm hÞ½mð2 þ hÞ þ h
d1 ¼ 2m
ð5:57Þ
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S. Kumar et al. / Applied Mathematics and Computation 218 (2012) 7382–7394
and
ðm2Þ 2m 1 mh d2 ¼ : mð2 þ hÞ þ h
ð5:58Þ
Some interesting particular cases of the family (5.56) are (i) For h ¼ 12 in (5.56), we get the formula
xnþ1 ¼ xn
2mHðxn Þf ðxn Þ ; d1 Hðxn Þf 0 ðxn Þ þ d2 m½2f 0 ðxn 0:5uÞ f 0 ðxn Þ
ð5:59Þ
where
n o 3 ðm1Þ 1 m 1 m ð2m 1Þ2 1 þ 2 1 2m 2m 1 2m 4 5 1 1 d1 ¼ 2m 2m ð2m 1Þð1 3mÞ 2
ð5:60Þ
and
d2 ¼
ðm2Þ 4m 1 1 : ð1 3mÞ 2m
ð5:61Þ
This is a new cubically convergent family of iterative methods for multiple roots. This is a modification over the well-known iterative formulas (8)–(12) in [2] for simple roots. Note that the family (5.33) can produce many more new cubically convergent families of multipoint iterative methods for multiple roots by choosing different values of the parameters. It is straight forward to see that per step these methods require three evaluations of function viz. two evaluations of f(x) and one of f 0 ðxÞ or one evaluation of f(x) and two of f 0 ðxÞ. In order to obtain an assessment of the efficiency of our methods, we shall make use of efficiency index defined in [1]. (5.25), (5.28), (5.50), (5.53) and pffiffiffi For our proposed iteration schemes,pnamely ffiffiffi (5.59), we find p = 3 and d = 3 yielding E ¼ 3 3 1:442 which is better than E ¼ 2 1:414, the efficiency index of the Newton’s method. 6. Numerical examples In this section, we shall present the numerical results obtained by employing the iterative methods namely modified Schröder’s method (3.6) (MSM), modified super-Halley method (3.12) (MSHM), modified Halley’s method (3.13) (MHM), modified Ostrowki’s square-root method (3.16) (MOM), method (3.14) (MM1), method (3.15) (MM2), modified Traub– Ostrowski’s method (5.25) (MTOM), modified Newton–Secant method (5.28) (MNSM), modified Jarratt’s method (5.53) (MJM), method (5.50) (MM3) and method (5.59) (MM4) to solve some nonlinear equations with known multiplicity m given in Table 6.1 and compare with classical exiting methods namely modified Newton’s method (MNM), Schröder’s method (SM), super-Halley method (SHM), Halley’s method (HM), Ostrowski’s square-root method (OM), Chen Dong method [15] (CDM), Victory and Neta method [13] (VNM), Traub–Ostrowski’s’ method (TOM), Newton–Secant method [11] (NSM) and Li et al. method [20] (LM). All the formulae are tested for a ¼ 12 and the results are summarized in Tables 6.2 and 6.3 (for one-point and multipoint iterative methods respectively). Computations have been performed using MATLABÒ version 7.5 (R2007b) in double precision arithmetic. We use ¼ 1015 as a tolerance error. The following stopping criteria are used for computer programs:
ðiÞ jxnþ1 xn j < ;
ðiiÞ jf ðxnþ1 Þj < :
Table 6.1 Test problems, multiplicity (m) and root ðr m Þ. Problems 6.1 6.2
Multiplicity (m) ðx3 þ 4x2 10Þ4 ¼ 0 3 sin2 ðxÞ x2 þ 1 ¼ 0
Root ðr m Þ
4
1.36523001341097
3
1.404491648215341
6.3
ðx2 expðxÞ 3x þ 2Þ3 ¼ 0
3
0.2575302854398607
6.4
ðcosðxÞ xÞ2 ¼ 0
2
0.7390851332151607
6.5
ððx 1Þ3 1Þ50 ¼ 0 x2 sinð4xÞ ¼ 0 4 expðx2 þ 7x 30Þ 1 ðx 5Þ ¼ 0
6.6 6.7 6.8 6.9
ð1 þ cosðxÞÞðexpðxÞ 2Þ2 ¼ 0 pffiffi2 sinðxÞ 22 ðx þ 1Þ ¼ 0
50
2.000000000000000
3 4
0.0000000000000000 3.000000000000000
2
0.6931471805599453
2
0.7853981633974483
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S. Kumar et al. / Applied Mathematics and Computation 218 (2012) 7382–7394 Table 6.2 Performance of one-point iterative methods (number of iterations required). Problems
MSM (3.6)
SHM
MSHM (3.12)
HM
MHM (3.13)
OM
MOM (3.16)
MM1 (3.14)
MM2 (3.15)
6.1
Initial guess 1 2
MNM 5 5
SM 5 5
5 5
3 3
3 3
3 3
3 3
3 3
3 3
3 3
3 3
6.2
1 2
6 5
6 5
6 5
3 3
3 3
4 4
4 4
3 4
3 4
3 3
4 3
6.3
0.1 0.5
4 4
4 4
4 4
3 3
3 3
3 3
3 3
3 3
3 3
3 3
3 3
6.4
0.25 1.7
5 5
5 5
5 4
4 4
3 3
3 5
3 3
3 4
3 3
3 3
3 4
6.5
2.5 3.5
4 6
4 6
4 6
2 2
2 2
2 3
2 3
2 3
2 3
2 2
3 3
6.6
0.28 0.35
4 4
4 4
4 4
3 3
3 3
4 4
3 4
3 4
3 4
3 4
3 3
6.7
2.9 3.15
7 7
6 10
6 9
5 6
4 6
4 4
4 4
4 4
3 4
4 5
5 Divergent
6.8
0 1.5
6 5
6 5
5 5
3 4
3 3
4 4
3 4
3 4
4 3
3 3
4 3
6.9
0.1 1.25
5 6
5 5
5 5
4 4
4 3
4 4
5 4
5 4
4 4
5 3
3 4
Table 6.3 Performance of multipoint iterative methods (number of iterations required). Problems
CDM
VNM
TOM
MTOM (5.25)
NSM
MNSM (5.28)
LM
MJM (5.53)
MM3 (5.50)
MM4 (5.59)
6.1
1 2
4 4
3 3
5 5
5 4
5 3
3 3
3 6
3 3
3 8
6 3
6.2
1 2
5 5
4 4
9 5
7 4
5 4
4 5
4 6
5 4
3 3
5 4
6.3
0.1 0.5
3 6
3 4
5 5
3 4
5 5
6 6
4 4
3 4
4 3
3 3
6.4
0.1 0.25
5 6
4 4
3 3
4 4
3 3
4 4
3 3
4 3
7 7
3 3
1.7 2.5 3.5
3 3 4
3 3 4
4 3 4
4 3 4
4 3 4
4 3 4
3 3 3
3 3 3
6 3 4
4 3 4
6.6
0.28 0.35
3 3
3 4
3 4
3 4
4 4
3 4
3 3
3 4
3 3
3 4
6.7
2.9 3.15
3 5
6 4
6 4
5 4
4 7
4 6
7 8
8 5
6 4
8 4
6.8
0 1.5
5 3
4 4
4 4
5 4
4 4
5 4
5 4
5 3
5 4
5 6
6.9
0.1 1.25
4 4
4 4
5 4
3 5
5 4
3 5
4 3
4 4
4 4
3 6
6.5
Initial guess
7. Conclusions This work proposes a new family of Schröder-type methods for finding multiple roots of nonlinear equations based on nonlinear means. Proposed family of Schröder-type methods (3.10) unifies some of the most known third-order iterative methods for solving nonlinear equations and also provide many more unknown processes. Further, we have also presented many new third-order multipoint iterative methods free from second-order derivative for multiple roots. Numerical
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S. Kumar et al. / Applied Mathematics and Computation 218 (2012) 7382–7394
examples presented here prove that the proposed one-point as well as multipoint iterative methods can compete with any of the existing classical iterative methods for multiple roots. A reasonably close initial guess is necessary for the multipoint methods to converge. This condition, however, applies to practically all the iterative methods for solving equations. Acknowledgements We would like to record our sincerest thanks to Professor B. Neta, Associate editor and anonymous reviewer for their constructive suggestions which have considerably contributed to the readability of this paper. References [1] A.M. Ostrowski, Solution of Equations in Euclidean and Banach Space, third ed., Academic Press, New York, 1973. [2] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, 1964. [3] J.E. Dennis, R.B. Schnable, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1983. [4] E. Schröder’s, Über unendlich viele Algorithmen Zur Auflösung der Gleichungen, Math. Ann. 2 (1870) 317–365. [5] W. Werner, Some improvement of classical methods for the solution of nonlinear equations, in numerical solution of nonlinear equations, Lect. Notes Math. 878 (1981) 426–440. [6] E. Hansen, M. Patrick, A family of root finding methods, Numer. Math. 27 (1977) 257–269. [7] L.D. Petkovic´, M.S. Petkovic´, D. Zˇivkovic´, Hansen–Patrick’s family is of Laguerre’s type, Novi SAD J. Math. 33 (2003) 109–115. [8] L.B. Rall, Convergence of Newton’s process to multiple solutions, Numer. Math. 9 (1966) 23–37. [9] C. Chun, B. Neta, A third-order modification of Newton’s method for multiple roots, Appl. Math. Comput. 211 (2009) 474–479. [10] Sanjeev Kumar, Vinay Kanwar, Sushil Kumar Tomar, Sukhjeet Singh, Geometrically constructed families of Newton’s method for unconstrained optimization and nonlinear equations, Int. J. Math. Math. Sci. 2011 (2011) 9, doi:10.1155/2011/972537. Article ID 972537. [11] Y.I. Kim, S.D. Lee, A third-order variant of Newton–Secant method finding a multiple zero, J. Chungcheong Math. Soc. 23 (4) (2010) 845–852. [12] P.S. Bullen, The Power Means, Hand Book of Means and Their Inequalities, Kluwer Dordrecht, Netherlands, 2003. [13] H.D. Victory, B. Neta, A higher order method for multiple zeros of nonlinear functions, Int. J. Comput. Math. 12 (1983) 329–335. [14] C. Dong, A basic theorem of constructing an iterative formula of higher order for computing multiple roots of an equation, Math. Numer. Sin. 11 (1982) 445–450. [15] C. Dong, A family of multipoint iterative functions for finding the multiple roots of equations, Int. J. Comput. Math. 21 (1987) 363–367. [16] N. Osada, An optimal multiple root-finding method of order three, J. Comput. Appl. Math. 51 (1994) 131–133. [17] B. Neta, New third order nonlinear equation solvers for multiple roots, Appl. Math. Comput. 202 (2008) 162–170. [18] B. Neta, Extension of Murakami’s high-order nonlinear solver to multiple roots, Int. J. Comput. Math. 8 (2010) 1023–1031. [19] C. Chun, H.J. Bae, B. Neta, New families of nonlinear solvers for finding multiple roots, Comput. Math. Appl. 57 (2009) 1574–1582. [20] S. Li, X. Liao, L. Cheng, A new fourth-order iterative method for finding the multiple roots of nonlinear equations, Appl. Math. Comput. 215 (2009) 1288–1292. [21] S.G. Li, L.Z. Cheng, B. Neta, Some fourth-order nonlinear solvers with closed formulae for multiple roots, Comput. Math. Appl. 59 (2010) 126–135. [22] B. Neta, Anthony N. Johnson, High-order nonlinear solver for multiple roots, Comput. Math. Appl. 55 (2008) 2012–2017. [23] A.Y. Özban, Some new variants of Newton’s method, Appl. Math. Lett. 17 (2004) 677–682. [24] P. Jarratt, Some fourth order multipoint iterative methods for solving equations, Math. Comput. 20 (1966) 434–437. [25] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach. 21 (1974) 643–651.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On some modified families of multipoint iterative methods for multiple roots of nonlinear equations Sanjeev Kumar a,⇑, V. Kanwar b,⇑, Sukhjit Singh c a
Department of Mathematics, Maharishi Markandeshwar University, Sadopur, Ambala 134 007, Haryana, India University Institute of Engineering and Technology, Panjab University, Chandigarh 160 014, India c Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148 106, Punjab, India b
a r t i c l e
i n f o
Keywords: Newton’s method Modified Newton’s method Schröder’s method Multiple roots Power means Order of convergence
a b s t r a c t In this paper, we propose a new one-parameter family of Schröder’s method for finding the multiple roots of nonlinear equations numerically. Further, we derive many new cubically convergent families of Schröder-type methods. Proposed families are derived from the modified Newton’s method for multiple roots and one-parameter family of Schröder’s method. Furthermore, we introduce new families of third-order multipoint iterative methods for multiple roots free from second-order derivative by semi discrete modifications of the above proposed methods. One of the families requires two evaluations of the function and one evaluation of its first-order derivative and the other family requires one evaluation of the function and two evaluations of its first-order derivative per iteration. Numerical examples are also presented to demonstrate the performance of proposed iterative methods. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Solving nonlinear equations is a common and important problem in science and engineering [1–25]. Analytical methods for solving such equations are almost non-existent and therefore, it is only possible to obtain approximate solutions by relying on numerical methods based on iterative procedures. With the advancement of computers, the problem of solving nonlinear equations by numerical methods gained more importance than before. In this paper, we consider iterative methods for finding a multiple root rm of multiplicity m > 1, i.e. f ðrm Þ ¼ f 0 ðrm Þ ¼ ¼ f ðm1Þ ðrm Þ ¼ 0 and f ðmÞ ðr m Þ – 0, of a nonlinear equation:
f ðxÞ ¼ 0;
ð1:1Þ
where f : I R ! R be a nonlinear smooth and continuous function on an open interval I. A lot of research work [1–9,11,13–22] has been carried out to compute the roots of nonlinear equation (1.1). The design of iterative methods for solving such equations is very important and interesting tasks in computational mathematics. Newton’s method for multiple roots appears in the work of Schröder [4], which is given as:
xnþ1 ¼ xn
f ðxn Þf 0 ðxn Þ : f 02 ðxn Þ f ðxn Þf 00 ðxn Þ
⇑ Corresponding authors. E-mail addresses: [email protected] (S. Kumar), [email protected] (V. Kanwar). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.12.081
ð1:2Þ
S. Kumar et al. / Applied Mathematics and Computation 218 (2012) 7382–7394
7383
This method has second-order convergence, including the case of multiple roots. It may be obtained by applying Newton’s method to the function uðxÞ ¼ ff0ðxðxnnÞÞ, which has simple roots in each multiple root of f(x). Another quadratically convergent method for multiple roots is the modified Newton’s method [8]:
xnþ1 ¼ xn m
f ðxn Þ : f 0 ðxn Þ
ð1:3Þ
More recently, Kumar et al. [10] have developed a family of Newton’s method for finding the simple roots of a nonlinear equation (1.1) and is given by
xnþ1 ¼ xn
f 0 ðxn Þ
2f ðxn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; f 02 ðxn Þ þ 4a2 f 2 ðxn Þ
ð1:4Þ
where a 2 R. In (1.4), the sign in the denominator should be chosen so that the denominator is the largest in magnitude. The beauty of this method is that it converges quadratically and moreover, has the same error equation as Newton’s method. Therefore, this method is an efficient alternative to the classical Newton’s method. In this paper, we further extend our work [10] and construct a modified family of Schröder’s method. We propose new cubically convergent families of Schröder-type methods based on power means, when multiplicity m of the root is known in advance. Thereafter by discretizing second-order derivative, we obtain many new second derivative free cubically convergent families of multipoint iterative methods for multiple roots. It is also shown that some well-known methods like super-Halley method [5], Halley’s method [1–7], Ostrowski’s square-root method [1–7], Newton–Secant method [11], etc. can be regarded as particular cases of the proposed family. 2. Review of definition of various means For a given finite real number c, the cth- power mean Pc [12] of positive scalars a and b, is defined as follows:
c c 1 a þb c Pc ¼ : 2
ð2:1Þ
It is easy to see that:
For c ¼ 1; For c ¼
1 ; 2
2ab ðHarmonic meanÞ; P 1 ¼ aþb (pffiffiffi pffiffiffi)2 aþ b P1 ¼ ; 2 2 aþb ðArithmetic meanÞ; s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
For c ¼ 1;
P1 ¼
For c ¼ 2;
P2 ¼
For c ! 0;
we have P 0 ¼ limc!0 P c ¼
a2 þ b 2
ð2:2Þ ð2:3Þ ð2:4Þ
2
ðRoot mean squareÞ;
ð2:5Þ
and
pffiffiffiffiffiffi ab ðGeometric meanÞ:
ð2:6Þ
For given positive scalars a and b, some other well-known means are defined as:
pffiffiffiffiffiffi ab þ b ðHeronian meanÞ; Ho M ¼ 3 2 2 a þb CoM ¼ ðContra-harmonic meanÞ; aþb 2 2 a2 þ ab þ b ðCentroidal meanÞ; CeM ¼ 3ð a þ b Þ aþ
ð2:7Þ ð2:8Þ ð2:9Þ
and
Lo M ¼
ab logðaÞ logðbÞ
ðLogarithmic meanÞ:
ð2:10Þ
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3. Development of the iterative methods 3.1. Modified family of Schröder’s method The family of Newton’s method (1.4) for finding simple root of a nonlinear equation (1.1) can be written as:
xnþ1 ¼ xn
2f ðxn Þ #: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi f ðxn Þ 2 0 f ðxn Þ 1 1 þ 4a f 0 ðxn Þ "
ð3:1Þ
Now consider the factor:
4a2
2 f ðxn Þ : f 0 ðxn Þ
ð3:2Þ
Since the scaling parameter a appears in the numerator of (3.2), it is clear that there exits some real values of a such that:
2 2 f ðxn Þ 1; 4a f 0 ðxn Þ
ð3:3Þ
holds. With this assumption, the binomial theorem is applicable in Eq. (3.1) and one can get the following formula free from square root term as:
xnþ1 ¼ xn
f ðxn Þf 0 ðxn Þ : 2 2 n Þ þ a f ðxn Þ
ð3:4Þ
f 02 ðx
Further, applying the well known Newton’s method:
xnþ1 ¼ xn
f ðxn Þ ; f 0 ðxn Þ
ð3:5Þ 0
n Þf ðxn Þ to the modified function uf ðxÞ ¼ f 02 ðxf ðxn Þþ a2 f 2 ðxn Þ of formula (3.4), we obtain:
xnþ1 ¼ xn Hðxn Þ
f ðxn Þf 0 ðxn Þ ; f 02 ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:6Þ
where
Hðxn Þ ¼
f 02 ðxn Þ þ a2 f 2 ðxn Þ : f 02 ðxn Þ a2 f 2 ðxn Þ
ð3:7Þ
This is a one-parameter modified family of Schröder’s method [4] for an equation having multiple root of multiplicity m > 1 unknown. It is interesting to note that by ignoring the term a, method (3.6) reduces to classical Schröder’s method. 3.2. Schröder-type cubically convergent iterative methods Using the modified Newton’s formulae (1.3) and (3.6), one can obtain:
xnþ1 ¼ xn
1 f ðxn Þ f ðxn Þf 0 ðxn Þ m 0 þ Hðxn Þ 02 : 2 f ðxn Þ f ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:8Þ
This is a new one-parameter family of super-Halley type methods for the case of multiple roots of multiplicity m > 1 known. It is interesting to note that by ignoring the term a, method (3.8) reduces to super-Halley method [5] for the case of multiple roots. Method (3.8) can further be rewritten as:
xnþ1
n o3 2 f 02 ðxn Þ f ðxn Þ 4m þ Hðxn Þ f 02 ðxn Þf ðxn Þf 00 ðxn Þ 5 : ¼ xn 0 2 f ðxn Þ
Now we shall generalize the formula (3.9) by cth-power mean. For this, we take:
a ¼ m and b ¼
Hðxn Þ
f 02 ðxn Þ : f 02 ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:9Þ
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Approximating the functions in (3.9) as follows, we have:
xnþ1
2 n oc 31c f 02 ðxn Þ c f ðxn Þ 6m þ Hðxn Þ f 02 ðxn Þf ðxn Þf 00 ðxn Þ 7 ¼ xn 0 4 5: f ðxn Þ 2
ð3:10Þ
Suppose that initial guess xn a is reasonably good approximation to the multiple roots of multiplicity m > 1 so that the quantity:
f ðxn Þf 00 ðxn Þ f 02 ðx Þ 1: n
ð3:11Þ
It can be seen that the quantities a and b are already positive in view of (3.3) and (3.11). This family may be called the cthpower mean iterative family of Schröder-type methods for multiple roots. Special cases: It is interesting to note that for different values of c, various well-known methods can be deduced from formula (3.10) as follows: (i) For c = 1 (arithmetic mean), formula (3.10) corresponds to cubically convergent super-Halley type methods [5] for multiple roots:
xnþ1 ¼ xn
1 f ðxn Þ f 02 ðxn Þ m þ Hðxn Þ 02 : 0 00 2 f ðxn Þ f ðxn Þ f ðxn Þf ðxn Þ
ð3:12Þ
(ii) For c = 1 (harmonic mean), formula (3.10) corresponds to the well-known Halley’s type methods [1–7] for multiple roots:
xnþ1 ¼ xn
2mHðxn Þf ðxn Þf 0 ðxn Þ : m½f 02 ðxn Þ f ðxn Þf 00 ðxn Þ þ Hðxn Þf 02 ðxn Þ
ð3:13Þ
(iii) For c ¼ 12, formula (3.10) reduces to a new method for multiple roots given by
xnþ1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ffi#2 1 f ðxn Þ pffiffiffiffiffi f 02 ðxn Þ mþ Hðxn Þ 02 ¼ xn : 4 f 0 ðxn Þ f ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:14Þ
(iv) For c = 2 (root-mean-square), formula (3.10) reduces to a new method for multiple roots given by
xnþ1
1 f ðxn Þ ¼ xn pffiffiffi 0 2 f ðxn Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f 02 ðxn Þ : m2 þ Hðxn Þ 02 f ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:15Þ
(v) For c ! 0 (geometric mean), formula (3.10) reduces to the well-known Ostrowski’s square-root type methods [1–7] given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m : xnþ1 ¼ xn f ðxn Þ Hðxn Þ 02 f ðxn Þ f ðxn Þf 00 ðxn Þ
ð3:16Þ
It is interesting to note that by ignoring the term a, methods (3.12), (3.13) and (3.16) reduce to super-Halley, Halley’s and Ostrowski’s square-root methods for multiple roots. In addition to this, if m = 1 the methods (3.12), (3.13) and (3.16) reduce to super-Halley, Halley’s and Ostrowski’s square-root methods for simple roots [1–7]. Some other new cubically convergent iterative methods based on Heronian mean, contra-harmonic mean, centroidal mean, logarithmic mean, etc. can also be obtained from formula (3.10) respectively. 4. Convergence analysis Now, we shall present the mathematical proof for the order of convergence of the proposed iterative family (3.6) and (3.10). Theorem 4.1. Let f : I # R ! R be a sufficiently differentiable function defined on an open interval I, enclosing a multiple root of f (x), say x ¼ r m 2 I with multiplicity m > 1. Assume that initial guess x ¼ x0 is sufficiently close to rm , then for a 2 R, the iteration scheme (3.6) has quadratic order of convergence and satisfies the following error equation:
enþ1 ¼
C2 2 e þ Oðe3n Þ: m n
ð4:1Þ
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Proof. Since f(x) is sufficiently differentiable function, therefore, expanding f ðxn Þ about x ¼ r m by Taylor’s expansion and using f ðrm Þ ¼ f 0 ðr m Þ ¼ f ðm1Þ ðr m Þ ¼ 0 and f ðmÞ ðrm Þ – 0 (a condition for x ¼ r m to be root of multiplicity m), we have:
f ðxn Þ ¼
f ðmÞ ðrm Þ m en 1 þ C 2 en þ C 3 e2n þ Oðe3n Þ ; m!
ð4:2Þ
f ðmþk1Þ ðr
mÞ where en ¼ xn r m and C k ¼ ðmþ1Þðmþ2Þ...ðmþk1Þf ðmÞ ðr Þ ; k ¼ 2; 3; . . .. m
Similarly for f 0 ðxn Þ and f 00 ðxn Þ, it may be shown that:
f 0 ðxn Þ ¼
f ðmÞ ðrm Þ m1 ðm þ 1Þ ðm þ 2Þ en C 2 en þ C 3 e2n þ Oðe3n Þ ; 1þ ðm 1Þ! m m
ð4:3Þ
f 00 ðxn Þ ¼
f ðmÞ ðr m Þ ðm2Þ mþ1 ðm þ 1Þðm þ 2Þ en C 2 en þ C 3 e2n þ Oðe3n Þ : 1þ ðm 2Þ! m1 mðm 1Þ
ð4:4Þ
and
From (4.2)–(4.4), we obtain:
" # f ðxn Þ en C2 fðm þ 1Þ2 C 2 2mC 3 g 2 3 u¼ 0 en þ Oðen Þ ; ¼ 1 en þ f ðxn Þ m m2 m
ð4:5Þ
and
" # f 00 ðxn Þ ðm 1Þ ðm þ 1Þ f2mðm þ 2ÞC 3 ðm þ 1Þ2 C 22 g 2 3 C 2 en þ 1þ ¼ en þ Oðen Þ : f ðxn Þ en mðm 1Þ m2 ðm 1Þ
ð4:6Þ
Further, Eq. (3.6) can be rewritten as:
xnþ1 ¼ xn
1 1 2 2 f ðxn Þ f ðxn Þf 00 ðxn Þ 2 f ðxn Þ 2 f ðxn Þ : 1 a 1 þ a 1 f 0 ðxn Þ f 02 ðxn Þ f 02 ðxn Þ f 02 ðxn Þ
ð4:7Þ
From (4.5) and (4.6), we get:
1 f ðxn Þf 00 ðxn Þ 1 2 e2n þ Oðe3n Þ; 1 ¼ m þ 2C e þ 6C þ C 3 þ 2 n 3 2 f 02 ðxn Þ m f 2 ðxn Þ a2 1 þ a2 02 ¼ 1 þ 2 e2n þ Oðe3n Þ f ðxn Þ m
ð4:8Þ ð4:9Þ
and
1 f 2 ðxn Þ a2 1 a2 02 ¼ 1 þ 2 e2n þ Oðe3n Þ: f ðxn Þ m
ð4:10Þ
Substituting (4.5), (4.8)–(4.10) in Eq. (4.7) and simplifying, we finally obtain:
enþ1 ¼
C2 2 e þ Oðe3n Þ: m n
ð4:11Þ
This completes the proof of the theorem. h Theorem 4.2. Let f : I # R ! R be a sufficiently differentiable function defined on an open interval I, enclosing a multiple root of f(x), say x ¼ r m 2 I with multiplicity m > 1. Assume that initial guess x ¼ x0 is sufficiently close to rm , then for a; c 2 R, the iteration scheme (3.10) has cubic order of convergence and satisfies the following error equation:
h enþ1 ¼
mðC 22 2C 3 Þ ðcC 22 þ 2a2 Þ 2m2
i e3n þ Oðe4n Þ:
ð4:12Þ
Proof. Further, formula (3.10) can be rewritten as:
1c
xnþ1 ¼ xn 2
" c c c #1c 2 2 f ðxn Þ f ðxn Þf 00 ðxn Þ c 2 f ðxn Þ 2 f ðxn Þ 1 a 02 1 : m þ 1 þ a 02 f 0 ðxn Þ f ðxn Þ f ðxn Þ f 02 ðxn Þ
ð4:13Þ
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Now using (4.5) and (4.6), we obtain:
c f 2 ðxn Þ a2 c 2C 2 a2 c 3 1 þ a2 02 ¼ 1 þ 2 e2n en þ Oðe4n Þ; f ðxn Þ m m3 c f 2 ðxn Þ a2 c 2C 2 a2 c 3 ¼ 1 þ 2 e2n en þ O e4n ; 1 a2 02 f ðxn Þ m m3
ð4:14Þ ð4:15Þ
and
c i f ðxn Þf 00 ðxn Þ 2cC 2 c h 1 ¼ mc 1 þ en þ 2 6mC 3 3ðm þ 1ÞC 22 þ 2ðc þ 1ÞC 22 e2n þ Oðe3n Þ : 02 f ðxn Þ m m
ð4:16Þ
Substituting (4.5), (4.14)–(4.16) in Eq. (4.13) and simplifying, we finally obtain:
h enþ1 ¼
mðC 22 2C 3 Þ ðcC 22 þ 2a2 Þ 2m2
i
e3n þ O e4n :
ð4:17Þ
This completes the proof of the theorem. h
5. Generalized families of multipoint iterative methods The main difficulty associated with recently proposed third-order iterative methods is the evaluation of second-order derivative. In recent years, many modifications of Newton’s method for multiple roots have been proposed by Kim and Lee [11], Victory and Neta [13], Dong [14,15], Neta [17,18], Chun et al. [19], Li et al. [20,21], Neta and Johnson [22] and the references cited there in. All these modifications are targeted at increasing the local order of convergence with the view of increasing their efficiency index [1]. Some of these multipoint iterative methods are of order three [11,13–15,17,19], while others are of order four [20–22]. The fourth order methods proposed by Li et al. in [20] and last two formulae in [21] have optimal order of convergence [25], since they require three evaluations per step, namely one evaluation of function and two evaluations of the first-order derivative. All these modifications of Newton’s method require the prior knowledge of multiplicity m. Here, we also intend to develop and unify general class of multipoint iterative methods for multiple roots. The main idea of the proposed methods lies in the discretization of second-order derivative involved in the cth-power mean family of Schröder-type iterative methods (3.10). In this section, we have derived two families by discretizing second-order derivative involved in the family (3.10). 5.1. First family Expanding the function f ðxn huÞ where h – 0 2 R, but finite, about the point x ¼ xn with f ðxn Þ – 0, we have: 0
f ðxn huÞ ¼ f ðxn Þ huf ðxn Þ þ Let us take u ¼
f ðxn Þ f 0 ðxn Þ
f ðxn Þf 00 ðxn Þ
h2 u2 00 f ðxn Þ þ Oðu3 Þ: 2
ð5:1Þ
and inserting this into equation (5.1), we obtain:
2f 02 ðxn Þ h2 f ðxn Þ
ff ðxn huÞ ð1 hÞf ðxn Þg:
ð5:2Þ
Using this approximate value of f ðxn Þf 00 ðxn Þ into the formula (3.10), we have:
xnþ1 ¼ xn
0c 1=c f ðxn Þ a0c þ b ; f 0 ðxn Þ 2
ð5:3Þ
where
( 0
a0 ¼ m and b ¼
Hðxn Þ
h2 f ðxn Þ
)
ðh2 þ 2 2hÞf ðxn Þ 2f ðxn huÞ
:
The iteration formula (5.3) requires two evaluations of function and one of first-order derivative per iteration. Further, family (5.3) forms the basis of the proposed family of schemes which may be described through the following theorem: Theorem 5.1. Let f : I # R ! R be a sufficiently differentiable function defined on an open interval I, enclosing a multiple root of f(x), say x ¼ rm 2 I with multiplicity m > 1. Assume that initial guess x ¼ x0 is sufficiently close to rm , then for a; c 2 R, (i) The iteration scheme:
xnþ1 ¼ xn
0c 1=c f ðxn Þ k1 a0c þ k2 b ; f 0 ðxn Þ 2
ð5:4Þ
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where h – m, will have third-order convergence in the vicinity of rm , if:
k1 ¼ lm
ðh mÞ þ 2lm ; a0
ð5:5Þ
and ð1þcÞ
k2 ¼ mc ðm hÞlm a0
:
ð5:6Þ
(ii) The proposed iterative scheme (5.4) satisfies the following error equation:
enþ1 ¼
2 m2
(
3 ) 2 m a1 h þ a2 h2 þ a3 h þ a4 2 3 al 2ðh mÞ þ C 3 þ a0 C 2 en þ Oðe4n Þ; a0 ðm hÞh2
ð5:7Þ
where
l¼ 1 a0 ¼
h ; m
ð5:8Þ
h2 2
2 þ h 2h 2lm
;
a1 ¼ ð1 þ 2m þ cÞ; m
ð5:9Þ ð5:10Þ
2
a2 ¼ 2 2ð1 þ cÞl þ m þ 6m þ mc þ 2c;
ð5:11Þ
a3 ¼ 2 þ ð2 þ 4m þ 2cÞlm 2m2 8m 2mc 2c;
ð5:12Þ
a4 ¼ ð4m 2m2 2mcÞlm þ 2m2 þ 4m þ 2mc:
ð5:13Þ
and
Proof. Further, using Eq. (4.5), we obtain:
$ m2 þ 2h þ mð3 þ 2hÞ 2 2 2 f ðmÞ ðr m Þ m m ðm2 mh þ h2 Þ C 2 en þ f ðxn huÞ ¼ l en 1 þ h C 2 en m! mðm hÞ 2m2 ðm hÞ2 % 4 m 2m3 h þ 4m2 h2 4mh3 þ h4 C 3 e2n þ Oðe3n Þ : þ m2 ðm hÞ2
ð5:14Þ
Substituting (4.2)–(4.5) and (5.14) in Eq. (5.4), we obtain:
enþ1 ¼ K 1 en þ K 2 e2n þ K 3 e3n þ Oðe4n Þ;
ð5:15Þ
where
c 1=c 1 mc k1 þ a0 k2 ; m 2 " #
c 1=c ð1þcÞ c 2lm a0 k2 C 2 m k1 þ a0 k2 K2 ¼ 2 ; 1 m 2 ðm hÞ mc k1 þ ac0 k2
K1 ¼ 1
K 3 ¼ M 1 C 3 þ M 2 C 22 ; M1 ¼
c 1=c k2 ac0 ð3m hÞa0 lm a2 ðm hÞ 2 mc k1 þ a0 k2 m þ ; m3 2 ðm hÞðmc k1 þ ac0 k2 Þ
and
c 1=c " ð1þcÞ m k1 þ ac0 k2 2lm a0 k2 a20 M2 ¼ ðm þ 1Þ þ 2 c c 2 4 3 2 ðm hÞ mc k1 þ ac0 k2 m h ðm hÞ m k1 þ a0 k2 ( mc k1 þ ac0 k2 h2 lm ð3mðm þ 1Þ þ 2ð1 þ 2mÞhÞð2 þ hðh 2ÞÞ ac0 k2 2ð1 þ cÞh4 l2m ac0 k2 þ ðm!Þ2 )!# 2a2 ðm hÞ2 h4 ðm!Þ2 : þ 2lm 3m2 þ mð3 4hÞ þ hð2 þ h þ chÞ ðm!Þ2 þ a20 The proposed scheme (5.4) for multiple roots will be cubically convergent if K 1 ¼ 0 and K 2 ¼ 0 simultaneously.
ð5:16Þ ð5:17Þ ð5:18Þ
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Hence after simplification, we obtain:
mc k1 þ ac0 k2 ¼ 2mc
ð5:19Þ
ð1þcÞ ðm hÞ mc k1 þ ac0 k2 2lm a0 k2 ¼ 0:
ð5:20Þ
and
Solving Eqs. (5.19) and (5.20), k1 and k2 are determined as follows:
k1 ¼ lm
ðh mÞ þ 2l m a0
ð5:21Þ
and ð1þcÞ
k2 ¼ mc ðm hÞlm a0
ð5:22Þ
:
Clearly, for finite and non-zero values of k1 and k2 ; h – m. This completes the proof of the first assertion. For the second part of the theorem, we need to compute the term K3. While doing so, substitute the values of k1 and k2 from Eqs. (5.21) and (5.22) into Eq. (5.18), we get:
2 K3 ¼ 2 m
(
2ðh mÞ
a2 lm a0
þ C 3 þ a0
a1 h3 þ a2 h2 þ a3 h þ a4 ðm hÞh2
) C 22
:
ð5:23Þ
Finally from Eq. (5.15), we obtain:
enþ1
2 ¼ 2 m
(
3 ) 2 m a1 h þ a2 h2 þ a3 h þ a4 2 3 al 2ðh mÞ þ C 3 þ a0 C 2 en þ Oðe4n Þ: a0 ðm hÞh2
ð5:24Þ
This completes the proof of the second assertion. For m = 1 (simple roots), k1 and k2 given by Eqs. (5.21) and (5.22) are both unity respectively. h Special cases: For different specific values of parameters c and h, the following various families of multipoint iterative methods for multiple roots can be derived from formula (5.4). (i) For (c, h) = (1, 1), we get the formula:
xnþ1 ¼ xn
1 f ðxn Þ f ðxn Þ m þ k Hðx Þ k ; 1 2 n 2 f 0 ðxn Þ f ðxn Þ 2f ðxn uÞ
ð5:25Þ
where
ðm1Þ 1 k1 ¼ 2m m 1 m
ð5:26Þ
m 2 ðm1Þ 1 1 k2 ¼ m2 1 2 1 1 : m m
ð5:27Þ
and
This is a new cubically convergent Traub–Ostrowski’s type family of iterative methods for multiple roots. This is a modification over the well-known Traub–Ostrowski’s formula for simple roots. (ii) For (c, h) = (1, 1), we get the formula:
xnþ1 ¼ xn
f ðxn Þ 2mHðxn Þf ðxn Þ ; f 0 ðxn Þ k1 Hðxn Þf ðxn Þ þ k2 m½f ðxn Þ 2f ðxn uÞ
ð5:28Þ
where
ðm1Þ 1 k1 ¼ 2m m 1 m
ð5:29Þ
ðm1Þ 1 k2 ¼ 1 : m
ð5:30Þ
and
This is a new cubically convergent Newton–Secant type family for multiple roots. This is a modification over the Newton– Secant formula derived by Kim and Lee [11].
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Note that the family (5.4) can produce many more new cubically convergent families of multipoint iterative methods for multiple roots by choosing different values of the parameters. 5.2. Second family Replacing the second-order derivative in Eq. (3.10) by following definition:
f 0 ðxn Þ f 0 ðxn huÞ ; hu
f 00 ðxn Þ
h – 0 2 R;
ð5:31Þ
we get the following new generalized family as:
xnþ1 ¼ xn
00c 1=c f ðxn Þ a00c þ b ; f 0 ðxn Þ 2
ð5:32Þ
where 00
a00 ¼ m and b ¼
Hðxn Þ
hf 0 ðxn Þ : ðh 1Þf 0 ðxn Þ þ f 0 ðxn huÞ
The iteration formula (5.32), requires one evaluation of function and two of first-order derivatives per iteration. Further, formula (5.32) forms the basis of the proposed family of schemes, which may be described through the following theorem: Theorem 5.2. Let f : I # R ! R be a sufficiently differentiable function defined on an open interval I, enclosing a multiple root of f(x), say x ¼ rm 2 I with multiplicity m > 1. Assume that initial guess x ¼ x0 is sufficiently close to rm , then for a; c 2 R: (i) The iteration scheme:
xnþ1 ¼ xn
00c 1=c f ðxn Þ d1 a00c þ d2 b ; f 0 ðxn Þ 2
where h – m and h –
" d1 ¼ 2m
2m , mþ1
ð5:33Þ
will have third-order convergence in the vicinity of r m , if:
ðm hÞ2 ð1 þ h þ lm Þ mhð1 hÞlm hðm hÞ½mð2 þ hÞ þ h
#
lðm1Þ
ð5:34Þ
and
d2 ¼ 2
ðh 1Þðm hÞ þ mlm ð1þcÞ c ðm1Þ b0 l : m h½mð2 þ hÞ þ h
ð5:35Þ
(ii) The proposed iterative scheme (5.33) satisfies the following error equation:
enþ1
( ) 2ðm hÞ3 d1 þ d2 h þ d3 h2 þ C 22 b0 b1 þ b2 h þ b3 h2 þ b4 h3 þ b5 h4 e3n þ Oðe4n Þ; ¼2 hm2 ðm hÞ2 ½mð2 þ hÞ þ h
ð5:36Þ
where
h ; m h ; b0 ¼ 1 þ h þ lðm1Þ
l¼ 1
ð5:37Þ ð5:38Þ
b1 ¼ 2m3 ðc þ m þ 2m2 Þð1 lm Þ; 2
m
ð5:39Þ 3
m
2
m
b2 ¼ 7m ½1 þ 2m ð1 þ cÞl m ð1 þ cÞð3 l Þ m ð5c 8ml Þ; b3 ¼ 4mð1 þ cÞ þ 18m2 þ 13m3 þ m4 þ 7m2 c þ m3 c lm 5mð1 þ cÞ þ 7m2 þ 5m2 c þ 2m3 ;
ð5:41Þ
b4 ¼ 1 c 7m 5mc 13m2 2m2 c 3m3 þ lm ð1 þ cÞð1 þ mÞ2 ;
ð5:42Þ
2
b5 ¼ 1 þ 3m þ 2m þ mð1 þ cÞ; 2
1
ð5:40Þ
ð5:43Þ
d1 ¼ 2ma ð1 l Þ;
ð5:44Þ
d2 ¼ 2C 3 þ 2a2 ðm þ 1Þlm ;
ð5:45Þ
d3 ¼ 2C 3 C 3 m 2a2 lm :
ð5:46Þ
and
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Proof. On the similar lines as in the previous theorem. h Special cases: For different specific values of parameters c and h, the following various families of multipoint iterative methods for multiple roots can be derived from family (5.33). (I) For c = 1 in formula (5.33), we obtain the family based on arithmetic mean given by
xnþ1 ¼ xn
1 f ðxn Þ hf 0 ðxn Þ m þ d Hðx Þ d ; 1 2 n 2 f 0 ðxn Þ ðh 1Þf 0 ðxn Þ þ f 0 ðxn huÞ
ð5:47Þ
where
ðm hÞlm ððm hÞðh 1Þ þ mlm Þ d1 ¼ 2 1 þ hðhðm þ 1Þ 2mÞ
ð5:48Þ
and
d2 ¼
2mlm ððm hÞðh 1Þ þ mlm Þ2 h2 ðhðm þ 1Þ 2mÞ
ð5:49Þ
:
Some interesting particular cases of the family (5.47) are: (i) For h ¼ 1 in (5.47), we get the formula:
xnþ1 ¼ xn
1 f ðxn Þ f 0 ðxn Þ m þ d Hðx Þ d ; 1 2 n 2 f 0 ðxn Þ f 0 ðxn uÞ
ð5:50Þ
where
d1 ¼ 2ðm 1Þ
ð5:51Þ
ðm1Þ 1 d2 ¼ 2m2 1 : m
ð5:52Þ
and
This is a new cubically convergent family of iterative methods for multiple roots. This is a modification over the Özban’s formula (5) in [20] for simple roots. 2m (ii) For h ¼ mþ2 in Eq. (5.47), we get the formula:
xnþ1
2 m f ðxn Þ 4 2f 0 ðxn Þ ¼ xn d1 þ d2 Hðxn Þ 2 f 0 ðxn Þ ðm 2Þf 0 ðxn Þ þ ðm þ 2Þf 0 xn
3
5;
2m u mþ2
ð5:53Þ
where
m3 þ 4m2 8 þ m2 ðm 2Þ d1 ¼
m mþ2
m ð5:54Þ
2ðm þ 2Þ
and
d2 ¼
m mþ2
m h
m i2 m ðm þ 2Þ2 þ mðm 2Þ mþ2 4ðm þ 2Þ
:
ð5:55Þ
This is a new cubically convergent Jarratt-type family of iterative methods for multiple roots. This is a modification over the well known Jarratt’s formula [21] for simple roots. (II) For c = 1 in (5.33), we obtain the family based on harmonic mean given by
xnþ1 ¼ xn
2mhHðxn Þf ðxn Þ ; d1 hHðxn Þf 0 ðxn Þ þ d2 m½ðh 1Þf 0 ðxn Þ þ f 0 ðxn huÞ
ð5:56Þ
where
" # ðm hÞ2 ð1 þ h þ lm Þ mhð1 hÞlm ðm1Þ l hðm hÞ½mð2 þ hÞ þ h
d1 ¼ 2m
ð5:57Þ
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and
ðm2Þ 2m 1 mh d2 ¼ : mð2 þ hÞ þ h
ð5:58Þ
Some interesting particular cases of the family (5.56) are (i) For h ¼ 12 in (5.56), we get the formula
xnþ1 ¼ xn
2mHðxn Þf ðxn Þ ; d1 Hðxn Þf 0 ðxn Þ þ d2 m½2f 0 ðxn 0:5uÞ f 0 ðxn Þ
ð5:59Þ
where
n o 3 ðm1Þ 1 m 1 m ð2m 1Þ2 1 þ 2 1 2m 2m 1 2m 4 5 1 1 d1 ¼ 2m 2m ð2m 1Þð1 3mÞ 2
ð5:60Þ
and
d2 ¼
ðm2Þ 4m 1 1 : ð1 3mÞ 2m
ð5:61Þ
This is a new cubically convergent family of iterative methods for multiple roots. This is a modification over the well-known iterative formulas (8)–(12) in [2] for simple roots. Note that the family (5.33) can produce many more new cubically convergent families of multipoint iterative methods for multiple roots by choosing different values of the parameters. It is straight forward to see that per step these methods require three evaluations of function viz. two evaluations of f(x) and one of f 0 ðxÞ or one evaluation of f(x) and two of f 0 ðxÞ. In order to obtain an assessment of the efficiency of our methods, we shall make use of efficiency index defined in [1]. (5.25), (5.28), (5.50), (5.53) and pffiffiffi For our proposed iteration schemes,pnamely ffiffiffi (5.59), we find p = 3 and d = 3 yielding E ¼ 3 3 1:442 which is better than E ¼ 2 1:414, the efficiency index of the Newton’s method. 6. Numerical examples In this section, we shall present the numerical results obtained by employing the iterative methods namely modified Schröder’s method (3.6) (MSM), modified super-Halley method (3.12) (MSHM), modified Halley’s method (3.13) (MHM), modified Ostrowki’s square-root method (3.16) (MOM), method (3.14) (MM1), method (3.15) (MM2), modified Traub– Ostrowski’s method (5.25) (MTOM), modified Newton–Secant method (5.28) (MNSM), modified Jarratt’s method (5.53) (MJM), method (5.50) (MM3) and method (5.59) (MM4) to solve some nonlinear equations with known multiplicity m given in Table 6.1 and compare with classical exiting methods namely modified Newton’s method (MNM), Schröder’s method (SM), super-Halley method (SHM), Halley’s method (HM), Ostrowski’s square-root method (OM), Chen Dong method [15] (CDM), Victory and Neta method [13] (VNM), Traub–Ostrowski’s’ method (TOM), Newton–Secant method [11] (NSM) and Li et al. method [20] (LM). All the formulae are tested for a ¼ 12 and the results are summarized in Tables 6.2 and 6.3 (for one-point and multipoint iterative methods respectively). Computations have been performed using MATLABÒ version 7.5 (R2007b) in double precision arithmetic. We use ¼ 1015 as a tolerance error. The following stopping criteria are used for computer programs:
ðiÞ jxnþ1 xn j < ;
ðiiÞ jf ðxnþ1 Þj < :
Table 6.1 Test problems, multiplicity (m) and root ðr m Þ. Problems 6.1 6.2
Multiplicity (m) ðx3 þ 4x2 10Þ4 ¼ 0 3 sin2 ðxÞ x2 þ 1 ¼ 0
Root ðr m Þ
4
1.36523001341097
3
1.404491648215341
6.3
ðx2 expðxÞ 3x þ 2Þ3 ¼ 0
3
0.2575302854398607
6.4
ðcosðxÞ xÞ2 ¼ 0
2
0.7390851332151607
6.5
ððx 1Þ3 1Þ50 ¼ 0 x2 sinð4xÞ ¼ 0 4 expðx2 þ 7x 30Þ 1 ðx 5Þ ¼ 0
6.6 6.7 6.8 6.9
ð1 þ cosðxÞÞðexpðxÞ 2Þ2 ¼ 0 pffiffi2 sinðxÞ 22 ðx þ 1Þ ¼ 0
50
2.000000000000000
3 4
0.0000000000000000 3.000000000000000
2
0.6931471805599453
2
0.7853981633974483
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S. Kumar et al. / Applied Mathematics and Computation 218 (2012) 7382–7394 Table 6.2 Performance of one-point iterative methods (number of iterations required). Problems
MSM (3.6)
SHM
MSHM (3.12)
HM
MHM (3.13)
OM
MOM (3.16)
MM1 (3.14)
MM2 (3.15)
6.1
Initial guess 1 2
MNM 5 5
SM 5 5
5 5
3 3
3 3
3 3
3 3
3 3
3 3
3 3
3 3
6.2
1 2
6 5
6 5
6 5
3 3
3 3
4 4
4 4
3 4
3 4
3 3
4 3
6.3
0.1 0.5
4 4
4 4
4 4
3 3
3 3
3 3
3 3
3 3
3 3
3 3
3 3
6.4
0.25 1.7
5 5
5 5
5 4
4 4
3 3
3 5
3 3
3 4
3 3
3 3
3 4
6.5
2.5 3.5
4 6
4 6
4 6
2 2
2 2
2 3
2 3
2 3
2 3
2 2
3 3
6.6
0.28 0.35
4 4
4 4
4 4
3 3
3 3
4 4
3 4
3 4
3 4
3 4
3 3
6.7
2.9 3.15
7 7
6 10
6 9
5 6
4 6
4 4
4 4
4 4
3 4
4 5
5 Divergent
6.8
0 1.5
6 5
6 5
5 5
3 4
3 3
4 4
3 4
3 4
4 3
3 3
4 3
6.9
0.1 1.25
5 6
5 5
5 5
4 4
4 3
4 4
5 4
5 4
4 4
5 3
3 4
Table 6.3 Performance of multipoint iterative methods (number of iterations required). Problems
CDM
VNM
TOM
MTOM (5.25)
NSM
MNSM (5.28)
LM
MJM (5.53)
MM3 (5.50)
MM4 (5.59)
6.1
1 2
4 4
3 3
5 5
5 4
5 3
3 3
3 6
3 3
3 8
6 3
6.2
1 2
5 5
4 4
9 5
7 4
5 4
4 5
4 6
5 4
3 3
5 4
6.3
0.1 0.5
3 6
3 4
5 5
3 4
5 5
6 6
4 4
3 4
4 3
3 3
6.4
0.1 0.25
5 6
4 4
3 3
4 4
3 3
4 4
3 3
4 3
7 7
3 3
1.7 2.5 3.5
3 3 4
3 3 4
4 3 4
4 3 4
4 3 4
4 3 4
3 3 3
3 3 3
6 3 4
4 3 4
6.6
0.28 0.35
3 3
3 4
3 4
3 4
4 4
3 4
3 3
3 4
3 3
3 4
6.7
2.9 3.15
3 5
6 4
6 4
5 4
4 7
4 6
7 8
8 5
6 4
8 4
6.8
0 1.5
5 3
4 4
4 4
5 4
4 4
5 4
5 4
5 3
5 4
5 6
6.9
0.1 1.25
4 4
4 4
5 4
3 5
5 4
3 5
4 3
4 4
4 4
3 6
6.5
Initial guess
7. Conclusions This work proposes a new family of Schröder-type methods for finding multiple roots of nonlinear equations based on nonlinear means. Proposed family of Schröder-type methods (3.10) unifies some of the most known third-order iterative methods for solving nonlinear equations and also provide many more unknown processes. Further, we have also presented many new third-order multipoint iterative methods free from second-order derivative for multiple roots. Numerical
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examples presented here prove that the proposed one-point as well as multipoint iterative methods can compete with any of the existing classical iterative methods for multiple roots. A reasonably close initial guess is necessary for the multipoint methods to converge. This condition, however, applies to practically all the iterative methods for solving equations. Acknowledgements We would like to record our sincerest thanks to Professor B. Neta, Associate editor and anonymous reviewer for their constructive suggestions which have considerably contributed to the readability of this paper. References [1] A.M. Ostrowski, Solution of Equations in Euclidean and Banach Space, third ed., Academic Press, New York, 1973. [2] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, 1964. [3] J.E. Dennis, R.B. Schnable, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1983. [4] E. Schröder’s, Über unendlich viele Algorithmen Zur Auflösung der Gleichungen, Math. Ann. 2 (1870) 317–365. [5] W. Werner, Some improvement of classical methods for the solution of nonlinear equations, in numerical solution of nonlinear equations, Lect. Notes Math. 878 (1981) 426–440. [6] E. Hansen, M. Patrick, A family of root finding methods, Numer. Math. 27 (1977) 257–269. [7] L.D. Petkovic´, M.S. Petkovic´, D. Zˇivkovic´, Hansen–Patrick’s family is of Laguerre’s type, Novi SAD J. Math. 33 (2003) 109–115. [8] L.B. Rall, Convergence of Newton’s process to multiple solutions, Numer. Math. 9 (1966) 23–37. [9] C. Chun, B. Neta, A third-order modification of Newton’s method for multiple roots, Appl. Math. Comput. 211 (2009) 474–479. [10] Sanjeev Kumar, Vinay Kanwar, Sushil Kumar Tomar, Sukhjeet Singh, Geometrically constructed families of Newton’s method for unconstrained optimization and nonlinear equations, Int. J. Math. Math. Sci. 2011 (2011) 9, doi:10.1155/2011/972537. Article ID 972537. [11] Y.I. Kim, S.D. Lee, A third-order variant of Newton–Secant method finding a multiple zero, J. Chungcheong Math. Soc. 23 (4) (2010) 845–852. [12] P.S. Bullen, The Power Means, Hand Book of Means and Their Inequalities, Kluwer Dordrecht, Netherlands, 2003. [13] H.D. Victory, B. Neta, A higher order method for multiple zeros of nonlinear functions, Int. J. Comput. Math. 12 (1983) 329–335. [14] C. Dong, A basic theorem of constructing an iterative formula of higher order for computing multiple roots of an equation, Math. Numer. Sin. 11 (1982) 445–450. [15] C. Dong, A family of multipoint iterative functions for finding the multiple roots of equations, Int. J. Comput. Math. 21 (1987) 363–367. [16] N. Osada, An optimal multiple root-finding method of order three, J. Comput. Appl. Math. 51 (1994) 131–133. [17] B. Neta, New third order nonlinear equation solvers for multiple roots, Appl. Math. Comput. 202 (2008) 162–170. [18] B. Neta, Extension of Murakami’s high-order nonlinear solver to multiple roots, Int. J. Comput. Math. 8 (2010) 1023–1031. [19] C. Chun, H.J. Bae, B. Neta, New families of nonlinear solvers for finding multiple roots, Comput. Math. Appl. 57 (2009) 1574–1582. [20] S. Li, X. Liao, L. Cheng, A new fourth-order iterative method for finding the multiple roots of nonlinear equations, Appl. Math. Comput. 215 (2009) 1288–1292. [21] S.G. Li, L.Z. Cheng, B. Neta, Some fourth-order nonlinear solvers with closed formulae for multiple roots, Comput. Math. Appl. 59 (2010) 126–135. [22] B. Neta, Anthony N. Johnson, High-order nonlinear solver for multiple roots, Comput. Math. Appl. 55 (2008) 2012–2017. [23] A.Y. Özban, Some new variants of Newton’s method, Appl. Math. Lett. 17 (2004) 677–682. [24] P. Jarratt, Some fourth order multipoint iterative methods for solving equations, Math. Comput. 20 (1966) 434–437. [25] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach. 21 (1974) 643–651.
