A third-order modification of Newton’s method for multiple roots

Applied Mathematics and Computation 211 (2009) 474–479

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A third-order modification of Newton’s method for multiple roots Changbum Chun a, Beny Neta b,* a b

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943, United States

a r t i c l e

i n f o

a b s t r a c t In this paper, we present a new third-order modification of Newton’s method for multiple roots, which is based on existing third-order multiple root-finding methods. Numerical examples show that the new method is competitive to other methods for multiple roots. Published by Elsevier Inc.

Keywords: Newton’s method Multiple roots Iterative methods Nonlinear equations Order of convergence Root-finding

1. Introduction Solving nonlinear equations is one of the most important problems in numerical analysis. In this paper, we consider iterative methods to find a multiple root a of multiplicity m, i.e., f ðjÞ ðaÞ ¼ 0; j ¼ 0; 1; . . . ; m  1 and f ðmÞ ðaÞ – 0, of a nonlinear equation f ðxÞ ¼ 0. Newton’s method is only of first order unless it is modified to gain the second order of convergence, see Schröder [1]. This . modification requires a knowledge of the multiplicity. Traub [2] has suggested to use any method for f ðmÞ ðxÞ or gðxÞ ¼ ff0ðxÞ ðxÞ Any such method will require higher derivatives than the corresponding one for simple zeros. Also the first one of those methods require the knowledge of the multiplicity m. In such a case, there are several other methods developed by Hansen and Patrick [3], Victory and Neta [4], Dong [5,6], Neta and Johnson [7], Neta [8] and Werner [9]. See also Neta [10]. Since in general one does not know the multiplicity, Traub [2] suggested a way to approximate it during the iteration. The way it is done is by evaluating the quotient

xn2  xn xn2  xn1 and rounding the number up. For example, the quadratically convergent modified Newton’s method is (see [1])

xnþ1 ¼ xn  m

fn fn0

ð1Þ

and the cubically convergent Halley’s method [11] is a special case of the Hansen and Patrick’s method [3]

xnþ1 ¼ xn 

fn mþ1 0 f 2m n



fn fn00 2fn0

;

ðiÞ

ð2Þ

where fn is short for f ðiÞ ðxn Þ. Another third-order method was developed by Victory and Neta [4] and based on King’s fifth order method (for simple roots) [12] * Corresponding author. E-mail addresses: [email protected] (C. Chun), [email protected] (B. Neta). 0096-3003/$ - see front matter Published by Elsevier Inc. doi:10.1016/j.amc.2009.01.087

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475

yn ¼ xn  un ; ð3Þ

f ðyn Þ fn þ Af ðyn Þ ; fn0 fn þ Bf ðyn Þ

xnþ1 ¼ yn  where

A ¼ l2m  lmþ1 ; lm ðm  2Þðm  1Þ þ 1 ; B¼ ðm  1Þ2 m l¼ m1

ð4Þ ð5Þ

and

un ¼

fn : fn0

ð6Þ

Dong [5] has developed two third-order methods requiring two evaluations of f and one evaluation of f 0

8 pffiffiffiffiffi < yn ¼ xn  m un ;  1m : xnþ1 ¼ yn  m 1  p1ffiffiffi m 8 < yn ¼ xn  un ; : xnþ1 ¼ yn þ f ðy

un f ðyn Þ

n Þ

ð1m1 Þ

m1

fn

f ðyn Þ ; fn0

;

ð7Þ

ð8Þ

where un is given by (6). Yet two other third-order methods developed by Dong [6], both require the same information and both based on a family of fourth order methods (for simple roots) due to Jarratt [13]:

8 < yn ¼ xn  un ; fn : xnþ1 ¼ yn  ð m Þmþ1 f 0 ðyn Þþmm2 1fn0 ; m1 ðm1Þ2 8 m un ; < yn ¼ xn  mþ1 : xnþ1 ¼ yn 

m f mþ1 n m f 0 ðy

ð1þm1 Þ

0 n Þfn

;

ð9Þ

ð10Þ

where un is given by (6). Osada [14] has developed a third-order method using the second derivative,

1 1 f0 xnþ1 ¼ xn  mðm þ 1Þun þ ðm  1Þ2 n00 ; 2 2 fn

ð11Þ

where un is given by (6). Neta and Johnson [7] have developed a fourth order method requiring one function- and three derivative-evaluation per step. The method is based on Jarratt’s method [15] given by the iteration

xnþ1 ¼ xn 

fn ; a1 fn0 þ a2 f 0 ðyn Þ þ a3 f 0 ðgn Þ

ð12Þ

where un is given by (6) and

yn ¼ xn  aun ;

vn ¼

fn ; f 0 ðyn Þ

ð13Þ

gn ¼ xn  bun  cv n : Neta and Johnson [7] give a table of values for the parameters a; b; c; a1 ; a2 ; a3 for several values of m. In the case m ¼ 2 they found a method that will require only two derivative-evaluations (a3 ¼ 0). This was not possible for higher m. Neta [8] has developed a fourth order method requiring one function- and three derivative-evaluation per step. The method is based on Murakami’s method [16] given by the iteration

xnþ1 ¼ xn  a1 un  a2 v n  a3 w3 ðxn Þ  wðxn Þ; where un is given by (6),

ð14Þ

v n ; yn , and gn are given by (13) and

fn ; w3 ðxn Þ ¼ 0 f ðgn Þ fn wðxn Þ ¼ : b1 fn0 þ b2 f 0 ðyn Þ

ð15Þ

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Neta [8] gives a table of values for the parameters a; b; c; a1 ; a2 ; a3 ; b1 ; b2 for several values of m. A method of order 1.5 requiring two function- and one derivative-evaluation is given by Werner [9]. It is only for double roots

y n ¼ xn  u n ;

ð16Þ

xnþ1 ¼ xn  sn un ; where

sn ¼

8 <

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ; if f ðyn Þ=fn 6 14 ;

1þ

14f ðyn Þ=fn

: 1 f =f ðy Þ; n 2 n

otherwise:

Later we give a table comparing the efficiency of all known methods for multiple roots including our new ones we develop here. The informational efficiency, E, of a method of order p using d function/derivative evaluations is defined [2] as

p E¼ : d

ð17Þ

The efficiency index, I, is defined as

I ¼ p1=d :

ð18Þ

It can be seen in Table 1 that our new method is competitive with the previously developed schemes. There is an approach in [17] that uses any pair of existing methods in constructing new iterative methods or families of the same order in the case of simple roots. Now, a simple question arises; does the approach also work for multiple roots case? The answer indeed is affirmative as will be seen in the following section. To this end and for the sake of illustration, two methods for multiple roots are considered. One is Osada’s third-order method (11) and the other is the Euler–Chebyshev method of order three [2]

xnþ1 ¼ xn 

mð3  mÞ f ðxn Þ m2 f ðxn Þ2 f 00 ðxn Þ :  2 f 0 ðxn Þ 2 f 0 ðxn Þ3

ð19Þ

The methods (11) and (19) are used to obtain a new modification of Newton’s method for multiple roots. The new method is shown to be cubically convergent by analysis of convergence, its performance and practical utility are demonstrated by numerical examples. 2. Development of method and convergence analysis Now, we approximately equate the correcting terms of both methods (11) and (19) to obtain the following approximate expression:

1 f ðxn Þ 1 f 0 ðxn Þ mð3  mÞ f ðxn Þ m2 f ðxn Þ2 f 00 ðxn Þ  mðm þ 1Þ 0 ; þ ðm  1Þ2 00   2 f ðxn Þ 2 f ðxn Þ 2 f 0 ðxn Þ 2 f 0 ðxn Þ3

ð20Þ

Table 1 Comparison of methods for multiple roots. Algorithm

p

d

E

I

f0

Werner [9] (16) m ¼ 2 Schröder [1] (1) Hansen and Patrick [3] Halley (2) Laguerre Hansen and Patrick [3] Victory and Neta [4] (3) Dong [5] (7), (8) Dong [6] (9), (10) Osada [14] Neta and Johnson [7] (12) m – 2 Neta and Johnson [7] (12) m ¼ 2 Neta [8] Neta [19] m – 3 Neta [19] m ¼ 3 Neta [19] Neta [19] Neta [19] Chun and Neta (22)

1.5 2 3 3 3 3 3 3 3 3 4 4 4 3 2 3 2.732 2.732 3

3 2 3 3 3 4 3 3 3 3 4 3 4 3 3 3 2 2 3

0.5 1 1 1 1 .75 1 1 1 1 1 1.333 1 1 .667 1 1.366 1.366 1

1.145 1.414 1.442 1.442 1.442 1.316 1.442 1.442 1.442 1.442 1.414 1.587 1.414 1.442 1.259 1.442 1.653 1.653 1.442

1 1 1 1 1 1 1 1 2 1 3 2 1 1 1 1 1 1 1

f 00

f 000

1 1 1 1

1

1

3 1 1

1

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477

this gives a new approximation

f ðxn Þf 00 ðxn Þ m  1 0 1 ðm  1Þ2 f 0 ðxn Þ3 f ðxn Þ   : 0 2f ðxn Þ m 2 m2 f ðxn Þf 00 ðxn Þ

ð21Þ

If we apply the approximation (21) to Halley’s method (2), then we obtain a new method

xnþ1 ¼ xn 

2m2 f ðxn Þ2 f 00 ðxn Þ mð3  mÞf ðxn Þf 0 ðxn Þf 00 ðxn Þ þ ðm  1Þ2 f 0 ðxn Þ3

:

ð22Þ

For (22), we have Theorem 1. Let a 2 I be a multiple root of multiplicity m of a sufficiently differentiable function f : I ! R for an open interval I. If x0 is sufficiently close to a, then the method defined by (22) has third-order convergence, and satisfies the error equation

enþ1 ¼

ðm2 þ 3ÞC 21  2mðm  1ÞC 2 3 en þ Oðe4n Þ; 2m2 ðm  1Þ

m! where en ¼ xn  a and C j ¼ ðmþjÞ!

ð23Þ

f ðmþjÞ ðaÞ . f ðmÞ ðaÞ

Proof. Using Taylor expansion of f ðxn Þ about a, we have

 f ðmÞ ðaÞ m  en 1 þ C 1 en þ C 2 e2n þ Oðe3n Þ ; m!   f ðmÞ ðaÞ m1 mþ1 mþ2 1þ f 0 ðxn Þ ¼ en C 1 en þ C 2 e2n þ Oðe3n Þ ; ðm  1Þ! m m " # ( )

2 2 ðmÞ f ðaÞ mþ1 mþ1 mþ2 2m2 2 3 e f 0 ðxn Þ2 ¼ e 1 þ 2 e þ þ 2 þ Oðe Þ ; C C C 1 n 1 2 n n n m m m ½ðm  1Þ!2   f ðmÞ ðaÞ m2 mþ1 ðm þ 1Þðm þ 2Þ 1þ en C 1 en þ C 2 e2n þ Oðe3n Þ ; f 00 ðxn Þ ¼ ðm  2Þ! m1 mðm  1Þ f ðxn Þ ¼

m! where C j ¼ ðmþjÞ!

f ðmþjÞ ðaÞ f ðmÞ ðaÞ

ð24Þ ð25Þ ð26Þ ð27Þ

and en ¼ xn  a.

Dividing (24) by (25) gives us

" # f ðxn Þ en 1 ðm þ 1ÞC 21  2mC 2 2 3 e þ e þ Oðe Þ : C ¼ 1  1 n n n f 0 ðxn Þ m m m2

ð28Þ

We now use Maple [18] to collect all these expansions into (22) to have 2

 2 2ðm  1Þ þ ð2m2  m þ 1ÞC 1 en 2 m ððm  1Þ!Þ  3  2m þ m2 þ 1 2 4m2  2m þ 10 2 þ C þ C 2 en 1 m2 m  3  m þ m2 þ m þ 1 m2  m þ 6 3 C þ4 C C þ 1 2 3 en ; m2 m

mð3  mÞf ðxn Þf 00 ðxn Þ þ ðm  1Þ2 ðf 0 ðxn ÞÞ ¼

2m2 f ðxn Þf 00 ðxn Þ ¼

en2m2 2

e2m2 n

ð29Þ

n h i 2mðm  1Þ þ 4m2 C 1 en þ 2mðm þ 1ÞC 21 þ 4ðm2 þ m þ 1ÞC 2 e2n

ððm  1Þ!Þ h i o þ4 ðm2 þ 2m þ 3ÞC 3 þ ðm þ 1Þ2 C 1 C 2 e3n :

ð30Þ

Now divide (30) by (29) and multiply by (28) and collect terms we have

enþ1 ¼

ðm2 þ 3ÞC 21  2mðm  1ÞC 2 3 en þ Oðe4n Þ; 2m2 ðm  1Þ

ð31Þ

which indicates that the order of convergence of the methods defined by (22) is at least three. This completes the proof. h Note that the method requires one evaluation of the function, the first and the second derivative at each step, therefore the informational efficiency is E ¼ 1 and the efficiency index is I ¼ 1:442. In a similar fashion, the proposed approach can be continuously applied to produce various types of new approximations from the other iterative methods for multiple roots, which can in turn be freely employed to find many new iterative methods or families for multiple roots.

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3. Numerical examples We present some numerical test results for various third-order multiple root-finding methods as well as our new methods and the Newton method in Table 3. Compared were the Newton method (1) (NM), Halley-like method (2) (HM), Osada’s method (11) (OM), the Euler–Chebyshev method (19) (ECM), and the method (22) (CM) introduced in this contribution. All computations were done using MAPLE with 128 digit floating point arithmetics (Digits :¼ 128). Displayed in Table 3 are the number of iterations (IT) required such that jf ðxn Þj < 1032 , and the value of jf ðxn Þj after the required iterations. The following functions are used for the comparison and we display the approximate zeros x found up to the 28th decimal places

f1 ðxÞ ¼ ðx3 þ 4x2  10Þ3 ; 2

x ¼ 1:3652300134140968457608068290;

2

2

f2 ðxÞ ¼ ðsin x  x þ 1Þ ;

x ¼ 1:4044916482153412260350868178;

f3 ðxÞ ¼ ðx2  ex  3x þ 2Þ5 ; 3

f4 ðxÞ ¼ ðcos x  xÞ ;

x ¼ 0:73908513321516064165531208767;

3

6

f5 ðxÞ ¼ ððx  1Þ  1Þ ; 2

x ¼ 0:25753028543986076045536730494;

x ¼ 2;

2

f6 ðxÞ ¼ ðxex  sin x þ 3 cos x þ 5Þ4 ; 2

f7 ðxÞ ¼ ðsin x  x=2Þ ;

x ¼ 1:2076478271309189270094167584;

x ¼ 1:8954942670339809471440357381:

We also ran the second case with various initial points and average the number of iterations required for convergence. These results are given in Table 2. The results presented in Tables 2 and 3 show that for the functions we tested, the new method introduced here have at least equal performance as compared to the other multiple root-finding methods of the same order, and also converge more rapidly than Newton’s method for multiple roots.

Table 2 Comparison of various third-order multiple root-finding methods and Newton’s method. The last row gives the average number of iterations required for convergence when starting from various initial guesses. x0

NM

HM

OM

ECM

CM

0.0001 1 2 3 4 5 6 7 8 9

46 7 7 7 7 8 8 8 9 9 11.6

23 5 5 5 6 5 6 6 6 9 7.6

37 6 5 6 5 6 7 7 7 7 9.3

68 6 5 6 5 6 7 6 6 6 12.1

35 6 5 6 5 6 7 6 7 6 8.9

Table 3 Comparison of various third-order multiple root-finding methods and Newton’s method. f ðxÞ

IT ðjf ðxn ÞjÞ NM

HM

OM

ECM

CM

f1 ; x 0 ¼ 2 f1 ; x0 ¼ 1

6(8.49e54) 6(4.91e62)

4(7.06e49) 4(3.38e57)

4(6.47e33) 5(5.40e84)

4(4.01e38) 4(1.94e38)

4(4.01e38) 4(1.94e38)

f2 ; x0 ¼ 2:3 f2 ; x0 ¼ 2

7(7.31e52) 7(5.11e64)

5(4.84e57) 5(7.43e77)

5(2.07e38) 5(3.53e51)

5(1.73e47) 5(1.53e63)

5(4.55e42) 5(4.09e56)

f3 ; x0 ¼ 0 f3 ; x0 ¼ 1

4(1.03e55) 4(3.46e52)

3(1.68e53) 4(1.39e85)

3(5.83e62) 4(2.01e91)

3(4.31e58) 4(2.24e89)

3(1.71e55) 4(1.93e87)

f4 ; x0 ¼ 1:7 f4 ; x0 ¼ 1

5(6.04e47) 5(1.22e60)

4(9.12e43) 4(1.78e85)

4(1.17e39) 4(1.42e78)

4(5.25e41) 4(1.43e81)

4(5.25e41) 4(-1.43e81)

f5 ; x0 ¼ 3 f5 ; x0 ¼ 1

6(2.70e45) 10(5.23e49)

4(7.44e45) 11(2.22e65)

5(3.12e85) 24(7.70e44)

5(1.89e94) 23(1.87e52)

4(3.55e37) 5(2.67e77)

f6 ; x0 ¼ 2 f6 ; x0 ¼ 1

8(5.60e37) 6(5.61e60)

5(1.60e61) 3(4.75e35)

6(5.09e45) 5(1.56e103)

6(3.21e64) 4(1.47e47)

6(2.83e82) 4(9.70e58)

f7 ; x0 ¼ 1:7 f7 ; x0 ¼ 2

6(3.80e57) 5(2.09e40)

4(7.40e47) 4(1.55e65)

5(1.81e76) 4(3.45e53)

4(1.01e37) 4(1.67e59)

5(1.03e92) 4(8.23e56)

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