A New Family of Iterative Methods Based on an Exponential Model for Solving Nonlinear Equations

Abstract

We present two new families of iterative methods for obtaining simple roots of nonlinear equations. The first family is developed by fitting the model m ( x ) = [superscript] e p x [/superscript] ( A [superscript] x 2 [/superscript] + B x + C ) to the function f ( x ) and its derivative [superscript] f [variant prime] [/superscript] ( x ) , [superscript] f '' [/superscript] ( x ) at a point [subscript] x n [/subscript] . In order to remove the second derivative of the first methods, we construct the second family of iterative methods by approximating the equation f ( x ) = 0 around the point ( [subscript] x n [/subscript] , f ( [subscript] x n [/subscript] ) ) by the quadratic equation. Analysis of convergence shows that the new methods have third-order or higher convergence. Numerical experiments show that new iterative methods are effective and comparable to those of the well-known existing methods.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Tianbao Liu 1 and Hengyan Li 2 and Zaixiang Pang 3

Academic Editor:Yongkun Li

1, School of Basic Science, Changchun University of Technology, Changchun 130012, China
2, College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
3, Engineering Training Center, Changchun University of Technology, Changchun 130012, China

Received 17 September 2013; Accepted 2 November 2013

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, we consider iterative methods to find a simple root α , that is, f ( α ) = 0 and ^{f [variant prime]} ( α ) ...0; 0 , of a nonlinear equation: [figure omitted; refer to PDF] where f : I ⊂ R [arrow right] R for an open interval I is a scalar function.

Many of the complex problems in science and engineering contains the function of nonlinear and transcendental nature in (1), so finding the simple roots of the nonlinear equation is one of the most important problems in numerical analysis. Numerical iterative methods are often used to obtain the approximate solution of such problems because it is not always possible to obtain its exact solution by usual algebraic process. We all know that Newton's method is an important and basic approach for solving nonlinear equations [1, 2], its formulation is given by [figure omitted; refer to PDF] and this method converges quadratically. Earlier, many investigations [3-21] have been made to explain the root of nonlinear algebraic and transcendental equations.

The outline of the paper is as follows. In Section 2, we firstly describe a one-parameter family of third-order methods by fitting the model m ( x ) = ^{e p x} ( A ^{x 2} + B x + C ) to the function f ( x ) and its derivative ^{f [variant prime]} ( x ) , ^{f [variant prime][variant prime]} ( x ) at a point _{x n} , and then we use a quadratic equation for approximating the equation f ( x ) = 0 to obtain a four-parameter family of second-derivative-free iterative methods. In Section 3, we obtain some different iterative methods by taking several parameters. In Section 4, different numerical tests confirm the theoretical results, and the new methods are comparable with other known methods and give better results in many cases. Finally, we infer some conclusions.

2. Development of Methods and Convergence Analysis

Consider the exponential model: [figure omitted; refer to PDF] where A , B , C , and p are parameters. We construct a new iteration scheme by fitting model (3) to the function f ( x ) and its derivative ^{f [variant prime]} ( x ) and ^{f [variant prime][variant prime]} ( x ) at a point _{x n} . Imposing the conditions as follows [figure omitted; refer to PDF] at the point ( _{x n} , m ( _{x n} ) ) and then solving (3) and (4) for A , B , and C , we have [figure omitted; refer to PDF] From (3), (4), and (5), we take the values of A , B , and C into (3) and give [figure omitted; refer to PDF] At the root estimate _{x n + 1} , it follows that m ( _{x n + 1} ) = 0 . We consider reducing (8) as follows [figure omitted; refer to PDF] Thus from (9) we develop the iteration formula: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The square root is required in (10); however, this may cost expensively and even fail in the case 1 - _{L f , p} ( _{x n} ) < 0 . In order to avoid the calculation of the square roots, we will derive some forms free from square roots by Taylor approximation [5].

It is easy to know that Taylor approximation of 1 - _{L f , p} ( _{x n} ) is [figure omitted; refer to PDF]

Using (12) in (10), we can obtain the following form: [figure omitted; refer to PDF]

We have the convergence analysis of the methods by (13).

Theorem 1.

Let α ∈ I be a simple zero of sufficiently differentiable function f : I ⊂ R [arrow right] R for an open interval I . If _{x 0} is sufficiently close to α , for m ...5; 1 , the methods defined by (13) are cubically convergent.

Proof.

Let _{e n} = _{x n} - α , and we use the following Taylor expansions: [figure omitted; refer to PDF] where _{c k} = ( 1 / k ! ) ( ^{f ( k )} ( α ) / ^{f [variant prime]} ( α ) ) ; furthermore, we have [figure omitted; refer to PDF] Since (15), we obtain [figure omitted; refer to PDF] Since (14), we get [figure omitted; refer to PDF] From (14) and (15), we also easily have [figure omitted; refer to PDF] We obtain the following expression by taking into account (14) and (16): [figure omitted; refer to PDF] From (18), (19), and (20), we obtain [figure omitted; refer to PDF] From (17) and (21), we have [figure omitted; refer to PDF] From (14) and (15), we have [figure omitted; refer to PDF] Using (14), (15), and (16), we have [figure omitted; refer to PDF] We know that [figure omitted; refer to PDF] From (15) and (25), we obtain [figure omitted; refer to PDF] From (23) and (26), we have [figure omitted; refer to PDF] Using (13), (24), and (27), we have [figure omitted; refer to PDF] From _{e n + 1} = _{x n + 1} - α , we have [figure omitted; refer to PDF] which completes the proof.

The family methods given by (13) are novel third-order methods, but the methods depend on the second derivatives in computing process, and therefore their practical applications are restricted in some cases. In recent years, several methods with free second derivatives have been developed; see [4-15] and references therein.

In order to avoid the calculation of the second derivatives, we consider approximating the equation f ( x ) = 0 around the point ( _{x n} , f ( _{x n} ) ) by the quadratic equation in x and y in the following form [8]: [figure omitted; refer to PDF] where _{a i} ∈ R , i = 1,2 , ... , 6 , are parameters. We impose the tangency conditions [figure omitted; refer to PDF] where _{x n} is n th iterate and [figure omitted; refer to PDF] From (30) and (31), we have [figure omitted; refer to PDF] where i = 1,2 , 3 .

From (33) we can approximate [figure omitted; refer to PDF] Using _{L a i , f , p} ( _{x n} , _{w n} ) instead of _{L f , p} ( _{x n} ) (11), we obtain a new four-parameter family of methods free from second derivative: [figure omitted; refer to PDF] where _{a i} ( i = 1,2 , 3 ), p ∈ R , m ...5; 0 .

We also have the convergence analysis of the methods by (35).

Theorem 2.

Let α ∈ I be a simple zero of sufficiently differentiable function f : I ⊂ R [arrow right] R for an open interval I . If _{x 0} is sufficiently close to α , for m ...5; 1 , _{a i} ( i = 1,2 , 3 ), p ∈ R , the methods defined by (35) are at least cubically convergent; as particular cases, if m ...5; 2 , _{a 2} = _{a 3} = p = 0 , _{a 1} ∈ R and the methods have convergence order four.

Proof.

Let _{e n} = _{x n} - α , and we use the following Taylor expansions: [figure omitted; refer to PDF] where _{c k} = ( 1 / k ! ) ( ^{f ( k )} ( α ) / ^{f [variant prime]} ( α ) ) ; furthermore, we have [figure omitted; refer to PDF] Dividing (36) by (38) [figure omitted; refer to PDF] From (39), we get [figure omitted; refer to PDF] Expanding f ( _{w n} ) in Taylor's Series about α and using (40), we get [figure omitted; refer to PDF] From (36) and (41), we have [figure omitted; refer to PDF] From (38), we obtain [figure omitted; refer to PDF] From (38), (41), and (43) we also easily obtain [figure omitted; refer to PDF] Substituting (36), (37), (38), (41), and (43) in the denominator of _{L a i} ( _{x n} , _{w n} ) , we obtain [figure omitted; refer to PDF] Using (44) and (45), we have [figure omitted; refer to PDF] From (36), (37), and (46), we have [figure omitted; refer to PDF] Since (43) and (47), we get [figure omitted; refer to PDF] Using (48), we write [figure omitted; refer to PDF] Using (36), (38), and (46), we obtain [figure omitted; refer to PDF] Taking into account (36), (38), (49), and (50), we finally obtain [figure omitted; refer to PDF] Taking into account the last expression (51) and _{e n + 1} = _{x n + 1} - α , we have [figure omitted; refer to PDF]

This means that the methods defined by (35) are at least of order three for any _{a i} ( i = 1,2 , 3 ), p ∈ R . Furthermore, we consider that if m ...5; 2 , _{a 2} = _{a 3} = p = 0 , then the methods defined by (35) are shown to converge the order four.

3. Some Special Cases

From (33)-(35), we have [figure omitted; refer to PDF] where _{a 1} , _{a 2} , _{a 3} ∈ R . From (33) we can approximate [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] where _{a 1} , _{a 2} , _{a 3} , p ∈ R , m ...5; 0 .

^{1 0} :: If _{a 1} = _{a 2} = _{a 3} = p = 1 , and m = 3 , from (55), we obtain a third-order method (LM1): [figure omitted; refer to PDF]

^{2 0} :: If _{a 1} = _{a 2} = 0 , _{a 3} = p = 1 , and m = 3 , from (55) we also obtain a third-order method (LM2): [figure omitted; refer to PDF]

^{3 0} :: If _{a 1} = 1 , _{a 2} = 0 , _{a 3} = p = 1 , and m = 3 , from (55) we obtain a new third-order method (LM3): [figure omitted; refer to PDF]

^{4 0} :: If _{a 1} = - 1 , _{a 2} = _{a 3} = 0 , p = 1 , and m = 3 , from (55) we obtain a third-order method (LM4): [figure omitted; refer to PDF]

^{5 0} :: If _{a 1} = _{a 2} = 0 , _{a 3} = - 1 , p = 1 , and m = 3 , from (55) we obtain a new third-order method (LM5): [figure omitted; refer to PDF]

4. Numerical Examples

In this section, some numerical examples commonly used in the literature are presented in Table 1 to check the effectiveness of the new methods. The following methods were compared: Newton method (NM), the method of Weerakoon and Fernando [10] (WF), the method of Potra and Pták (PP) [11], Chebyshev's method (CHM) [12, 13], Halley's method (HM) [12], and our new methods (56) (LM1), (57) (LM2), (58) (LM3), (59) (LM4), and (60) (LM5). Displayed in Table 1 are the number of iterations (IT), the number of function evaluations (NFE) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative, the value f ( _{x n + 1} ) , the computing time (TIME, the unit of time is one second), and the distance of two consecutive approximations δ = | _{x n + 1} - _{x n} | . All computations were done using Matlab 7.1 environment with a ADM athlon (tm) II X2 250-3.01 GHz based PC. We accept an approximate solution rather than the exact root, depending on the precision ... of the computer. We use the following stopping criteria for computer programs: | f ( _{x n + 1} ) | < ... , we used the fixed stopping criterion ... = 1 ^{0 - 15} .

Table 1: Comparison of various third-order methods and Newton's method.

	IT	NFE	f ( x n + 1 )	Time	δ
f 1 : x 0 = 1
NM	5	10	0	0.062577	2.126987475037367 e - 011
WF	3	9	0	0.018731	2.284722713019605 e - 006
PP	4	12	0	0.044520	1.558753126573720 e - 013
CHM	4	12	0	0.040102	1.643130076445232 e - 014
HM	3	9	0	0.019876	3.698649917449615 e - 007
LM1	4	12	0	0.029137	1.043609643147647 e - 014
LM2	3	9	0	0.022242	1.788683664960544 e - 006
LM3	3	9	0	0.023550	1.295199573814188 e - 006
LM4	3	9	0	0.019227	3.091731315407742 e - 010
LM5	3	9	0	0.020131	1.788683664960544 e - 006
f 1 : x 0 = 2
NM	5	10	0	0.057231	5.020497351182485 e - 010
WF	4	12	0	0.039384	4.440892098500626 e - 016
PP	4	12	0	0.044239	7.949196856316121 e - 014
CHM	4	12	0	0.042782	2.065014825802791 e - 014
HM	3	9	0	0.017733	3.107350415199051 e - 006
LM1	4	12	0	0.032638	2.706235235905297 e - 011
LM2	4	12	0	0.037830	5.551115123125783 e - 015
LM3	4	12	0	0.034087	3.108624468950438 e - 015
LM4	4	12	0	0.038399	2.220446049250313 e - 016
LM5	4	12	0	0.035809	5.551115123125783 e - 015
f 2 : x 0 = 0
NM	4	8	0	0.035702	2.665312415217613 e - 012
WF	3	9	0	0.019699	7.801814749797131 e - 012
PP	3	9	0	0.021596	1.219191414492116 e - 012
CHM	3	9	0	0.023312	8.906764215055318 e - 013
HM	3	9	0	0.017431	7.374600929921371 e - 012
LM1	3	9	0	0.016351	9.004856806882344 e - 007
LM2	3	9	0	0.017242	8.067152316715287 e - 007
LM3	3	9	0	0.018978	8.334935165388302 e - 007
LM4	3	9	0	0.016178	7.334822825222354 e - 007
LM5	3	9	0	0.016795	8.067152316715287 e - 007
f 2 : x 0 = 0.5
NM	4	8	0	0.044392	1.791899961745003 e - 013
WF	3	9	0	0.023132	6.424749621203318 e - 012
PP	3	9	0	0.024293	4.607425552194400 e - 014
CHM	3	9	0	0.026689	3.087480271446452 e - 011
HM	3	9	0	0.022199	4.208039472430869 e - 011
LM1	3	9	0	0.024120	2.850281847210923 e - 007
LM2	3	9	4.440892098500626 e - 016	0.021035	2.744941123289379 e - 007
LM3	3	9	0	0.021310	2.778229745148408 e - 007
LM4	3	9	0	0.020886	2.655207446689012 e - 007
LM5	3	9	4.440892098500626 e - 016	0.021592	2.744941123289379 e - 007
f 3 : x 0 = 1
NM	7	14	3.537126081266182 e - 024	0.085178	1.085848323840232 e - 012
WF	4	12	2.621304391538411 e - 016	0.030341	4.330310691887267 e - 006
PP	5	15	8.196910187379942 e - 033	0.063971	8.806888499109001 e - 012
CHM	5	15	6.352230116524407 e - 022	0.053064	2.520356663650445 e - 011
HM	5	15	7.257520328033309 e - 029	0.055315	8.459855063117184 e - 015
LM1	5	15	5.355132874954819 e - 017	0.041370	4.825251676864564 e - 007
LM2	4	12	6.348042447121620 e - 018	0.026722	5.276208195236892 e - 013
LM3	3	9	1.559423551656343 e - 017	0.019476	5.545966966175517 e - 011
LM4	4	12	6.485252546346663 e - 018	0.030513	1.317913979899399 e - 009
LM5	4	12	6.348042447121620 e - 018	0.031575	5.276208195236892 e - 013
f 3 : x 0 = 0.5
NM	6	12	5.905159674954809 e - 020	0.071738	1.402992074360412 e - 010
WF	4	12	1.764824578467612 e - 017	0.039948	4.200981459101664 e - 009
PP	4	12	1.694834561079519 e - 019	0.056046	2.767209186089879 e - 007
CHM	4	12	8.798671206634291 e - 017	0.036458	3.059650585008672 e - 007
HM	4	12	3.044907255908736 e - 017	0.072238	5.518067735074518 e - 009
LM1	4	12	1.293991275513200 e - 017	0.033751	1.322689611777333 e - 013
LM2	4	12	2.828488650436813 e - 016	0.032997	4.626219030783813 e - 006
LM3	4	12	6.789745028898642 e - 018	0.036425	1.912594303370619 e - 014
LM4	4	12	1.506225012219799 e - 017	0.034137	2.365541379047147 e - 011
LM5	4	12	2.828488650436813 e - 016	0.035087	4.626219030783813 e - 006
f 4 : x 0 = 2.5
NM	6	12	0	0.071353	1.154631945610163 e - 014
WF	4	12	0	0.034140	7.314593375440381 e - 012
PP	4	12	0	0.042598	4.221685223626537 e - 010
CHM	4	12	0	0.036365	9.853584614916144 e - 011
HM	4	12	0	0.038080	4.662936703425658 e - 014
LM1	3	9	0	0.018372	9.336336148635382 e - 010
LM2	4	12	0	0.035562	1.610311883837312 e - 010
LM3	4	12	0	0.036985	2.312174895990893 e - 010
LM4	4	12	0	0.035463	5.127454016928823 e - 012
LM5	4	12	0	0.033362	1.610311883837312 e - 010
f 4 : x 0 = 3.5
NM	7	14	0	2.151288	2.877564853065451 e - 011
WF	5	15	0	0.054803	6.550315845288424 e - 013
PP	5	15	0	0.063380	4.512221707386743 e - 010
CHM	5	15	0	0.061753	4.188738245147761 e - 011
HM	4	12	0	0.046608	4.485352507632712 e - 006
LM1	5	15	6.661338147750939 e - 016	0.050909	2.781577124189028 e - 008
LM2	5	15	0	0.048688	9.426681657487279 e - 011
LM3	10	30	0	0.111640	1.081885248055414 e - 009
LM4	4	12	0	0.040109	2.154232348061669 e - 011
LM5	5	15	0	0.043482	9.426681657487279 e - 011
f 5 : x 0 = 0
NM	5	10	0	0.050123	1.701233598438989 e - 010
WF	3	9	0	0.023397	7.792236328407753 e - 007
PP	4	12	0	0.038373	1.500558566291943 e - 010
CHM	4	12	0	0.037498	5.327979279989847 e - 009
HM	4	12	0	0.035701	1.121325254871408 e - 014
LM1	4	12	3.330669073875470 e - 016	0.028362	4.678117765610779 e - 006
LM2	4	12	0	0.028666	7.181037887660224 e - 007
LM3	6	18	0	0.053924	6.676881270095691 e - 013
LM4	4	12	0	0.028247	2.163133006050089 e - 010
LM5	4	12	0	0.029150	7.181037887660224 e - 007
f 5 : x 0 = 1
NM	4	8	0	0.039328	1.701233598438989 e - 010
WF	2	6	4.440892098500626 e - 016	0.003092	2.674277017133964 e - 005
PP	3	9	0	0.020461	9.809075773858922 e - 011
CHM	3	9	0	0.019378	1.600380383770528 e - 009
HM	3	9	0	0.018186	6.624212289807474 e - 010
LM1	3	9	0	0.018199	7.500783549829748 e - 008
LM2	3	9	1.110223024625157 e - 016	0.017732	5.330569952111119 e - 008
LM3	3	9	0	0.013523	8.804162354714151 e - 008
LM4	3	9	0	0.013782	3.531516445942629 e - 008
LM5	3	9	1.110223024625157 e - 016	0.016581	5.330569952111119 e - 008
f 6 : x 0 = 1
NM	6	12	3.330669073875470 e - 016	0.065520	3.059774655866931 e - 013
WF	4	12	4.440892098500626 e - 016	0.030929	1.793023507445923 e - 010
PP	16	48	4.440892098500626 e - 016	0.266014	1.531728257564424 e - 007
CHM	5	15	4.440892098500626 e - 016	0.061006	6.883094094689568 e - 010
HM	4	12	4.440892098500626 e - 016	0.043655	2.686739719592879 e - 013
LM1	4	12	3.330669073875470 e - 016	0.030877	1.659126835917846 e - 008
LM2	4	12	3.330669073875470 e - 016	0.031156	9.103828801926284 e - 014
LM3	4	12	3.330669073875470 e - 016	0.033076	1.261090962767497 e - 006
LM4	5	15	4.440892098500626 e - 016	0.041102	8.215650382226158 e - 014
LM5	4	12	3.330669073875470 e - 016	0.030923	9.103828801926284 e - 014
f 6 : x 0 = 2.5
NM	6	12	3.330669073875470 e - 016	0.064979	1.404654170755748 e - 012
WF	4	12	3.330669073875470 e - 016	0.038404	4.229505634611996 e - 012
PP	4	12	3.330669073875470 e - 016	0.042164	1.030850205196998 e - 008
CHM	4	12	3.330669073875470 e - 016	0.043437	1.475204565171140 e - 007
HM	4	12	4.440892098500626 e - 016	0.043896	9.462626682221753 e - 009
LM1	5	15	3.330669073875470 e - 016	0.053132	6.201483770951199 e - 012
LM2	4	12	3.330669073875470 e - 016	0.036191	4.450250368215336 e - 008
LM3	6	18	3.330669073875470 e - 016	0.064485	8.881784197001252 e - 016
LM4	4	12	3.330669073875470 e - 016	0.035393	5.568381311604753 e - 009
LM5	4	12	3.330669073875470 e - 016	0.036946	4.450250368215336 e - 008
f 7 : x 0 = 3.25
NM	8	16	0	0.114189	9.720393379097914 e - 010
WF	6	18	0	0.065645	1.691979889528739 e - 013
PP	6	18	0	0.073469	1.131490456884876 e - 010
CHM	6	18	0	0.082677	2.398081733190338 e - 014
HM	5	15	0	0.055630	3.082423205569285 e - 012
LM1	4	12	0	0.036191	2.613820271335499 e - 011
LM2	5	15	0	0.049599	4.440892098500626 e - 016
LM3	4	12	0	0.038172	1.546353226355990 e - 006
LM4	6	18	0	0.051577	4.440892098500626 e - 016
LM5	5	15	0	0.048340	4.440892098500626 e - 016
f 7 : x 0 = 3.45
NM	11	22	0	0.147308	4.008793297316515 e - 011
WF	8	24	0	0.106276	7.105427357601002 e - 015
PP	8	24	0	0.116999	2.160227552394645 e - 010
CHM	7	21	0	0.100725	1.268533917908599 e - 006
HM	6	18	0	0.062876	1.694565332499565 e - 008
LM1	6	18	0	0.060920	4.440892098500626 e - 016
LM2	6	18	0	0.060951	1.753264200488047 e - 011
LM3	6	18	0	0.060627	1.518252190635394 e - 011
LM4	7	21	0	0.087873	9.144596191390519 e - 011
LM5	6	18	0	0.060607	1.753264200488047 e - 011

We used the following test functions and display the computed approximate zero ^{x *} [16]: [figure omitted; refer to PDF]

5. Conclusions

In this paper, we presented two new families of iterative methods for solving nonlinear equations. One is developed by fitting the model m ( x ) = ^{e p x} ( A ^{x 2} + B x + C ) to the function f ( x ) and its derivative ^{f [variant prime]} ( x ) , ^{f [variant prime][variant prime]} ( x ) at a point _{x n} . The other family of iterative methods was constructed by approximating the equation f ( x ) = 0 around the point ( _{x n} , f ( _{x n} ) ) with the quadratic equation to avoid the calculation of the second derivatives. Analysis of convergence shows that the new methods have third-order or higher convergence: if m ...5; 2 , _{a 2} = _{a 3} = p = 0 , then the methods defined by (35) are shown to converge the order four. We observed from numerical examples that the proposed methods are efficient and demonstrate equal or better performance as compared with other well-known methods.

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