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An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

  • Tianbao Liu EMAIL logo , Xiwen Qin and Qiuyue Li
Published/Copyright: December 31, 2019
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Open Mathematics
From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

Keywords: iterative methods; Newton’s method; Cauchy’s method; order of convergence; Padé approximant
MSC 2010: 65H05; 37F10

1 Introduction

In this paper, we consider iterative methods to find a simple root α, i.e, f(α) = 0 and f′(α) ≠ 0, of a nonlinear equation

f(x)=0, (1)

where f : I ⊂ ℝ → ℝ for an open interval I is a scalar function.

Finding the simple root of the nonlinear equation (1) is a common and important problems in numerical analysis of science and engineering, and iterative methods are usually used to approximate a solution of these equations. We know that Newton’s method is an important and basic approach for solving nonlinear equations [1, 2], and its formulation is given by

xn+1=xn−f(xn)f′(xn), (2)

this method converges quadratically.

The classical Cauchy’s method [2] is expressed as

xn+1=xn−21+1−2Lf(xn)f(xn)f′(xn), (3)

where

Lf(xn)=f″(xn)f(xn)f′2(xn). (4)

This family methods given by (3) is a well-known third-order method. However, the method depends on the second derivatives in computing process, and therefore their practical applications are restricted rigorously. In recent years, several methods with free second derivatives have been developed, see [3, 4, 5, 6, 7, 8, 9, 10] and references therein.

In this paper, we will improve the family defined by (3) and obtain third and optimal fourth order family of second-derivative-free variants of Cauchy’s methods by using Padé approximant. The rest of the paper is organized as follows: In Section 2, we present a new third order family of modified Cauchy method and show the order of convergence of this family; In Section 3, different numerical tests confirm the theoretical results, and the new methods are comparable with other known methods and give better results in many cases; In Section 4, based on the family of third-order method, a new optimal fourth-order family of iterative methods is obtained by using weight function; In Section 5, numerical tests and the comparison with the existing optimal fourth-order methods are included to confirm our theoretical results; In Section 6, the basins of attraction of the existing optimal fourth-order methods and our methods are presented and compared to illustrate their performances. Finally, we infer some conclusions.

2 Development of the third order method and its convergence analysis

In order to avoid the evaluation of the second derivatives f″(xn) of Cauchy’s method (3), we consider approximating it by the derivative y″(xn) of the following second degree Padé approximant:

y(t)=a1+a2(t−wn)+a3(t−wn)21+a4(t−wn), (5)

where a1, a2, a3 and a4 are real parameters. We impose the tangency conditions

y(xn)=f(xn),y′(xn)=f′(xn),y(wn)=f(wn), (6)

where xn is nth iterate and

wn=xn−f(xn)f′(xn). (7)

By using the tangency conditions from (6), we obtain the value of a1, a2, a4, and a4 is determined in terms of a3 in the following

a1=f(wn),a2=f′(xn)−2f′(xn)f(wn)f(xn),a4=a3f′(xn)−f′(xn)f(wn)f2(xn). (8)

From (5), we also have

y″(t)=2[a3−a2a4+a1a42][1+a4(t−wn)]3. (9)

Substituting (8) into (9) yields

f″(xn)≈y″(xn)=2f′4(xn)f(wn)f(xn)[f′2(xn)f(xn)+a3f2(xn)−f′2(xn)f(wn)]. (10)

Using (10) we can approximate

Lf(xn)=f″(xn)f(xn)f′2(xn)≈2f′2(xn)f(wn)[f′2(xn)f(xn)+a3f2(xn)−f′2(xn)f(wn)]. (11)

We define

Lf, μ(xn,wn)=2f′2(xn)f(wn)[f′2(xn)f(xn)+μf2(xn)−f′2(xn)f(wn)]. (12)

Using Lf,μ(xn, wn) instead of Lf(xn), we obtain a new one-parameter family of modified Cauchy method free from second derivative

xn+1=xn−21+1−2Lf, μ(xn,wn)f(xn)f′(xn), (13)

where μ ∈ R. Similar to the classical Cauchy’s method, a square root is required in (13). However, this may cost expensively, even fail in the case 1 – 2Lf,μ(xn, wn) < 0.In order to avoid the calculation of the square roots, we will derive some forms free from square roots by Taylor approximation [4].

It is easy to know that Taylor approximation of 1−2Lf, μ(xn,wn) is

1−2Lf, μ(xn,wn)=∑k≥0m(12k)(−2Lf, μ(xn,wn))k, (14)

where m > 0.

Using (14) in (13), we can obtain the following form

xn+1=xn−21+∑k≥0m(12k)(−2Lf, μ(xn,wn))kf(xn)f′(xn), (15)

where μ ∈ R.

On the other hand, it is clear that

21+1−2Lf, μ(xn,wn)=1−1−2Lf, μ(xn,wn)Lf, μ(xn,wn)=∑k≥0m(12k+1)(−1)k2k+1Lf, μ(xn,wn)k (16)

Then, Using (16) in (13), we also can construct a new family of iterative methods as follows:

xn+1=xn−(∑k≥0m(12k+1)(−1)k2k+1Lf, μ(xn,wn)k)f(xn)f′(xn), (17)

where μ ∈ R, m > 0.

We have the convergence analysis of the methods by (17).

Theorem 2.1

Let α ∈ I be a simple zero of sufficiently differentiable function f : I ⊂ ℝ → ℝ for an open interval I. If x0 is sufficiently close to α, then the order of convergence of the methods defined by (17) is three, and the error equation

en+1=[c2μf′(α)−c22]en3+O(en4). (18)

Proof

Let en = xn – α, we use the following Taylor expansions:

f(xn)=f′(α)[en+c2en2+c3en3+c4en4+O(en5)], (19)

where ck=1k!f(k)(α)f′(α). Furthermore, we have

f′(xn)=f′(α)[1+2c2en+3c3en2+4c4en3+5c5en4+O(en5)]. (20)

Dividing (19) by (20),

f(xn)f′(xn)=en−c2en2+2(c22−c3)en3+(7c2c3−4c23−3c4)en4+O(en5). (21)

From (21), we get

wn=xn−f(xn)f′(xn)=α+c2en2−2(c22−c3)en3−(7c2c3−4c23−3c4)en4+O(en5). (22)

Expanding f(wn) in Taylor’s Series about α and using (22), we get

f(wn)=f′(α)[wn−α+c2(wn−α)2+c3(wn−α)3+c4(wn−α)4+⋯]=f′(α)[c2en2+2(c3−c22)en3+(5c23+3c4−7c2c3)en4+O(en5)]. (23)

Since (20), we obtain

f′2(xn)=f′2(α)[1+4c2en+(6c3+4c22)en2+(8c4+12c2c3)en3+(10c5+16c2c4+9c32)en4+O(en5)]. (24)

Because of (19), we get

f2(xn)=f′2(α)[en2+2c2en3+(2c3+c22)en4+O(en5)]. (25)

From (23) and (24), we get

f′2(xn)f(wn)=f′3(α)[c2en2+(2c3+2c22)en3+(7c2c3+3c4+c23)en4+(4c5+4c3c22+10c2c4+6c32)en5+O(en6)]. (26)

From (20), (24), (25) and (26), we obtain

Lf, μ(xn,wn)=2f′2(xn)f(wn)[f′2(xn)f(xn)+μf2(xn)−f′2(xn)f(wn)]=2c2en+(4c3−4c22−2μc2f′(α))en2+(6c4−12c2c3+6c23+8μc22−4μc3f′(α)+2μ2c2f′2(α))en3+(8c5+22c3c22−16c2c4−8c32−6c24+26c2c3μ−20μc23−6c4μf′(α)+4c3μ2−12μ2c22f′2(α)−2μ3c2f′3(α))en4+O(en5). (27)

Furthermore, from (27) we have

∑k≥0m(12k+1)(−1)k2k+1Lf, μ(xn,wn)k=1+12Lf, μ(xn,wn)+12Lf, μ(xn,wn)2+58Lf, μ(xn,wn)3+78Lf, μ(xn,wn)4+⋯=1+c2en+(2c3−μc2f′(α))en2+(3c4+2c2c3−2μc3f′(α)+μ2c2f′2(α))en3+(8c32−10c3c22−16μc2c3f′(α)−10c24+9μc23f′(α)+6μ2c22f′2(α)+12c2c4)en4+O(en5). (28)

Since (17) and (28), we have

xn+1=xn−(∑k≥0m(12k+1)(−1)k2k+1Lf, μ(xn,wn)k)f(xn)f′(xn)=xn−en−(c22−μc2f′(α))en3+O(en4), (29)

from en+1 = xn+1 – α, we have

en+1=−(c22−μc2f′(α))en3+O(en4). (30)

Then the methods defined by (17) is shown to converge of the order three.

Similar to the proof of Theorem 2.1, we can prove that the methods defined by (12) and (15) are third-order methods.

Some special cases

10 : If μ = 0, from (12) and (17) we obtain

Lf,0(xn,wn)=2f(wn)[f(xn)−f(wn)], (31)
xn+1=xn−(∑k≥0m(12k+1)(−1)k2k+1Lf,0(xn,wn)k)f(xn)f′(xn), (32)

where m > 0. For m = 2, we obtain a third-order method(LM1)

xn+1=xn−(1+12Lf,0(xn,wn)+12Lf,0(xn,wn)2)f(xn)f′(xn). (33)

For m = 3, we obtain from (17) a third-order method(LM2)

xn+1=xn−(1+12Lf,0(xn,wn)+12Lf,0(xn,wn)2+58Lf,0(xn,wn)3)f(xn)f′(xn). (34)

20 : If μ = 1, from (12) we obtain

Lf,1(xn,wn)=2f′2(xn)f(wn)[f′2(xn)f(xn)+f2(xn)−f′2(xn)f(wn)]. (35)

For m = 2, we obtain from (17) a third-order method(LM3)

xn+1=xn−(1+12Lf,1(xn,wn)+12Lf,1(xn,wn)2)f(xn)f′(xn). (36)

30 : If μ = – 12 , from (12) we obtain

Lf,−12(xn,wn)=2f′2(xn)f(wn)[f′2(xn)f(xn)−12f2(xn)−f′2(xn)f(wn)]. (37)

For m = 2, we obtain from (17) a third-order method(LM4)

xn+1=xn−(1+12Lf,−12(xn,wn)+12Lf,−12(xn,wn)2)f(xn)f′(xn). (38)

40 : If μ = –1, for m = 2, we obtain a third-order method (LM5) from (12) and (17)

xn+1=xn−(1+12Lf,−1(xn,wn)+12Lf,−1(xn,wn)2)f(xn)f′(xn). (39)

50 : If μ = 12 , from (12) and (15) we obtain some iterative methods as follows:

For m = 1, we obtain a third-order method

xn+1=xn−22−Lf,12(xn,wn)f(xn)f′(xn). (40)

For m = 2, we obtain a third-order method(LM6)

xn+1=xn−44−2Lf,12(xn,wn)−Lf,12(xn,wn)2f(xn)f′(xn). (41)

For m = 3, we obtain a third-order method

xn+1=xn−44−2Lf,12(xn,wn)−Lf,12(xn,wn)2−Lf,12(xn,wn)3f(xn)f′(xn). (42)

3 Numerical examples of the third order methods

In this section, we present the results of numerical simulations in Table 2 to compare the efficiencies of the methods. The considered methods are Newton method (NM), the method of Weerakoon and Fernando [8] (WF), the method of Potra and Pták(PP) [9], Chebyshev’s method (CHM) [11, 12], Halley’s method (HM) [11], and our new methods (33) (LM1), (34) (LM2), (36) (LM3), (38) (LM4), (39) (LM5) and (41) (LM6). Displayed in Table 2 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 absolute residual error of the corresponding function value (|f(xn)|), the computing time (TIME, the unit of time is one second) and the distance of two consecutive approximations δ = |xn – xn–1|. 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 ϵ of the computer. We use the following stopping criteria for computer programs: |f(xn)| < ϵ, we used the fixed stopping criterion ϵ = 10–15. ”–” is divergence. We used the following test functions and display the computed approximate zero x* in Table 1 [13].

Table 1

Test functions and display the computed approximate zero x*.

Test functions x*
f1(x) = x3 + 4x2 – 10 1.3652300134140969
f2(x) = x2 – ex – 3x + 2 0.25753028543986076
f3(x) = sin(x)ex + ln(1 + x2) 0
f4(x) = (x – 1)3 – 1 2
f5(x) = cosx – x 0.73908513321516067
f6(x) = sin2x – x2 + 1 1.4044916482153411
f7(x) = ex2+7x–30 – 1 3

Table 2

Comparison of various third-order methods and Newton’s method.

IT NFE |f(xn)| TIME δ
f1 : x0 = 1
NM 5 10 0 0.046897 2.126987475037367e-011
WF 3 9 0 0.016892 2.284722713019605e-006
PP 4 12 0 0.033053 1.558753126573720e-013
CHM 4 12 0 0.033136 1.643130076445232e-014
HM 3 9 0 0.018876 3.698649917449615e-007
LM1 3 9 0 0.017555 7.656778435505274e-006
LM2 3 9 0 0.017310 6.519974116159233e-009
LM3 3 9 0 0.016915 6.545952378145259e-006
LM4 4 12 0 0.033269 2.220446049250313e-016
LM5 4 12 0 0.032977 2.220446049250313e-016
LM6 3 9 0 0.016579 1.582211815787105e-006
f1 : x0 = 2
NM 5 10 0 0.048387 5.020497351182485e-010
WF 4 12 0 0.021366 4.440892098500626e-016
PP 4 12 0 0.033571 7.949196856316121e-014
CHM 4 12 0 0.033689 2.065014825802791e-014
HM 3 9 0 0.016968 3.107350415199051e-006
LM1 3 9 0 0.017158 1.870204660026076e-007
LM2 4 12 0 0.033365 2.220446049250313e-016
LM3 3 9 0 0.016895 3.063923381674272e-008
LM4 3 9 0 0.016990 3.636202978718472e-007
LM5 3 9 0 0.017591 6.449765455052159e-007
LM6 4 12 0 0.033007 2.220446049250313e-016
f2 : x0 = 0
NM 4 8 0 0.026519 2.665312415217613e-012
WF 3 9 0 0.016351 7.801814749797131e-012
PP 3 9 0 0.016522 1.219191414492116e-012
CHM 3 9 0 0.018237 8.906764215055318e-013
HM 3 9 0 0.017423 7.374600929921371e-012
LM1 3 9 0 0.017213 1.014743844507393e-013
LM2 3 9 0 0.017442 1.497690860219336e-013
LM3 3 9 0 0.017380 3.035904860837491e-013
LM4 3 9 0 0.016920 2.620348382720295e-012
LM5 3 9 0 0.016538 1.656591530618812e-011
LM6 2 6 0 0.000403 1.015871229748111e-005
f2 : x0 = 0.5
NM 4 8 0 0.040693 1.791899961745003e-013
WF 3 9 0 0.017298 6.424749621203318e-012
PP 3 9 0 0.016290 4.607425552194400e-014
CHM 3 9 0 0.017653 3.087480271446452e-011
HM 3 9 0 0.018192 4.208039472430869e-011
LM1 3 9 0 0.017986 1.054711873393899e-015
LM2 3 9 0 0.017973 7.216449660063518e-016
LM3 3 9 0 0.017978 1.497135748707024e-013
LM4 3 9 0 0.017530 9.942047185518277e-014
LM5 3 9 0 0.016987 7.234768339969833e-013
LM6 3 9 0 0.016680 1.887379141862766e-015
f3 : x0 = 1
NM 7 14 3.537126081266182e-024 0.076967 1.085848323840232e-012
WF 4 12 2.621304391538411e-016 0.031335 4.330310691887267e-006
PP 5 15 8.196910187379942e-033 0.050273 8.806888499109001e-012
CHM 5 15 6.352230116524407e-022 0.051221 2.520356663650445e-011
HM 5 15 7.257520328033309e-029 0.054265 8.459855063117184e-015
LM1 4 12 8.271806125530277e-025 0.035949 3.479185746468363e-009
LM2 4 12 0 0.034977 1.804501237019987e-013
LM3 4 12 0 0.033510 3.732361177500448e-009
LM4 4 12 0 0.033691 6.861191797005728e-010
LM5 4 12 0 0.032995 3.650246134545045e-015
LM6 4 12 0 0.033798 2.539428973634579e-010
f3 : x0 = 0.5
NM 6 12 5.905159674954809e-020 0.060337 1.402992074360412e-010
WF 4 12 1.764824578467612e-017 0.032156 4.200981459101664e-009
PP 4 12 1.694834561079519e-019 0.034667 2.767209186089879e-007
CHM 4 12 8.798671206634291e-017 0.033659 3.059650585008672e-007
HM 4 12 3.044907255908736e-017 0.035225 5.518067735074518e-009
LM1 4 12 0 0.034032 4.539201791677570e-014
LM2 4 12 0 0.035989 5.535242064507416e-014
LM3 4 12 0 0.032770 2.221451036901571e-013
LM4 3 9 1.163082115698561e-019 0.016919 2.852561908633870e-007
LM5 4 12 0 0.034210 6.743878120329082e-014
LM6 3 9 0 0.017372 1.053838990356347e-011
f4 : x0 = 2.5
NM 6 12 0 0.055351 1.154631945610163e-014
WF 4 12 0 0.031820 7.314593375440381e-012
PP 4 12 0 0.032233 4.221685223626537e-010
CHM 4 12 0 0.033641 9.853584614916144e-011
HM 4 12 0 0.033511 4.662936703425658e-014
LM1 3 9 6.661338147750939e-016 0.017800 1.544537542308433e-008
LM2 4 12 0 0.033585 2.244870955792067e-013
LM3 3 9 0 0.016430 6.254473188249676e-007
LM4 3 9 0 0.016473 1.067152234579538e-006
LM5 4 12 0 0.032690 4.440892098500626e-016
LM6 4 12 0 0.032958 2.042810365310288e-014
f4 : x0 = 3.5
NM 7 14 0 0.086241 2.877564853065451e-011
WF 5 15 0 0.049605 6.550315845288424e-013
PP 5 15 0 0.048335 4.512221707386743e-010
CHM 5 15 0 0.049367 4.188738245147761e-011
HM 4 12 0 0.033641 4.485352507632712e-006
LM1 4 12 0 0.035118 1.079692646399622e-008
LM2 4 12 0 0.034134 7.838174553853605e-013
LM3 4 12 0 0.035020 3.379705404427114e-010
LM4 4 12 0 0.033277 1.283696526854783e-008
LM5 4 12 0 0.033268 8.547096808086963e-009
LM6 4 12 0 0.032565 8.725183908708800e-009
f5 : x0 = 0
NM 5 10 0 0.049831 1.701233598438989e-010
WF 3 9 0 0.017922 7.792236328407753e-007
PP 4 12 0 0.032661 1.500558566291943e-010
CHM 4 12 0 0.033834 5.327979279989847e-009
HM 4 12 0 0.032642 1.121325254871408e-014
LM1 4 12 0 0.033695 3.819167204710539e-014
LM2 3 9 0 0.017802 8.247395144600489e-008
LM3 4 12 0 0.037679 1.818811767861917e-011
LM4 4 12 0 0.033281 8.344436253082677e-013
LM5 4 12 0 0.033370 8.471505719143124e-010
LM6 4 12 0 0.032211 2.348121697082206e-013
f5 : x0 = 1
NM 4 8 0 0.030217 1.701233598438989e-010
WF 2 6 4.440892098500626e-016 0.003077 2.674277017133964e-005
PP 3 9 0 0.016561 9.809075773858922e-011
CHM 3 9 0 0.018428 1.600380383770528e-009
HM 3 9 0 0.018336 6.624212289807474e-010
LM1 3 9 0 0.017117 2.252753539266905e-012
LM2 3 9 0 0.016831 4.671929509925121e-012
LM3 3 9 0 0.018593 2.668459120336308e-009
LM4 3 9 0 0.016778 1.749711486809247e-012
LM5 3 9 0 0.016682 1.375262126401822e-010
LM6 3 9 0 0.017519 3.148793448204401e-010
f6 : x0 = 1
NM 6 12 3.330669073875470e-016 0.064253 3.059774655866931e-013
WF 4 12 4.440892098500626e-016 0.032786 1.793023507445923e-010
PP 16 48 4.440892098500626e-016 0.246502 1.531728257564424e-007
CHM 5 15 4.440892098500626e-016 0.050039 6.883094094689568e-010
HM 4 12 4.440892098500626e-016 0.038356 2.686739719592879e-013
LM1 4 12 3.330669073875470e-016 0.034670 1.042735342515755e-008
LM2 4 12 3.330669073875470e-016 0.034475 7.038286398142191e-009
LM3 4 12 4.440892098500626e-016 0.034466 3.420013050536852e-008
LM4 4 12 3.330669073875470e-016 0.033590 1.918714076509787e-010
LM5 4 12 4.440892098500626e-016 0.034618 1.002852445530778e-007
LM6 4 12 3.330669073875470e-016 0.033031 6.483733550055604e-010
f6 : x0 = 2.5
NM 6 12 3.330669073875470e-016 0.060863 1.404654170755748e-012
WF 4 12 3.330669073875470e-016 0.033203 4.229505634611996e-012
PP 4 12 3.330669073875470e-016 0.033607 1.030850205196998e-008
CHM 4 12 3.330669073875470e-016 0.034233 1.475204565171140e-007
HM 4 12 4.440892098500626e-016 0.033904 9.462626682221753e-009
LM1 4 12 4.440892098500626e-016 0.034679 1.265654248072679e-014
LM2 3 9 3.330669073875470e-016 0.016425 1.176158348492606e-008
LM3 4 12 4.440892098500626e-016 0.033903 4.662936703425658e-015
LM4 4 12 3.330669073875470e-016 0.033077 1.501021529293212e-013
LM5 4 12 4.440892098500626e-016 0.033523 8.837375276016246e-014
LM6 4 12 3.330669073875470e-016 0.033035 2.375877272697835e-014
f7 : x0 = 3.25
NM 8 16 0 0.102321 9.720393379097914e-010
WF 6 18 0 0.066815 1.691979889528739e-013
PP 6 18 0 0.068323 1.131490456884876e-010
CHM 6 18 0 0.069897 2.398081733190338e-014
HM 5 15 0 0.052332 3.082423205569285e-012
LM1 4 12 0 0.034222 1.781058993621798e-007
LM2 4 12 0 0.033930 5.758273724509877e-008
LM3 4 12 0 0.033957 2.094845084066321e-007
LM4 4 12 0 0.033676 1.635068742622536e-007
LM5 4 12 0 0.033192 1.496285984003976e-007
LM6 - - - - -
f7 : x0 = 3.45
NM 11 22 0 0.152806 4.008793297316515e-011
WF 8 24 0 0.101062 7.105427357601002e-015
PP 8 24 0 0.102346 2.160227552394645e-010
CHM 7 21 0 0.096244 1.268533917908599e-006
HM 6 18 0 0.062830 1.694565332499565e-008
LM1 6 18 0 0.067983 3.221867217462204e-012
LM2 5 15 0 0.051391 7.682743330406083e-014
LM3 6 18 0 0.066789 3.313793683901167e-012
LM4 6 18 0 0.066753 3.173461493588548e-012
LM5 6 18 0 0.068585 3.125055769714891e-012
LM6 - - - - -

4 Development of the optimal fourth order method and its convergence analysis

Corresponding to the well known Traub’s method (see [14]), This scheme (17) with order of convergence three, is not optimal in the sense of Kung-Traub conjecture [14]. In this section, we introduce parametric weight functions and the well-known technique of undetermined coefficients to the family of iterative methods (17) to increase the order of convergence to four.

We consider using a weight function H(μ(xn, wn, γi)) instead of μ in the operator (12), and consider the well-known technique of undetermined coefficients to design an new operator Lf,H,μ̃ (xn, wn) as follows

Lf,H,μ~(xn,wn)=2μ1f′2(xn)f(wn)[μ2f′2(xn)f(xn)+μ3H(μ(xn,wn,γi))f2(xn)−μ4f′2(xn)f(wn)], (43)

where H(μ(xn, wn, γi)) is a function of real variable

μ(xn,wn,γi)=γ1f(xn)+γ2f(wn)1+γ3f(xn)+γ4f(wn), (44)

γi (i = 1, …, 4) and μj(j = 1, …, 4) are real parameters. Then, using (43) in (17), we also can construct two new optimal fourth-order family of modified Cauchy methods as follows:

xn+1=xn−21+∑k≥0m(12k)(−2Lf,H,μ~(xn,wn))kf(xn)f′(xn), (45)

and

xn+1=xn−(∑k≥0m(12k+1)(−1)k2k+1Lf,H,μ~(xn,wn)k)f(xn)f′(xn), (46)

where m > 0.

In the following result, we present the conditions that the weight function H(μ(xn, wn, γi)) and the parameters must satisfy for obtaining two families of iterative methods with fourth-order of convergence, which becoming optimal schemes by Kung-Traub conjecture.

Theorem 3.1

Let f : I ⊆ ℝ → ℝ be a sufficiently differentiable function in an open interval I, such that α ∈ I is a simple solution of the nonlinear equation f(x) = 0. Let H : ℝ → ℝ be any sufficiently differentiable function satisfying H(0) = 0, |H′(0)| < ∞, |H″(0)| < ∞. If x0 is close enough to α, μ1 = μ2 ≠ 0, and μ4 = 0, then the method defined by (46) has fourth-order of convergence and its error equation is:

en+1=1μ2c2(μ3H′(0)γ1−μ2c3)en4+O(en5). (47)

Proof

Let en = xn – α, because of the Taylor series expansions of f(xn) and f(wn), we have

μ(xn,wn,γi)=γ1f(xn)+γ2f(wn)1+γ3f(xn)+γ4f(wn)=f′(α)γ1en+(f′(α)(γ1c2+γ2c2)−f′2(α)γ1γ3)en2+(12f′(α)(2γ1c3−4γ2c22+4γ2c3)−f′2(α)(γ1c2+γ2c2)γ3+f′3(α)γ1γ32−12f′(α)γ1(2γ3f′(α)c2+2γ4f′(α)c2))en3+(−16f′(α)γ1(6γ3f′(α)c3+12γ4f′(α)c3−12γ4f′(α)c22)+16f′(α)(30γ2c23+6γ1c4−42γ2c2c3+18γ2c4)−12f′2(α)(2γ1c3−4γ2c22+4γ2c3)γ3+f′3(α)(γ1c2+γ2c2)γ32−f′4(α)γ1γ33+f′2(α)γ1γ3(2γ3f′(α)c2+2γ4f′(α)c2)−12f′(α)(γ1c2+γ2c2)(2γ3f′(α)c2+2γ4f′(α)c2))en4+O(en5). (48)

Taking into account the expansion of μ(xn, wn, γi), and by using Taylor series expansion of H(μ(xn, wn, γi)) around 0, we obtain

H(μ(xn,wn,γi))=H(0)+H′(0)μ(xn,wn,γi)+H″(0)2!μ2(xn,wn,γi)+H‴(0)3!μ3(xn,wn,γi)+O(μ4(xn,wn,γi))=H(0)+f′(α)H′(0)γ1en+(H′(0)f′(α)γ1c2+12H″(0)f′2(α)γ12+H′(0)f′(α)γ2c2−H′(0)f′2(α)γ1γ3)en2+(H′(0)f′3(α)γ1γ32−2H′(0)f′(α)γ2c22+H′(0)f′(α)γ1c3+H″(0)f′2(α)γ2c2γ1−2H′(0)f′2(α)γ1γ3c2−H′(0)f′2(α)γ3γ2c2−H′(0)f′2(α)γ1γ4c2+H″(0)f′2(α)γ12c2+16H‴(0)f′3(α)γ13+2H′(0)f′(α)γ2c3−H″(0)f′3(α)γ12γ3)en3+O(en4). (49)

From (20), (24), (25), (26) and (43), we obtain

Lf,H,μ~(xn,wn)=2μ1f′2(xn)f(wn)[μ2f′2(xn)f(xn)+μ3H(μ(xn,wn,γi))f2(xn)−μ4f′2(xn)f(wn)]=2μ1μ2c2en+(8μ1μ2c22+2μ1μ2(2c3−2c22)+13μ1f′(α)μ22c2(6f′(α)μ4c2−30μ2f′(α)c2−6μ3H(0)))en2+(8μ1μ2c23+8μ1μ2(2c3−2c22)c2+43μ1f′(α)μ22c22(6f′(α)μ4c2−30μ2f′(α)c2−6μ3H(0))+12μ1μ2c2c3+μ1μ2(10c23−14c2c3+6c4)+13μ1f′(α)μ22(2c3−2c22)(6f′(α)μ4c2−30μ2f′(α)c2−6μ3H(0))+118μ1f′2(α)μ23c2(6f′(α)μ4c2−30μ2f′(α)c2−6μ3H(0))2+16μ1f′(α)μ22c2(−12f′(α)μ3H′(0)γ1−84f′(α)μ2c3−24μ3H(0)c2+24f′(α)μ4c22−96f′(α)μ2c22+24f′(α)μ4c3))en3+O(en4). (50)

Substituting (50) into (46), we have

xn+1=xn−(∑k≥0m(12k+1)(−1)k2k+1Lf,H,μ~(xn,wn)k)f(xn)f′(xn)=xn−(1+12Lf,H,μ~(xn,wn)+12Lf,H,μ~(xn,wn)2+58Lf,H,μ~(xn,wn)3+78Lf,H,μ~(xn,wn)4+⋯)f(xn)f′(xn)=xn−en−(μ1μ2c2−c2)en2−(3μ1μ2c22+μ1μ2(2c3−2c22)+16μ1f′(α)μ22c2(6f′(α)μ4c2−30f′(α)μ2c2−6μ3H(0))+2μ12μ22c22−2c3+2c22)en3−1f′2(α)μ23(c23f′2(α)(13μ1μ22+μ1μ42−14μ12μ2+4μ12μ4−4μ23+5μ13−7μ1μ2μ4)+f′2(α)μ2c2c3(7μ22−14μ1μ2+4μ1μ4+8μ12)+3c4f′(α)2μ22(−μ2+μ1)+μ1c2μ3(μ3H(0)2−f′2(α)μ2H′(0)γ1)−μ1c22f′(α)μ3H(0)(−7μ2+4μ1+2μ4)−2μ1f′(α)μ2c3μ3H(0))en4+O(en5). (51)

From en+1 = xn+1 – α, we consider that if H(0) = 0, μ1 = μ2, μ4 = 0, Then, we obtain the error equation of (46) in the form:

en+1=1μ2c2(μ3H′(0)γ1−μ2c3)en4+O(en5). (52)

Then the methods defined by (46) is shown to converge of the order four.□

Similar to the proof of Theorem 3.1, we can prove that the methods defined by (43) and (45) are fourth-order methods.

When H(0) = 0, μ1 = μ2, μ4 = 0, we obtain

Lf,H,μ~(xn,wn)=2μ1f′2(xn)f(wn)[μ1f′2(xn)f(xn)+μ3H(μ(xn,wn,γi))f2(xn)]. (53)

Let λ = μ3μ1 in (53), we have

Lf,H,λ(xn,wn)=2f′2(xn)f(wn)[f′2(xn)f(xn)+λH(μ(xn,wn,γi))f2(xn)]. (54)

From the expansion of Lf,H,λ(xn, wn) in (54), we can obtain following members of family (45) and (46).

Some special cases

10 : If we consider the following weight function H1 = H(μ(xn, wn, γi)) = 0, from (46) and (54) we obtain

Lf,H1,λ=2f(wn)f(xn), (55)
xn+1=xn−(∑k≥0m(12k+1)(−1)k2k+1Lf,H1,λ(xn,wn)k)f(xn)f′(xn), (56)

where m > 0. For m = 2, we obtain a recently developed fourth-order method by Khattri et al. (KM1) [13]

xn+1=xn−(1+12Lf,H1,λ(xn,wn)+12Lf,H1,λ(xn,wn)2)f(xn)f′(xn)=xn−(1+f(wn)f(xn)+2f2(wn)f2(xn))f(xn)f′(xn). (57)

For m = 3, we also get the existing optimal fourth-order method by Khattri et al. (KM2) [13]

xn+1=xn−(1+12Lf,H1,λ(xn,wn)+12Lf,H1,λ(xn,wn)2+58Lf,H1,λ(xn,wn)3)f(xn)f′(xn)=xn−(1+f(wn)f(xn)+2f2(wn)f2(xn)+5f3(wn)f3(xn))f(xn)f′(xn). (58)

For m = 4, we obtain the developed fourth-order method by Khattri et al. (KM3) [13], which is given by

xn+1=xn−(1+12Lf,H1,λ(xn,wn)+12Lf,H1,λ(xn,wn)2+58Lf,H1,λ(xn,wn)3+78Lf,H1,λ(xn,wn)4)f(xn)f′(xn)=xn−(1+f(wn)f(xn)+2f2(wn)f2(xn)+5f3(wn)f3(xn)+14f4(wn)f4(xn))f(xn)f′(xn). (59)

20 : Now, we consider the following weight function, which also satisfies all the conditions of Theorem 3.1. If H2 = H(μ(xn, wn, γi)) = μ(xn,wn,γi)21−μ(xn,wn,γi) , γ1 = 0, γ2 = γ3 = γ4 = 1 and λ = 1, from (43) and (54), we have

Lf,H2,1(xn,wn)=2f′2(xn)f(wn)[f′2(xn)f(xn)+H2f2(xn)]. (60)

For m = 2, from (46) and (60) we obtain a new fourth-order method (LTM1)

xn+1=xn−(1+12Lf,H2,1(xn,wn)+12Lf,H2,1(xn,wn)2)f(xn)f′(xn). (61)

For m = 3, we obtain a new fourth-order method (LTM2)

xn+1=xn−(1+12Lf,H2,1(xn,wn)+12Lf,H2,1(xn,wn)2+58Lf,H2,1(xn,wn)3)f(xn)f′(xn). (62)

For m = 4, we obtain a new fourth-order method

xn+1=xn−(1+12Lf,H2,1(xn,wn)+12Lf,H2,1(xn,wn)2+58Lf,H2,1(xn,wn)3+78Lf,H2,1(xn,wn)4)f(xn)f′(xn). (63)

For m = 2, from (44) and (60) we obtain a new fourth-order method (LTM3)

xn+1=xn−44−2Lf,H2,1(xn,wn)−Lf,H2,1(xn,wn)2f(xn)f′(xn). (64)

For m = 3, we obtain a new fourth-order method (LTM4)

xn+1=xn−44−2Lf,H2,1(xn,wn)−Lf,H2,1(xn,wn)2−Lf,H2,1(xn,wn)3f(xn)f′(xn). (65)

30 : If H3 = H(μ(xn, wn, γi)) = μ(xn, wn, γi), γ1 = –1, γ2 = 1, γ3 = –1, γ4 = 1 and λ = – 23 , from (43) and (54), we have

Lf,H3,−23(xn,wn)=2f′2(xn)f(wn)[f′2(xn)f(xn)−23H3f2(xn)]. (66)

For m = 2, from (45) and (66) we obtain a new fourth-order method (LTM5)

xn+1=xn−(1+12Lf,H3,−23(xn,wn)+12Lf,H3,−23(xn,wn)2)f(xn)f′(xn). (67)

For m = 3, we obtain a new fourth-order method (LTM6)

xn+1=xn−(1+12Lf,H3,−23(xn,wn)+12Lf,H3,−23(xn,wn)2+58Lf,H3,−23(xn,wn)3)f(xn)f′(xn). (68)

For m = 2, from (44) and (66) we obtain a new fourth-order method

xn+1=xn−44−2Lf,H3,−23(xn,wn)−Lf,H3,−23(xn,wn)2f(xn)f′(xn). (69)

For m = 3, we obtain a new fourth-order method

xn+1=xn−44−2Lf,H3,−23(xn,wn)−Lf,H3,−23(xn,wn)2−Lf,H3,−23(xn,wn)3f(xn)f′(xn). (70)

40 : If H4 = H(μ(xn, wn, γi)) = μ(xn,wn,γi)21−μ(xn,wn,γi) , γ1 = –1, γ2 = 1, γ3 = – 45 , γ4 = 2 and λ = 12 , from (43) and (54), we have

Lf,H4,12(xn,wn)=2f′2(xn)f(wn)[f′2(xn)f(xn)+12H4f2(xn)]. (71)

For m = 2, from (46) and (71) we obtain a new fourth-order method

xn+1=xn−(1+12Lf,H4,12(xn,wn)+12Lf,H4,12(xn,wn)2)f(xn)f′(xn). (72)

For m = 3, we obtain a new fourth-order method (LTM7)

xn+1=xn−(1+12Lf,H4,12(xn,wn)+12Lf,H4,12(xn,wn)2+58Lf,H4,12(xn,wn)3)f(xn)f′(xn). (73)

For m = 2, from (45) and (71) we obtain a new fourth-order method (LTM8)

xn+1=xn−44−2Lf,H4,12(xn,wn)−Lf,H4,12(xn,wn)2f(xn)f′(xn). (74)

For m = 3, we obtain a new fourth-order method

xn+1=xn−44−2Lf,H4,12(xn,wn)−Lf,H4,12(xn,wn)2−Lf,H4,12(xn,wn)3f(xn)f′(xn). (75)

5 Numerical examples of the optimal fourth order methods

This section is devoted to verify the validity and effectiveness of our theoretical results which we have proposed earlier. We present the numerical results obtained by applying the proposed methods on some scalar equations. We are going to compare LTM1-LTM8 with some fourth-order known methods as Kung-Traub scheme [14]

yn=xn−f(xn)f′(xn),xn+1=yn−f2(xn)(f(xn)−f(yn))2f(yn)f′(xn), (76)

denoted by KTM; three methods of Khattri et al. [13], denoted by KM1, KM2 and KM3 (see (57), (58) and (59) in special cases of fourth order methods); the fourth-order method by Chun [15] denoted by CM1; two fourth-order methods by Chun and Ham [16] denoted by CM2, CM3; the fourth-order method by Kou et al. [17] denoted by NSPP; the fourth-order method by Sharma and Bahl [18] denoted by SBM, which the method are applied to solve systems of nonlinear equations by Sharma et al. [19]. The CM1 is given as

yn=xn−f(xn)f′(xn),xn+1=xn−2f2(xn)f′(xn)[f(xn)−f(yn)]+f(xn)f′(xn)+f′(xn)f(yn)f2(xn)+f′2(xn). (77)

The CM2 is defined as

yn=xn−f(xn)f′(xn)f′2(xn)−f(xn),Ln=f′2(xn)−f(xn),xn+1=xn−[(f(yn)−f(xn))Ln2−f2(xn)f′2(xn)2(f(yn)−f(xn))Ln2+f(xn)f′2(xn)(f′2(xn)−2f(xn))]f(xn)f′(xn), (78)

and CM3 is

yn=xn−f(xn)f′(xn)f′2(xn)+100f(xn),Ln=f′2(xn)+100f(xn),xn+1=xn−[(f(yn)−f(xn))Ln2+100f2(xn)f′2(xn)2(f(yn)−f(xn))Ln2+f(xn)f′2(xn)(f′2(xn)+200f(xn))]f(xn)f′(xn). (79)

The NSPP is given as

yn=xn−f(xn)f′(xn),xn+1=xn−f2(xn)+f2(yn)f′(xn)(f(xn)−f(yn)). (80)

The SBM is defined as

yn=xn−23f(xn)f′(xn),xn+1=xn−[−12+98f′(xn)f′(yn)+38f′(yn)f′(xn)]f(xn)f′(xn). (81)

Displayed in Table 3 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 absolute residual error of the corresponding function value (|f(xn)|), the distance of two consecutive approximations δ = |xn – xn–1|, and the computational approximate order of convergence ρ, which is an approximation of the theoretical order of convergence, introduced in [20] as

p≈ρ=ln(|xk+1−xk|/|xk−xk−1|)ln(|xk−xk−1|/|xk−1−xk−2|). (82)

Table 3

Comparison of some different fourth-order methods.

Method IT NFE |f(xn)| δ ρ
f1(x) = 0 KTM 19 57 0 1.007297327770829e-006 3.57
x0 = –0.1 KM1 64 192 0 1.477826749862743e-009 3.76
KM2 45 135 0 4.900428897136600e-005 3.72
KM3 102 306 0 1.091067414571434e-005 4.42
CM1 6 18 0 9.497031339122941e-007 3.15
CM2 9 27 0 1.145895600629387e-009 3.80
CM3 45 135 0 2.819966482547898e-014 3.49
NSPP 14 42 0 2.220446049250313e-016 1.98
SBM 12 36 0 2.602835724729857e-010 3.81
LTM1 49 147 0 1.015936601511669e-008 3.71
LTM2 21 63 0 1.554312234475219e-015 4.09
LTM3 17 51 0 1.743050148661496e-013 3.90
LTM4 11 33 0 3.606588179216885e-007 4.10
LTM5 21 63 0 4.322075137341841e-006 4.31
LTM6 26 78 0 2.886579864025407e-015 3.62
LTM7 16 48 0 5.212047367475492e-006 4.00
LTM8 14 42 0 6.794564910705958e-014 4.18
f2(x) = 0 KTM 5 15 0 7.216449660063518e-016 3.89
x0 = 5 KM1 5 15 0 1.105560087921731e-011 4.13
KM2 5 15 0 1.152966611073225e-013 3.95
KM3 4 12 0 9.358037653284246e-006 4.12
CM1 4 12 8.881784197001252e-016 1.997754338076696e-004 6.61
CM2 5 15 0 2.425837308805967e-014 3.91
CM3 - - - - -
NSPP 5 15 0 2.473671267821942e-011 4.20
SBM 5 15 4.440892098500626e-016 9.436895709313831e-016 3.78
LTM1 5 15 0 5.860528728973691e-011 4.24
LTM2 5 15 0 6.933342788784103e-014 3.94
LTM3 4 12 4.440892098500626e-016 1.094604859936954e-004 5.29
LTM4 5 15 0 2.942091015256665e-015 3.96
LTM5 5 15 0 1.326020566683184e-007 3.54
LTM6 5 15 0 6.679101716144942e-013 3.51
LTM7 5 15 0 5.632716515435732e-013 3.54
LTM8 5 15 0 7.216449660063518e-016 3.64
f3(x) = 0 KTM 4 12 2.698906001294071e-024 1.217853159082307e-008 3.62
x0 = 1.9 KM1 4 12 0 2.148368599356326e-007 3.45
KM2 4 12 0 9.592177833278592e-011 4.65
KM3 4 12 0 1.846342610051233e-011 4.20
CM1 4 12 0 1.119123721644855e-010 3.75
CM2 5 15 0 1.656923030498442e-007 3.50
CM3 5 15 5.048709793414476e-029 2.684470484445867e-013 3.61
NSPP 4 12 1.985233470127266e-023 4.289648665705555e-008 3.55
SBM 4 12 2.418605025271162e-021 4.917930580759401e-011 3.78
LTM1 4 12 5.293955920339377e-023 2.170159571681513e-007 3.45
LTM2 4 12 0 9.881239316361645e-011 4.65
LTM3 4 12 6.617444900424221e-024 9.954034684185604e-009 3.61
LTM4 4 12 1.615587133892632e-027 1.428516766682764e-011 4.27
LTM5 4 12 3.308722450212111e-024 2.045966094113560e-008 3.59
LTM6 4 12 3.231174267785264e-027 2.309911940542863e-011 4.33
LTM7 4 12 0 7.432071191110223e-011 4.26
LTM8 4 12 4.135903062765138e-025 2.239952597536676e-009 3.65
f4(x) = 0 KTM 5 15 0 1.776356839400251e-015 3.95
x0 = 5 KM1 5 15 0 6.295657328792004e-010 3.78
KM2 4 12 0 5.425764586330928e-005 3.72
KM3 4 12 0 1.135485330872044e-006 3.82
CM1 4 12 0 1.661871202074394e-005 3.34
CM2 4 12 0 4.525455072901252e-006 3.51
CM3 4 12 0 1.644155478430776e-009 2.87
NSPP 5 15 0 7.429612480791548e-013 3.89
SBM 5 15 0 1.021405182655144e-014 3.93
LTM1 5 15 0 6.829425913679188e-010 3.78
LTM2 4 12 6.661338147750939e-016 5.805743723130696e-005 3.71
LTM3 5 15 0 4.440892098500626e-016 3.86
LTM4 4 12 6.661338147750939e-016 1.499435946517025e-007 3.89
LTM5 5 15 0 9.895031460871451e-010 3.86
LTM6 4 12 0 4.299091316228854e-005 3.24
LTM7 4 12 0 1.045922523612575e-005 4.31
LTM8 4 12 0 5.399981822362676e-005 3.37
f5(x) = 0 KTM 9 27 0 1.288968931589807e-013 3.80
x0 = 5.8 KM1 5 15 0 5.195843755245733e-014 3.30
KM2 4 12 0 1.789890458070431e-011 3.60
KM3 - - - - -
CM1 5 15 1.110223024625157e-016 6.661338147750939e-016 3.97
CM2 4 12 0 4.142503895465666e-009 3.84
CM3 4 12 1.110223024625157e-016 8.643999676372083e-006 2.93
NSPP 18 54 0 1.818157153921085e-005 4.93
SBM - - - - -
LTM1 5 15 0 1.394815624942147e-004 5.37
LTM2 4 12 1.110223024625157e-016 1.033639332229663e-004 2.87
LTM3 9 27 0 1.776356839400251e-015 4.10
LTM4 16 48 0 1.953010203822325e-004 5.98
LTM5 6 18 0 9.492295838242626e-011 3.68
LTM6 5 15 1.110223024625157e-016 2.079826453160738e-005 4.89
LTM7 6 18 0 7.094251031070087e-006 3.25
LTM8 6 18 1.110223024625157e-016 4.067230241489028e-007 4.21
f6(x) = 0 KTM 5 15 4.440892098500626e-016 2.725704772088555e-005 3.05
x0 = 30 KM1 6 18 4.440892098500626e-016 2.819966482547898e-014 3.90
KM2 5 15 3.330669073875470e-016 3.551058632700332e-006 3.61
KM3 5 15 3.330669073875470e-016 2.469841886565405e-009 4.11
CM1 5 15 4.440892098500626e-016 2.900702567032454e-008 4.28
CM2 5 15 3.330669073875470e-016 5.619937435419331e-006 3.47
CM3 ~ ~ ~ ~ ~
NSPP 5 15 8.881784197001252e-016 1.200137685715141e-004 2.70
SBM 6 18 4.440892098500626e-016 4.440892098500626e-016 3.97
LTM1 6 18 4.440892098500626e-016 2.287059430727823e-014 3.91
LTM2 5 15 3.330669073875470e-016 1.494727950301922e-006 3.74
LTM3 5 15 4.440892098500626e-016 7.169287327135621e-006 3.21
LTM4 5 15 4.440892098500626e-016 1.957256579032674e-011 4.06
LTM5 6 18 3.330669073875470e-016 2.093081263865315e-011 3.82
LTM6 6 18 4.440892098500626e-016 3.996802888650564e-015 4.03
LTM7 5 15 3.330669073875470e-016 6.713331008989520e-005 3.12
LTM8 5 15 4.440892098500626e-016 1.567170138416785e-005 3.77
f7(x) = 0 KTM 5 15 0 8.881784197001252e-016 3.96
x0 = 3.25 KM1 5 15 0 2.532513754260890e-009 3.80
KM2 5 15 0 4.440892098500626e-016 3.35
KM3 4 12 0 3.625560864861654e-007 3.72
CM1 5 15 0 1.288302797775032e-012 3.92
CM2 5 15 0 4.440892098500626e-016 1.39
CM3 4 12 0 8.305973686617563e-010 3.67
NSPP 5 15 0 1.444178110432404e-012 3.92
SBM 5 15 0 1.243449787580175e-014 3.95
LTM1 5 15 0 2.535360366096029e-009 3.80
LTM2 4 12 0 1.436867444937207e-005 4.49
LTM3 4 12 0 1.407228477168232e-005 3.71
LTM4 4 12 0 2.142859223397409e-010 3.57
LTM5 6 18 0 4.440892098500626e-016 1.19
LTM6 4 12 0 1.473763922721361e-005 4.46
LTM7 4 12 0 1.399004200797194e-005 4.51
LTM8 4 12 0 1.387654194484611e-005 3.71

In this section, the computations were done using Matlab 7.1 environment. We accept an approximate solution rather than the exact root, depending on the precision ϵ of the computer. We use the following stopping criteria for computer programs: |f(xn)| < ϵ, we used the fixed stopping criterion ϵ = 10–15. “–” is divergence. “~” means that it converges to other solutions. We used the test functions and display the computed approximate zero x* in Table 1.

From Table 3, it is clear that CM1, CM2, LTM4, SBM, LTM8, NSPP and LTM7 require less number of iterations (IT) and function evaluations (NFE) in the corresponding test function f1(x) compared with the other fourth-order methods, especially the method CM1 performs best in terms of convergence.

In test function f2(x), the methods KM3, CM1 and LTM3 have better performances. The numerical results also show that KM3 have smaller residual error in the corresponding function |f2(xn)| compared with LTM3. The existing method CM3 fails in convergence for the case f2(x).

Regarding the results of test function f3(x), we claim that our methods and the existing fourth-order methods have almost similar performance.

From the results of the test function f4(x), our methods LTM2, LTM4, LTM6, LTM7, LTM8 and the existing method KM2, KM3, CM1, CM2, CM3 require less number of iterations (IT) and function evaluations (NFE) than other methods, which demonstrate that several of our methods converge faster than some existing ones.

In test function f5(x), the methods KM2, CM2, CM3 and our method LTM2 have better performances in terms of the speed of convergence. The existing methods KM3 and SBM fails in convergence for the case f5(x). The results show that our fourth-order method LTM2 can compete with KTM, KM1, CM1, NSPP, SBM and KM3.

In test function f6(x), in terms of convergence, the methods KM1, SBM, LTM1, LTM5, LTM6 perform slightly worse, while the method CM3 performs the worst.

In test function f7(x), we also check the effectiveness of our methods when we consider the same nonlinear equation with same initial approximation. Then, we find that the methods KM3, CM3 and our methods LTM2, LTM3, LTM4, LTM6, LTM7, LTM8 perform better than KTM, KM1, KM2, CM1, CM2, NSPP and SBM in terms of speed of convergence for solving the nonlinear equations. However, in this particular case the method LTM5 don’t perform better than other methods.

Consequently, our fourth-order methods can compete with some fourth-order known methods, such as KTM, KM1, KM2, KM3, CM1, CM2, CM3, NSPP and SBM, especially the present methods LTM2, LTM3, LTM4, LTM7 and LTM8 perform equal or better than some existing methods in many aspects.

Application to a physical problem

We consider Planck’s radiation law problem [21, 22]

Φ(λ~)=8πcPλ~−5ecPλ~BT−1, (83)

which calculates the energy density within an isothermal blackbody. In the expression of formula (83), λ͠ is the wavelength of the radiation, T is the absolute temperature of the blackbody, B is Boltzmann’s constant, P is the Planck’s constant and c is the speed of light. In some cases, due to the needs of the application, it is often necessary to determine wavelength λ͠ which corresponds to maximum energy density Φ(λ͠). To find the critical points, we use the Chain Rule to differentiate the function of equation (83), and obtain

Φ′(λ~)=(8πcPλ~−6ecPλ~BT−1)(cPλ~BTecPλ~BTecPλ~BT−1−5), (84)

so to find the critical number of Φ for the maxima, we solve the equation

cPλ~BTecPλ~BTecPλ~BT−1−5=0. (85)

Consider the relationship between variables, if X=cPλ~BT, then the equation (85) is converted into the following nonlinear equation

F(X)=e−X+X5−1=0. (86)

The function (86) is continuous, and it has a solution X = 0, which is what we do not interest. We want to obtain positive roots of the nonlinear function, so that requires us to apply iterative method to get approximate solution of this equation. Here, our desired root is X* = 4.96511423174428. Keeping in view this fact, we apply KTM, KM1, KM2, KM3, CM1, CM2, CM3, NSPP, SBM, and our methods LTM1-LTM8 to the nonlinear equation (86) and compare. Displayed in Table 4 are the number of iterations (IT), the number of function evaluations (NFE), the absolute residual error of the corresponding function value (|F(Xn)|), the computing time (TIME, the unit of time is one second) and the distance of two consecutive approximations δ = |Xn – Xn–1|, where ”–” is divergence. We use the following stopping criteria for computer programs: |F(Xn)| < ϵ = 10–15. Note that in Table 4, in terms of iterations number (IT) and function evaluations (NFE), the fourth-order methods have the same performance. KTM, KM1, KM2, CM1, NSPP, SBM, and our methods LTM1, LTM2, LTM4, LTM5, LTM7, LTM8 have smaller residual error in the nonlinear function as compared to the other methods of fourth-order. Our methods is slightly better at computing time. Consequently, the roots of F(X) = 0 give the maximum wavelength of radiation λ͠ by means of the following relation:

λ~≈cPX∗BT=cP4.96511423174428BT. (87)

Table 4

Comparison of fourth-order methods for the physical problem.

Method IT NFE |F(Xn)| δ TIME
F(X) = 0 KTM 3 9 0 2.709308333237459e-010 0.022328
X0 = 3 KM1 3 9 0 6.467101520968299e-009 0.028262
KM2 3 9 0 2.692956968530780e-012 0.022563
KM3 3 9 2.220446049250313e-016 3.547384608282300e-010 0.022609
CM1 3 9 0 3.070780428959807e-004 0.023050
CM2 3 9 2.220446049250313e-016 5.524792026534442e-006 0.022492
CM3 3 9 2.220446049250313e-016 1.943731808839999e-005 0.023956
NSPP 3 9 0 9.101039921688425e-010 0.022708
SBM 3 9 0 1.925377191014377e-009 0.023061
LTM1 3 9 0 3.959470085135308e-009 0.022057
LTM2 3 9 0 9.615003904173136e-009 0.020803
LTM3 3 9 2.220446049250313e-016 1.243449787580175e-014 0.020864
LTM4 3 9 0 4.251994312198804e-010 0.021211
LTM5 3 9 0 4.753649580013786e-005 0.021086
LTM6 3 9 2.220446049250313e-016 4.648025434228487e-005 0.022313
LTM7 3 9 0 3.362910103312800e-004 0.020591
LTM8 3 9 0 3.350666193249197e-004 0.021832

6 Basin of attractions

In this section, we study some dynamical properties of the family of iterative methods (45) and (46) based on their basins of attraction when they are applied to the complex polynomial P(z). We investigate the structure of the basins of attraction for comparing convergence and stability of the family of iterative methods. Here we briefly introduce some necessary dynamical concepts and basic results to be used later. Most of them can be found in the classic works such as [20, 23, 24, 25, 26, 27, 28, 29, 30] and references therein. Let R : ℂ̂ → ℂ̂ be a rational map on the Riemann sphere. The orbit of a point z0 ∈ ℂ̂ is defined as the set {z0, R(z0), R2(z0), …, Rn(z0), …}. A point z0 ∈ ℂ̂ is a fixed point of the rational function R if satisfy R(z0) = z0. A periodic point z0 of period m > 1 is a point such that Rm(z0) = z0, where m is the smallest such integer. A point z0 is called attracting if satisfy |R′(z0)| < 1, repelling if satisfy |R′(z0)| > 1, and neutral if satisfy |R′(z0)| = 1. Moreover, if satisfy |R′(z0)| = 0, the fixed point is super attracting.

Let zf∗ be an attracting fixed point of the rational function R. The basin of attraction of the fixed point zf∗ is defined

A(zf∗)={z0∈C^:Rn(z0)→zf∗,n→∞}. (88)

The set of points whose orbits tends to an attracting fixed point zf∗ is defined as the Fatou set, 𝓕(R). The complementary set, the Julia 𝓙(R), is the closure of the set consisting of its repelling fixed points, and establishes the borders between the basins of attraction.

Some known and existing fourth-order methods and our fourth-order methods are considered, they are KTM (76), KM1 (57), KM2 (58), KM3 (59), CM1 (77), CM2 (78), CM3 (79), NSPP (80), SBM (81), LTM1 (61), LTM2 (62), LTM3 (64), LTM4 (65), LTM5 (67), LTM6 (68), LTM7 (73) and LTM8 (74). In our experiments, we take a square region D = [–2, 2] × [–2, 2] of the complex plane, with 400 × 400 points, and we apply the iterative methods starting in very z0 in the square. The iterative methods can converge to the root or, eventually, diverges. As an illustration, we consider the stopping criterium for convergence to be less than a tolerance ϵ = 10–7 and a maximum of 200 iterations. If a sequence {zn} with the residual |P(zn)| < ϵ, generated by the iterative method for the initial guess z0 within the maximum iteration, then we decide the iterative method converges for z0, otherwise we consider the method to be divergent. We take black color for denoting lack of convergence to any of the roots or convergence to the infinity.

Test problem 1

Let P1(z) = z3 – 1 having three simple zeros {−12−32i,−12+32i,1}. Based on the Figure 1-3, we observe that the method CM3 is the best method in terms of less chaotic behavior on the boundary points, the methods KTM, CM2, NSPP, SBM, LTM5 and LTM8 are better. From the three Figures, we find that, with the increase of the value m of (56), the chaotic behaviors of the methods KM1, KM2 and KM3 become more and more complex, which the feature of attraction basins is also reflected in our methods LTM1, LTM2, LTM3 and LTM4. In the next we have taken polynomials of increasing degree.

Figure 1 Basins of attraction of the methods KTM, KM1, KM2, KM3, CM1 and CM2 respectively for P1(z) = z3 – 1.
Figure 1

Basins of attraction of the methods KTM, KM1, KM2, KM3, CM1 and CM2 respectively for P1(z) = z3 – 1.

Figure 2 Basins of attraction of the methods CM3, NSPP, SBM, LTM1, LTM2 and LTM3 respectively for P1(z) = z3 – 1.
Figure 2

Basins of attraction of the methods CM3, NSPP, SBM, LTM1, LTM2 and LTM3 respectively for P1(z) = z3 – 1.

Figure 3 Basins of attraction of the methods LTM4, LTM5, LTM6, LTM7 and LTM8 respectively for P1(z) = z3 – 1.
Figure 3

Basins of attraction of the methods LTM4, LTM5, LTM6, LTM7 and LTM8 respectively for P1(z) = z3 – 1.

Test problem 2

Let P2(z) = z5 + z having five simple zeros {–0.7071067812 ± 0.7071067812i, 0, 0.707 1067812 ± 0.7071067812i}. We conclude based on Figure 4-6 that the methods CM2 and LTM3 outperform all the others, and the methods LTM8, KTM and SBM are better in terms of less chaotic behavior than other methods. However, the fractal picture of the method LTM8 has some non convergent points. The method CM3 has the most divergence points in Figure 5, so it performs worst in this test problem. Since the value of m is bigger, the method KM3 has the most complex behavior on the boundary points.

Figure 4 Basins of attraction of the methods KTM, KM1, KM2, KM3, CM1 and CM2 respectively for P2(z) = z5 + z.
Figure 4

Basins of attraction of the methods KTM, KM1, KM2, KM3, CM1 and CM2 respectively for P2(z) = z5 + z.

Figure 5 Basins of attraction of the methods CM3, NSPP, SBM, LTM1, LTM2 and LTM3 respectively for P2(z) = z5 + z.
Figure 5

Basins of attraction of the methods CM3, NSPP, SBM, LTM1, LTM2 and LTM3 respectively for P2(z) = z5 + z.

Figure 6 Basins of attraction of the methods LTM4, LTM5, LTM6, LTM7 and LTM8 respectively for P2(z) = z5 + z.
Figure 6

Basins of attraction of the methods LTM4, LTM5, LTM6, LTM7 and LTM8 respectively for P2(z) = z5 + z.

Test problem 3

Let P3(z) = z6 + z – 1 having six simple zeros {0.7780895987, –1.1347241384, –0.451 0551586 ± 1.0023645716i, 0.6293724285 ± 0.7357559530i}. In the fractal pictures from Figure 7-9, it is clear that the methods KTM, CM2, NSPP, SBM and our methods LTM1, LTM2 have the largest basins of attraction as compared to the other methods. In addition, although LTM3, LTM8 have a small amount of no convergence points, the two methods have less chaotic behavior on the boundary points than other methods, including the known and existing fourth-order methods KTM, KM1, KM2, KM3, CM1, NSPP and SBM. In terms of the dynamical behavior on the boundary points, the method KM3 is most complex, followed by the methods LTM6 and KM2. From Figures 6(a), the method LTM4 has more non-convergence point regions than other methods.

Figure 7 Basins of attraction of the methods KTM, KM1, KM2, KM3, CM1 and CM2 respectively for P3(z) = z6 + z – 1.
Figure 7

Basins of attraction of the methods KTM, KM1, KM2, KM3, CM1 and CM2 respectively for P3(z) = z6 + z – 1.

Figure 8 Basins of attraction of the methods CM3, NSPP, SBM, LTM1, LTM2 and LTM3 respectively for P3(z) = z6 + z – 1.
Figure 8

Basins of attraction of the methods CM3, NSPP, SBM, LTM1, LTM2 and LTM3 respectively for P3(z) = z6 + z – 1.

Figure 8 Basins of attraction of the methods LTM4, LTM5, LTM6, LTM7 and LTM8 respectively for P3(z) = z6 + z – 1.
Figure 8

Basins of attraction of the methods LTM4, LTM5, LTM6, LTM7 and LTM8 respectively for P3(z) = z6 + z – 1.

7 Conclusions

In this paper, we have designed and studied a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the methods was also considered, and the methods have convergence order three. Based on the family of third-order method, a new optimal fourth-order family of iterative methods (in the sense of Kung-Traub’s conjecture) is obtained by using weight function. We observed from numerical study that the proposed methods are efficient and demonstrate equal or better performance as compared with other well-known fourth-order methods. Finally, the dynamical analysis of this optimal fourth-order family and existing fourth-order methods have been made on some different polynomials, showing some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, the fractal graphics show the chaotic behaviors of our methods become more and more complex with the increase of the value m of the series in iterative methods, which also reflected in the existing fourth-order methods KM1, KM2 and KM3.

Acknowledgements

This work is financially supported by the National Natural Science Foundation of China (Nos. 11401046, 11301036), and Scientific Research Foundation of the Education Department of Jilin Province, China (No. JJKH20170536KJ, JJKH20170537KJ).

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Received: 2019-05-30
Accepted: 2019-10-19
Published Online: 2019-12-31

© 2019 Tianbao Liu et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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  30. A categorical approach to abstract convex spaces and interval spaces
  31. Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations
  32. Parity results for broken 11-diamond partitions
  33. A new fourth power mean of two-term exponential sums
  34. The new operations on complete ideals
  35. Soft covering based rough graphs and corresponding decision making
  36. Complete convergence for arrays of ratios of order statistics
  37. Sufficient and necessary conditions of convergence for ρ͠ mixing random variables
  38. Attractors of dynamical systems in locally compact spaces
  39. Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains
  40. Statistical approximation properties of λ-Bernstein operators based on q-integers
  41. An investigation of fractional Bagley-Torvik equation
  42. Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer
  43. On the hybrid power mean of two kind different trigonometric sums
  44. Embedding of Supplementary Results in Strong EMT Valuations and Strength
  45. On Diophantine approximation by unlike powers of primes
  46. A General Version of the Nullstellensatz for Arbitrary Fields
  47. A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces
  48. Random Polygons and Estimations of π
  49. The optimal pebbling of spindle graphs
  50. MBJ-neutrosophic ideals of BCK/BCI-algebras
  51. A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
  52. A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
  53. Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
  54. Stability property of the prey free equilibrium point
  55. Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
  56. Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
  57. Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
  58. Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts
  59. Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application
  60. Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems
  61. Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model
  62. Algebraic loop structures on algebra comultiplications
  63. Two weight estimates for a class of (p, q) type sublinear operators and their commutators
  64. Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
  65. 2-closures of primitive permutation groups of holomorph type
  66. Monotonicity properties and inequalities related to generalized Grötzsch ring functions
  67. Variation inequalities related to Schrödinger operators on weighted Morrey spaces
  68. Research on cooperation strategy between government and green supply chain based on differential game
  69. Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
  70. *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
  71. Some improved bounds on two energy-like invariants of some derived graphs
  72. Pricing under dynamic risk measures
  73. Finite groups with star-free noncyclic graphs
  74. A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
  75. S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain
  76. On Diophantine equations involving Lucas sequences
  77. A new way to represent functions as series
  78. Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system
  79. Some remarks on a pair of seemingly unrelated regression models
  80. Lyapunov stable homoclinic classes for smooth vector fields
  81. Stabilizers in EQ-algebras
  82. The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
  83. Spectrum perturbations of compact operators in a Banach space
  84. The non-commuting graph of a non-central hypergroup
  85. Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
  86. Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
  87. Dislocated quasi cone b-metric space over Banach algebra and contraction principles with application to functional equations
  88. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis
  89. Differential polynomials of L-functions with truncated shared values
  90. Exclusion sets in the S-type eigenvalue localization sets for tensors
  91. Continuous linear operators on Orlicz-Bochner spaces
  92. Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
  93. Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
  94. A quantitative obstruction to collapsing surfaces
  95. Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
  96. Coexistence for a kind of stochastic three-species competitive models
  97. Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
  98. On the two-term exponential sums and character sums of polynomials
  99. F-biharmonic maps into general Riemannian manifolds
  100. Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
  101. Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains
  102. Power graphs and exchange property for resolving sets
  103. On nearly Hurewicz spaces
  104. Least eigenvalue of the connected graphs whose complements are cacti
  105. Determinants of two kinds of matrices whose elements involve sine functions
  106. A characterization of translational hulls of a strongly right type B semigroup
  107. Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
  108. Lp estimates for maximal functions along surfaces of revolution on product spaces
  109. Path-induced closure operators on graphs for defining digital Jordan surfaces
  110. Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras
  111. Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
  112. Injective hulls of many-sorted ordered algebras
  113. Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3
  114. Global properties of virus dynamics with B-cell impairment
  115. The monotonicity of ratios involving arc tangent function with applications
  116. A family of Cantorvals
  117. An asymptotic property of branching-type overloaded polling networks
  118. Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
  119. Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
  120. L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
  121. L-topological-convex spaces generated by L-convex bases
  122. An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
  123. New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
  124. Hankel determinant of order three for familiar subsets of analytic functions related with sine function
  125. On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
  126. Results on existence for generalized nD Navier-Stokes equations
  127. Regular Banach space net and abstract-valued Orlicz space of range-varying type
  128. Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
  129. On a new convergence in topological spaces
  130. On a fixed point theorem with application to functional equations
  131. Coupled system of a fractional order differential equations with weighted initial conditions
  132. Rough quotient in topological rough sets
  133. Split Hausdorff internal topologies on posets
  134. A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
  135. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
  136. Special Issue on Graph Theory (GWGT 2019)
  137. The general position problem and strong resolving graphs
  138. Connected domination game played on Cartesian products
  139. On minimum algebraic connectivity of graphs whose complements are bicyclic
  140. A novel method to construct NSSD molecular graphs
https://doi.org/10.1515/math-2019-0122
Keywords for this article
iterative methods; Newton’s method; Cauchy’s method; order of convergence; Padé approximant
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Articles in the same Issue

  1. Regular Articles
  2. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one
  3. Centralizers of automorphisms permuting free generators
  4. Extreme points and support points of conformal mappings
  5. Arithmetical properties of double Möbius-Bernoulli numbers
  6. The product of quasi-ideal refined generalised quasi-adequate transversals
  7. Characterizations of the Solution Sets of Generalized Convex Fuzzy Optimization Problem
  8. Augmented, free and tensor generalized digroups
  9. Time-dependent attractor of wave equations with nonlinear damping and linear memory
  10. A new smoothing method for solving nonlinear complementarity problems
  11. Almost periodic solution of a discrete competitive system with delays and feedback controls
  12. On a problem of Hasse and Ramachandra
  13. Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge
  14. A note on the formulas for the Drazin inverse of the sum of two matrices
  15. Completeness theorem for probability models with finitely many valued measure
  16. Periodic solution for ϕ-Laplacian neutral differential equation
  17. Asymptotic orbital shadowing property for diffeomorphisms
  18. Modular equations of a continued fraction of order six
  19. Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas
  20. Stability Problems and Analytical Integration for the Clebsch’s System
  21. Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures
  22. On split Lie color triple systems
  23. Triangular Surface Patch Based on Bivariate Meyer-König-Zeller Operator
  24. Generators for maximal subgroups of Conway group Co1
  25. Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model
  26. Characterizations of Convex spaces and Anti-matroids via Derived Operators
  27. On Partitions and Arf Semigroups
  28. Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions
  29. A concise proof to the spectral and nuclear norm bounds through tensor partitions
  30. A categorical approach to abstract convex spaces and interval spaces
  31. Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations
  32. Parity results for broken 11-diamond partitions
  33. A new fourth power mean of two-term exponential sums
  34. The new operations on complete ideals
  35. Soft covering based rough graphs and corresponding decision making
  36. Complete convergence for arrays of ratios of order statistics
  37. Sufficient and necessary conditions of convergence for ρ͠ mixing random variables
  38. Attractors of dynamical systems in locally compact spaces
  39. Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains
  40. Statistical approximation properties of λ-Bernstein operators based on q-integers
  41. An investigation of fractional Bagley-Torvik equation
  42. Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer
  43. On the hybrid power mean of two kind different trigonometric sums
  44. Embedding of Supplementary Results in Strong EMT Valuations and Strength
  45. On Diophantine approximation by unlike powers of primes
  46. A General Version of the Nullstellensatz for Arbitrary Fields
  47. A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces
  48. Random Polygons and Estimations of π
  49. The optimal pebbling of spindle graphs
  50. MBJ-neutrosophic ideals of BCK/BCI-algebras
  51. A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉
  52. A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers
  53. Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization
  54. Stability property of the prey free equilibrium point
  55. Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem
  56. Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders
  57. Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching
  58. Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts
  59. Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application
  60. Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems
  61. Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model
  62. Algebraic loop structures on algebra comultiplications
  63. Two weight estimates for a class of (p, q) type sublinear operators and their commutators
  64. Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays
  65. 2-closures of primitive permutation groups of holomorph type
  66. Monotonicity properties and inequalities related to generalized Grötzsch ring functions
  67. Variation inequalities related to Schrödinger operators on weighted Morrey spaces
  68. Research on cooperation strategy between government and green supply chain based on differential game
  69. Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting
  70. *-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds
  71. Some improved bounds on two energy-like invariants of some derived graphs
  72. Pricing under dynamic risk measures
  73. Finite groups with star-free noncyclic graphs
  74. A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies
  75. S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain
  76. On Diophantine equations involving Lucas sequences
  77. A new way to represent functions as series
  78. Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system
  79. Some remarks on a pair of seemingly unrelated regression models
  80. Lyapunov stable homoclinic classes for smooth vector fields
  81. Stabilizers in EQ-algebras
  82. The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles
  83. Spectrum perturbations of compact operators in a Banach space
  84. The non-commuting graph of a non-central hypergroup
  85. Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation
  86. Positive solutions of the discrete Dirichlet problem involving the mean curvature operator
  87. Dislocated quasi cone b-metric space over Banach algebra and contraction principles with application to functional equations
  88. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis
  89. Differential polynomials of L-functions with truncated shared values
  90. Exclusion sets in the S-type eigenvalue localization sets for tensors
  91. Continuous linear operators on Orlicz-Bochner spaces
  92. Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity
  93. Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces
  94. A quantitative obstruction to collapsing surfaces
  95. Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species
  96. Coexistence for a kind of stochastic three-species competitive models
  97. Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1
  98. On the two-term exponential sums and character sums of polynomials
  99. F-biharmonic maps into general Riemannian manifolds
  100. Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn
  101. Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains
  102. Power graphs and exchange property for resolving sets
  103. On nearly Hurewicz spaces
  104. Least eigenvalue of the connected graphs whose complements are cacti
  105. Determinants of two kinds of matrices whose elements involve sine functions
  106. A characterization of translational hulls of a strongly right type B semigroup
  107. Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications
  108. Lp estimates for maximal functions along surfaces of revolution on product spaces
  109. Path-induced closure operators on graphs for defining digital Jordan surfaces
  110. Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras
  111. Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system
  112. Injective hulls of many-sorted ordered algebras
  113. Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3
  114. Global properties of virus dynamics with B-cell impairment
  115. The monotonicity of ratios involving arc tangent function with applications
  116. A family of Cantorvals
  117. An asymptotic property of branching-type overloaded polling networks
  118. Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
  119. Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion
  120. L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras
  121. L-topological-convex spaces generated by L-convex bases
  122. An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior
  123. New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices
  124. Hankel determinant of order three for familiar subsets of analytic functions related with sine function
  125. On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5
  126. Results on existence for generalized nD Navier-Stokes equations
  127. Regular Banach space net and abstract-valued Orlicz space of range-varying type
  128. Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means
  129. On a new convergence in topological spaces
  130. On a fixed point theorem with application to functional equations
  131. Coupled system of a fractional order differential equations with weighted initial conditions
  132. Rough quotient in topological rough sets
  133. Split Hausdorff internal topologies on posets
  134. A preconditioned AOR iterative scheme for systems of linear equations with L-matrics
  135. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering
  136. Special Issue on Graph Theory (GWGT 2019)
  137. The general position problem and strong resolving graphs
  138. Connected domination game played on Cartesian products
  139. On minimum algebraic connectivity of graphs whose complements are bicyclic
  140. A novel method to construct NSSD molecular graphs
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