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

Table 1

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

	IT	NFE	𝑓(𝑥𝑛+1)	Time	𝛿
𝑓1∶𝑥0=1
NM	5	10	0	0.062577	2.126987475037367𝑒−011
WF	3	9	0	0.018731	2.284722713019605𝑒−006
PP	4	12	0	0.044520	1.558753126573720𝑒−013
CHM	4	12	0	0.040102	1.643130076445232𝑒−014
HM	3	9	0	0.019876	3.698649917449615𝑒−007
LM1	4	12	0	0.029137	1.043609643147647𝑒−014
LM2	3	9	0	0.022242	1.788683664960544𝑒−006
LM3	3	9	0	0.023550	1.295199573814188𝑒−006
LM4	3	9	0	0.019227	3.091731315407742𝑒−010
LM5	3	9	0	0.020131	1.788683664960544𝑒−006
𝑓1∶𝑥0=2
NM	5	10	0	0.057231	5.020497351182485𝑒−010
WF	4	12	0	0.039384	4.440892098500626𝑒−016
PP	4	12	0	0.044239	7.949196856316121𝑒−014
CHM	4	12	0	0.042782	2.065014825802791𝑒−014
HM	3	9	0	0.017733	3.107350415199051𝑒−006
LM1	4	12	0	0.032638	2.706235235905297𝑒−011
LM2	4	12	0	0.037830	5.551115123125783𝑒−015
LM3	4	12	0	0.034087	3.108624468950438𝑒−015
LM4	4	12	0	0.038399	2.220446049250313𝑒−016
LM5	4	12	0	0.035809	5.551115123125783𝑒−015
𝑓2∶𝑥0=0
NM	4	8	0	0.035702	2.665312415217613𝑒−012
WF	3	9	0	0.019699	7.801814749797131𝑒−012
PP	3	9	0	0.021596	1.219191414492116𝑒−012
CHM	3	9	0	0.023312	8.906764215055318𝑒−013
HM	3	9	0	0.017431	7.374600929921371𝑒−012
LM1	3	9	0	0.016351	9.004856806882344𝑒−007
LM2	3	9	0	0.017242	8.067152316715287𝑒−007
LM3	3	9	0	0.018978	8.334935165388302𝑒−007
LM4	3	9	0	0.016178	7.334822825222354𝑒−007
LM5	3	9	0	0.016795	8.067152316715287𝑒−007
𝑓2∶𝑥0=0.5
NM	4	8	0	0.044392	1.791899961745003𝑒−013
WF	3	9	0	0.023132	6.424749621203318𝑒−012
PP	3	9	0	0.024293	4.607425552194400𝑒−014
CHM	3	9	0	0.026689	3.087480271446452𝑒−011
HM	3	9	0	0.022199	4.208039472430869𝑒−011
LM1	3	9	0	0.024120	2.850281847210923𝑒−007
LM2	3	9	4.440892098500626𝑒−016	0.021035	2.744941123289379𝑒−007
LM3	3	9	0	0.021310	2.778229745148408𝑒−007
LM4	3	9	0	0.020886	2.655207446689012𝑒−007
LM5	3	9	4.440892098500626𝑒−016	0.021592	2.744941123289379𝑒−007
𝑓3∶𝑥0=1
NM	7	14	3.537126081266182𝑒−024	0.085178	1.085848323840232𝑒−012
WF	4	12	2.621304391538411𝑒−016	0.030341	4.330310691887267𝑒−006
PP	5	15	8.196910187379942𝑒−033	0.063971	8.806888499109001𝑒−012
CHM	5	15	6.352230116524407𝑒−022	0.053064	2.520356663650445𝑒−011
HM	5	15	7.257520328033309𝑒−029	0.055315	8.459855063117184𝑒−015
LM1	5	15	5.355132874954819𝑒−017	0.041370	4.825251676864564𝑒−007
LM2	4	12	6.348042447121620𝑒−018	0.026722	5.276208195236892𝑒−013
LM3	3	9	1.559423551656343𝑒−017	0.019476	5.545966966175517𝑒−011
LM4	4	12	6.485252546346663𝑒−018	0.030513	1.317913979899399𝑒−009
LM5	4	12	6.348042447121620𝑒−018	0.031575	5.276208195236892𝑒−013
𝑓3∶𝑥0=0.5
NM	6	12	5.905159674954809𝑒−020	0.071738	1.402992074360412𝑒−010
WF	4	12	1.764824578467612𝑒−017	0.039948	4.200981459101664𝑒−009
PP	4	12	1.694834561079519𝑒−019	0.056046	2.767209186089879𝑒−007
CHM	4	12	8.798671206634291𝑒−017	0.036458	3.059650585008672𝑒−007
HM	4	12	3.044907255908736𝑒−017	0.072238	5.518067735074518𝑒−009
LM1	4	12	1.293991275513200𝑒−017	0.033751	1.322689611777333𝑒−013
LM2	4	12	2.828488650436813𝑒−016	0.032997	4.626219030783813𝑒−006
LM3	4	12	6.789745028898642𝑒−018	0.036425	1.912594303370619𝑒−014
LM4	4	12	1.506225012219799𝑒−017	0.034137	2.365541379047147𝑒−011
LM5	4	12	2.828488650436813𝑒−016	0.035087	4.626219030783813𝑒−006
𝑓4∶𝑥0=2.5
NM	6	12	0	0.071353	1.154631945610163𝑒−014
WF	4	12	0	0.034140	7.314593375440381𝑒−012
PP	4	12	0	0.042598	4.221685223626537𝑒−010
CHM	4	12	0	0.036365	9.853584614916144𝑒−011
HM	4	12	0	0.038080	4.662936703425658𝑒−014
LM1	3	9	0	0.018372	9.336336148635382𝑒−010
LM2	4	12	0	0.035562	1.610311883837312𝑒−010
LM3	4	12	0	0.036985	2.312174895990893𝑒−010
LM4	4	12	0	0.035463	5.127454016928823𝑒−012
LM5	4	12	0	0.033362	1.610311883837312𝑒−010
𝑓4∶𝑥0=3.5
NM	7	14	0	2.151288	2.877564853065451𝑒−011
WF	5	15	0	0.054803	6.550315845288424𝑒−013
PP	5	15	0	0.063380	4.512221707386743𝑒−010
CHM	5	15	0	0.061753	4.188738245147761𝑒−011
HM	4	12	0	0.046608	4.485352507632712𝑒−006
LM1	5	15	6.661338147750939𝑒−016	0.050909	2.781577124189028𝑒−008
LM2	5	15	0	0.048688	9.426681657487279𝑒−011
LM3	10	30	0	0.111640	1.081885248055414𝑒−009
LM4	4	12	0	0.040109	2.154232348061669𝑒−011
LM5	5	15	0	0.043482	9.426681657487279𝑒−011
𝑓5∶𝑥0=0
NM	5	10	0	0.050123	1.701233598438989𝑒−010
WF	3	9	0	0.023397	7.792236328407753𝑒−007
PP	4	12	0	0.038373	1.500558566291943𝑒−010
CHM	4	12	0	0.037498	5.327979279989847𝑒−009
HM	4	12	0	0.035701	1.121325254871408𝑒−014
LM1	4	12	3.330669073875470𝑒−016	0.028362	4.678117765610779𝑒−006
LM2	4	12	0	0.028666	7.181037887660224𝑒−007
LM3	6	18	0	0.053924	6.676881270095691𝑒−013
LM4	4	12	0	0.028247	2.163133006050089𝑒−010
LM5	4	12	0	0.029150	7.181037887660224𝑒−007
𝑓5∶𝑥0=1
NM	4	8	0	0.039328	1.701233598438989𝑒−010
WF	2	6	4.440892098500626𝑒−016	0.003092	2.674277017133964𝑒−005
PP	3	9	0	0.020461	9.809075773858922𝑒−011
CHM	3	9	0	0.019378	1.600380383770528𝑒−009
HM	3	9	0	0.018186	6.624212289807474𝑒−010
LM1	3	9	0	0.018199	7.500783549829748𝑒−008
LM2	3	9	1.110223024625157𝑒−016	0.017732	5.330569952111119𝑒−008
LM3	3	9	0	0.013523	8.804162354714151𝑒−008
LM4	3	9	0	0.013782	3.531516445942629𝑒−008
LM5	3	9	1.110223024625157𝑒−016	0.016581	5.330569952111119𝑒−008
𝑓6∶𝑥0=1
NM	6	12	3.330669073875470𝑒−016	0.065520	3.059774655866931𝑒−013
WF	4	12	4.440892098500626𝑒−016	0.030929	1.793023507445923𝑒−010
PP	16	48	4.440892098500626𝑒−016	0.266014	1.531728257564424𝑒−007
CHM	5	15	4.440892098500626𝑒−016	0.061006	6.883094094689568𝑒−010
HM	4	12	4.440892098500626𝑒−016	0.043655	2.686739719592879𝑒−013
LM1	4	12	3.330669073875470𝑒−016	0.030877	1.659126835917846𝑒−008
LM2	4	12	3.330669073875470𝑒−016	0.031156	9.103828801926284𝑒−014
LM3	4	12	3.330669073875470𝑒−016	0.033076	1.261090962767497𝑒−006
LM4	5	15	4.440892098500626𝑒−016	0.041102	8.215650382226158𝑒−014
LM5	4	12	3.330669073875470𝑒−016	0.030923	9.103828801926284𝑒−014
𝑓6∶𝑥0=2.5
NM	6	12	3.330669073875470𝑒−016	0.064979	1.404654170755748𝑒−012
WF	4	12	3.330669073875470𝑒−016	0.038404	4.229505634611996𝑒−012
PP	4	12	3.330669073875470𝑒−016	0.042164	1.030850205196998𝑒−008
CHM	4	12	3.330669073875470𝑒−016	0.043437	1.475204565171140𝑒−007
HM	4	12	4.440892098500626𝑒−016	0.043896	9.462626682221753𝑒−009
LM1	5	15	3.330669073875470𝑒−016	0.053132	6.201483770951199𝑒−012
LM2	4	12	3.330669073875470𝑒−016	0.036191	4.450250368215336𝑒−008
LM3	6	18	3.330669073875470𝑒−016	0.064485	8.881784197001252𝑒−016
LM4	4	12	3.330669073875470𝑒−016	0.035393	5.568381311604753𝑒−009
LM5	4	12	3.330669073875470𝑒−016	0.036946	4.450250368215336𝑒−008
𝑓7∶𝑥0=3.25
NM	8	16	0	0.114189	9.720393379097914𝑒−010
WF	6	18	0	0.065645	1.691979889528739𝑒−013
PP	6	18	0	0.073469	1.131490456884876𝑒−010
CHM	6	18	0	0.082677	2.398081733190338𝑒−014
HM	5	15	0	0.055630	3.082423205569285𝑒−012
LM1	4	12	0	0.036191	2.613820271335499𝑒−011
LM2	5	15	0	0.049599	4.440892098500626𝑒−016
LM3	4	12	0	0.038172	1.546353226355990𝑒−006
LM4	6	18	0	0.051577	4.440892098500626𝑒−016
LM5	5	15	0	0.048340	4.440892098500626𝑒−016
𝑓7∶𝑥0=3.45
NM	11	22	0	0.147308	4.008793297316515𝑒−011
WF	8	24	0	0.106276	7.105427357601002𝑒−015
PP	8	24	0	0.116999	2.160227552394645𝑒−010
CHM	7	21	0	0.100725	1.268533917908599𝑒−006
HM	6	18	0	0.062876	1.694565332499565𝑒−008
LM1	6	18	0	0.060920	4.440892098500626𝑒−016
LM2	6	18	0	0.060951	1.753264200488047𝑒−011
LM3	6	18	0	0.060627	1.518252190635394𝑒−011
LM4	7	21	0	0.087873	9.144596191390519𝑒−011
LM5	6	18	0	0.060607	1.753264200488047𝑒−011