A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations

Table 2

Numerical results from 𝑓₁ to 𝑓₁₀ and seventh-order schemes.

		HFM	SKM	NM	M7
𝑓1,𝑥0=0.9	|𝑥𝑘+1−𝑥𝑘|	9.2403𝑒−050		7.8995𝑒−104	1.9456𝑒−023
	|𝑓(𝑥𝑘+1)|	0		0	1.8101𝑒−159
	ACOC	—		7.0023	6.6629
	iter	6	≥100	4	3
	e-time	0.1126		0.1404	0.0927
𝑓2,𝑥0=1.2	|𝑥𝑘+1−𝑥𝑘|	1.1602𝑒−045	1.0203𝑒−036	6.3747𝑒−058	3.1050𝑒−029
	|𝑓(𝑥𝑘+1)|	0	0	0	1.0495𝑒−202
	ACOC	6.7965	7.2403	6.7108	6.8723
	iter	3	4	3	3
	e-time	0.0604	0.1445	0.1100	0.1075
𝑓3,𝑥0=2.1	|𝑥𝑘+1−𝑥𝑘|	1.8810𝑒−040	9.8612𝑒−037	1.4468𝑒−052	5.6495𝑒−024
	|𝑓(𝑥𝑘+1)|	0	0	0	3.7489𝑒−167
	ACOC	6.3228	6.2401	6.5225	7.0731
	iter	3	4	3	3
	e-time	0.0315	0.0969	0.0674	0.0468
𝑓4,𝑥0=2.2	|𝑥𝑘+1−𝑥𝑘|	5.1064𝑒−034	—	9.9985𝑒−042	3.4709𝑒−027
	|𝑓(𝑥𝑘+1)|	0	—	0	5.1781𝑒−184
	ACOC	6.8073	—	6.9177	6.8325
	iter	3	n.c.	3	3
	e-time	0.0532	—	0.0842	0.0930
𝑓5,𝑥0=2.3	|𝑥𝑘+1−𝑥𝑘|	2.6845𝑒−053	3.2107𝑒−025	1.2399𝑒−062	1.2638𝑒−030
	|𝑓(𝑥𝑘+1)|	3.1147𝑒−207	9.9146𝑒−167	3.1147𝑒−207	6.8463𝑒−207
	ACOC	6.9527	7.1651	6.9798	6.8181
	iter	3	5	3	3
	e-time	0.0406	0.0894	0.0961	0.0596
𝑓6,𝑥0=2	|𝑥𝑘+1−𝑥𝑘|	8.6332𝑒−124		1.0547𝑒−075	5.4741𝑒−023
	|𝑓(𝑥𝑘+1)|	0		0	9.2491𝑒−157
	ACOC	6.9990		6.9948	5.9331
	iter	4	≥100	4	3
	e-time	0.0911		0.1409	0.1226
𝑓7,𝑥0=0.5	|𝑥𝑘+1−𝑥𝑘|	8.9879𝑒−076	6.4335𝑒−045	1.6607𝑒−028	4.7872𝑒−034
	|𝑓(𝑥𝑘+1)|	3.8934𝑒−208	1.9467𝑒−208	1.1707𝑒−195	3.8434𝑒−208
	ACOC	7.0115	6.8734	7.2394	6.8055
	iter	4	3	3	3
	e-time	0.0747	0.0757	0.1214	0.0732
𝑓8,𝑥0=1.5	|𝑥𝑘+1−𝑥𝑘|	2.1424𝑒−053		8.0693𝑒−063	1.1249𝑒−030
	|𝑓(𝑥𝑘+1)|	0		0	9.3442𝑒−207
	ACOC	6.9528		6.9783	6.7788
	iter	3	≥100	3	3
	e-time	0.0730		0.0967	0.1170
𝑓9, 𝑥0=0.8	|𝑥𝑘+1−𝑥𝑘|	1.6408𝑒−027	1.3528𝑒−128	4.3711𝑒−039	6.1073𝑒−028
	|𝑓(𝑥𝑘+1)|	1.5509𝑒−190	2.4334𝑒−208	1.2167𝑒−208	1.6582𝑒−191
	ACOC	7.3243	6.9999	7.1316	6.7613
	iter	3	4	3	4
	e-time	0.0552	0.1158	0.1112	0.1232
𝑓10,𝑥0=0.6	|𝑥𝑘+1−𝑥𝑘|	2.0089𝑒−035	—	1.1046𝑒−051	2.7207𝑒−019
	|𝑓(𝑥𝑘+1)|	0	—	0	2.1785𝑒−167
	ACOC	10.9352	—	10.9217	8.7406
	iter	3	n.c.	3	3
	e-time	0.0367	—	0.0922	0.0749