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

Table 1

Numerical results from 𝑓₁ to 𝑓₁₀.

		SM	DHM	LZM	CTM	M7
𝑓1,𝑥0=0.9	|𝑥𝑘+1−𝑥𝑘|	3.9289𝑒−112	1.7589𝑒−055	3.7228𝑒−122	1.5049𝑒−124	1.9456𝑒−023
	|𝑓(𝑥𝑘+1)|	4.4514𝑒−223	2.8819𝑒−164	0	0	1.8101𝑒−159
	ACOC	2.0000	3.0000	4.0000	4.0000	6.6629
	iter	9	7	5	5	3
	e-time	0.1217	0.1891	0.1139	0.1033	0.0927
𝑓2,𝑥0=1.2	|𝑥𝑘+1−𝑥𝑘|	1.4587𝑒−149	7.6358𝑒−103	3.4035𝑒−138	2.6499𝑒−141	3.1050𝑒−029
	|𝑓(𝑥𝑘+1)|	2.0878𝑒−298	2.5787𝑒−306	0	0	1.0495𝑒−202
	ACOC	2.0000	3.0000	4.0000	4.0000	6.8723
	iter	9	11	5	5	3
	e-time	0.1282	0.2791	0.1161	0.1092	0.1075
𝑓3,𝑥0=2.1	|𝑥𝑘+1−𝑥𝑘|	8.3630𝑒−085	1.9786𝑒−059	1.0746e-143	1.4483e-112	5.6495e-024
	|𝑓(𝑥𝑘+1)|	1.7410𝑒−169	1.8682𝑒−177	0	0	3.7489𝑒−167
	ACOC	2.0000	3.0000	4.0000	4.0000	7.0731
	iter	8	6	5	5	3
	e-time	0.0746	0.0894	0.0874	0.0867	0.0468
𝑓4,𝑥0=2.2	|𝑥𝑘+1−𝑥𝑘|	1.9109𝑒−116	1.8976𝑒−072	3.3922𝑒−110	1.0118𝑒−116	3.4709𝑒−027
	|𝑓(𝑥𝑘+1)|	4.3820𝑒−231	1.5033𝑒−214	0	0	5.1781𝑒−184
	ACOC	2.0000	3.0000	4.0000	4.0000	6.8325
	iter	10	6	5	5	3
	e-time	0.1176	0.1443	0.1246	0.0914	0.0930
𝑓5,𝑥0=2.3	|𝑥𝑘+1−𝑥𝑘|	7.8747𝑒−085	1.2204𝑒−123	9.1432𝑒−142	8.5347𝑒−144	1.2638𝑒−030
	|𝑓(𝑥𝑘+1)|	5.9818𝑒−167	0	0	0	6.8463𝑒−207
	ACOC	2.0000	3.0000	4.0000	4.0000	6.8181
	iter	10	7	5	5	3
	e-time	0.1168	0.1353	0.0945	0.1059	0.0596
𝑓6,𝑥0=2	|𝑥𝑘+1−𝑥𝑘|	1.4558𝑒−087	—	2.1767𝑒−109	5.9067𝑒−112	5.4741𝑒−023
	|𝑓(𝑥𝑘+1)|	5.7398𝑒−174	—	0	0	9.2491𝑒−157
	ACOC	2.0000	—	4.0000	4.0000	5.9331
	iter	8	nc	5	5	3
	e-time	0.1245	—	0.1576	0.1418	0.1226
𝑓7,𝑥0=0.5	|𝑥𝑘+1−𝑥𝑘|	5.1639𝑒−127	3.2959𝑒−079	1.5312𝑒−050	3.3808𝑒−073	4.7872𝑒−034
	|𝑓(𝑥𝑘+1)|	9.3020𝑒−253	1.1763𝑒−235	4.6052𝑒−199	7.2079𝑒−290	9.9787𝑒−234
	ACOC	2.0000	3.0000	3.9999	4.0000	6.8055
	iter	11	6	5	5	3
	e-time	0.1671	0.1696	0.1426	0.1349	0.0732
𝑓8,𝑥0=1.5	|𝑥𝑘+1−𝑥𝑘|	1.0817𝑒−142	8.4333𝑒−122	6.9628𝑒−136	2.1376𝑒−137	1.1249𝑒−030
	|𝑓(𝑥𝑘+1)|	1.6591𝑒−282	0	0	0	7.6946𝑒−207
	ACOC	2.0000	3.0000	4.0000	4.0000	6.7788
	iter	11	7	5	5	3
	e-time	0.1886	0.2356	0.1797	0.1463	0.1170
𝑓9,𝑥0=0.8	|𝑥𝑘+1−𝑥𝑘|	2.2055𝑒−129	4.6810𝑒−128	2.9693𝑒−139	7.1679𝑒−140	6.1073𝑒−028
	|𝑓(𝑥𝑘+1)|	9.0498𝑒−257	0	0	0	1.6582𝑒−191
	ACOC	2.0000	3.0000	4.0000	4.0000	6.7613
	iter	15	6	7	8	4
	e-time	0.2368	0.1680	0.2684	0.1902	0.1232
𝑓10,𝑥0=0.6	|𝑥𝑘+1−𝑥𝑘|	2.4132𝑒−081	1.3419𝑒−130	6.2415𝑒−141	1.0766𝑒−031	2.7207𝑒−019
	|𝑓(𝑥𝑘+1)|	2.8106𝑒−242	0	0	1.9282𝑒−155	2.1785𝑒−167
	ACOC	3.0000	5.0000	5.0000	4.9922	8.7406
	iter	7	5	5	4	3
	e-time	0.0955	0.1471	0.0746	0.0733	0.0749