New Predictor-Corrector Methods with High Efficiency for Solving Nonlinear Systems

Table 2

Numerical results for functions 𝐹₁ to 𝐹₄.

Function	Method	Iter	Sol	‖𝑥^(𝑘)−𝑥^(𝑘−1)‖	‖𝐹(𝑥^(𝑘))‖	𝜌	𝑒-time (sec)
	NC	8	𝜉1	1.43𝑒−121	2.06𝑒−243	2.0000	8.6407
	JT	4	𝜉1	1.69𝑒−60	2.06𝑒−243	4.0000	3.9347
𝐹1	M4	4	𝜉1	1.69𝑒−60	2.06𝑒−243	4.0000	3.7813
𝑥^(0)=(0.8,…,0.8)	M6	4	𝜉1	6.94𝑒−193	4.33𝑒−1160	6.0000	5.3911
	M8	3	𝜉1	9.40𝑒−50	3.51𝑒−4011	8.0913	5.0065
	PsM10	3	𝜉1	1.28𝑒−91	9.54𝑒−921	10.0545	4.9061
	PsM14	3	𝜉1	4.65𝑒−164	0	14.0702	6.1018
	NC	17	𝜉1	3.37𝑒−340	1.14𝑒−340	—	9.2128
	JT	9	𝜉1	8.18𝑒−085	1.14𝑒−340	4.0000	10.1416
𝐹1	M4	9	𝜉1	8.18𝑒−085	1.14𝑒−340	4.0000	10.9104
𝑥^(0)=(0.0015,…,0.0015)	M6	7	𝜉1	1.40𝑒−035	9.46𝑒−216	—	12.3266
	M8	19	𝜉1	9.50𝑒−030	1.29𝑒−240	—	59.4832
	PsM10	6	𝜉1	3.02𝑒−102	5.23𝑒−1027	—	17.9957
	PsM14	5	𝜉1	1.84𝑒−162	0	—	22.6130
	NC	9	𝜉1	2.45𝑒−181	5.92𝑒−362	2.0148	0.2395
	JT	5	𝜉1	9.48𝑒−189	8.13𝑒−754	4.0279	0.3250
𝐹2	M4	5	𝜉1	9.48𝑒−189	8.13𝑒−754	4.0279	0.1841
𝑥^(0)=(−0.5,−0.5)	M6	4	𝜉1	1.34𝑒−146	2.14𝑒−878	5.9048	0.2744
	M8	3	𝜉1	1.90𝑒−038	1.23𝑒−302	7.8530	0.3718
	PsM10	3	𝜉1	6.72𝑒−72	2.68𝑒−714	9.9092	0.4674
	PsM14	3	𝜉1	2.13𝑒−122	1.95𝑒−1706	13.9829	0.3187
	NC	13	𝜉1	2.20𝑒−182	2.73𝑒−374	1.9917	0.3713
	JT	7	𝜉1	2.10𝑒−179	4.51𝑒−716	3.9925	0.4001
𝐹2	M4	7	𝜉1	2.10𝑒−179	4.51𝑒−716	3.9925	0.7535
𝑥^(0)=(−5,−3)	M6	8	𝜉1	2.55𝑒−036	5.81𝑒−216	—	0.9382
	M8	>5000
	PsM10	4	𝜉1	2.59𝑒−021	3.51𝑒−208	—	0.4363
	PsM14	29	𝜉2	9.45𝑒−020	5.05𝑒−273	—	7.8090
	NC	10	𝜉1	1.65𝑒−190	4.61𝑒−380	2.0000	1.4675
	JT	5	𝜉1	8.03𝑒−113	7.59𝑒−450	3.9995	0.3151
𝐹3	M4	5	𝜉1	8.03𝑒−113	7.59𝑒−450	3.9995	0.3034
𝑥^(0)=(2,−3)	M6	4	𝜉1	1.25𝑒−082	2.83𝑒−493	6.0015	0.3696
	M8	4	𝜉1	1.54𝑒−162	3.16𝑒−1296	7.9993	0.4463
	PsM10	3	𝜉1	5.59𝑒−044	1.40𝑒−436	9.4708	0.4682
	PsM14	3	𝜉1	3.46𝑒−068	3.45𝑒−948	13.1659	0.5925
	NC	35	𝜉1	3.71𝑒−177	2.33𝑒−253	—	1.4828
	JT	11	𝜉1	3.29𝑒−143	1.67𝑒−574	—	0.7781
𝐹3	M4	11	𝜉1	3.29𝑒−143	1.67𝑒−574	—	0.7535
𝑥^(0)=(0.2,0.1)	M6	9	𝜉1	1.31𝑒−064	3.61𝑒−385	—	0.8001
	M8	n.c.	𝜉1
	PsM10	5	𝜉1	6.85𝑒−156	1.06𝑒−1555	—	0.6352
	PsM14	8	𝜉2	7.87𝑒−155	0	—	1.1870
	NC	10	𝜉1	1.03𝑒−135	1.55𝑒−270	1.9995	2.3263
	JT	5	𝜉1	9.94𝑒−073	2.09𝑒−289	4.0066	0.5296
𝐹4	M4	5	𝜉1	9.94𝑒−073	2.09𝑒−289	4.0066	0.6340
𝑥^(0)=(1,−1.5,−0.5)	M6	4	𝜉1	9.31𝑒−057	4.86𝑒−338	5.9750	0.7443
	M8	4	𝜉1	4.43𝑒−046	1.08𝑒−364	—	0.8282
	PsM10	3	𝜉1	1.43𝑒−031	1.04𝑒−311	9.6674	0.8100
	PsM14	3	𝜉1	1.91𝑒−033	4.05𝑒−462	13.9954	1.0465
	NC	12	𝜉1	1.08𝑒−192	1.55𝑒−384	1.9996	2.7271
	JT	6	𝜉1	2.31𝑒−103	7.97𝑒−412	4.0090	0.7761
𝐹4	M4	6	𝜉1	2.31𝑒−103	7.97𝑒−412	4.0090	1.0301
𝑥^(0)=(7,−5,−5)	M6	5	𝜉1	2.99𝑒−086	4.69𝑒−515	—	1.0090
	M8	15	𝜉3	1.77𝑒−071	1.48𝑒−568	—	3.4007
	PsM10	4	𝜉1	6.86𝑒−067	1.25𝑒−666	—	1.0245
	PsM14	7	𝜉2	1.09𝑒−130	9.15𝑒−1825	—	1.8179