A Trust-Region-Based BFGS Method with Line Search Technique for Symmetric Nonlinear Equations

Table 2

Test Results For Problem 2.

(a) (Small-scales). Test results for Algorithm 2.1.

x0	(0.5,…,0.5)	(1,…,1)	(3,…,3)	(−0.75,…,−0.75)	(−2,…,−2)	(−3,…,−3)
Dim	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG
𝑛=10	25/44/9.720971e-07	21/32/4.889567e-07	92/103/2.475812e-08	21/32/8.691255e-07	46/59/5.860956e-07	86/105/6.348374e-08
𝑛=50	37/56/9.950345e-07	39/56/8.776379e-07	113/139/9.587026e-07	40/63/6.984106e-07	69/96/9.523480e-07	103/125/9.404211e-07
𝑛=99	42/59/9.725361e-07	41/60/7.374460e-07	113/135/7.909796e-07	40/55/8.380367e-07	117/489/9.805302e-07	97/129/7.975248e-07
x0	(0.5,0,0.5,0,…)	(1,0,1,0,…)	(3,0,3,0,…)	(−0.75,0,−0.75,0,…)	(−2,0,−2,0,…)	(−3,0,−3,0,…)
Dim	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG
𝑛=10	24/35/4.711749e-07	21/30/3.147507e-07	44/65/3.529113e-07	27/48/4.004367e-07	39/76/8.503415e-07	29/42/7.623619e-07
𝑛=50	36/57/8.776354e-07	36/57/8.287552e-07	54/77/8.491652e-07	41/64/9.492805e-07	42/69/9.029472e-07	58/77/9.752703e-07
𝑛=99	36/61/8.265146e-07	37/56/9.507706e-07	60/93/5.373087e-07	42/73/8.247653e-07	50/79/9.217390e-07	62/88/8.307004e-07

(b) (Large-scales). Test results for Algorithm 2.1

x0	(0.5,…,0.5)	(1,…,1)	(3,…,3)	(−0.75,…,−0.75)
Dim	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG
𝑛=200	40/57/7.464372e-007	41/58/4.921097e-007	112/130/6.229759e-007	40/65/7.785598e-007
𝑛=500	36/57/7.887407e-007	41/60/3.538433e-007	113/135/8.871522e-007	38/69/9.785814e-007
𝑛=1000	42/65/7.382939e-007	40/59/7.463210e-007	120/146/6.044161e-007	40/69/4.563405e-007
x0	(0.5,0,0.5,0,…)	(1,0,1,0,…)	(3,0,3,0,…)	(−0.75,0,−0.75,0,…)
Dim	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG
𝑛=200	41/64/6.671246e-007	36/61/9.977774e-007	63/94/8.153527e-007	39/64/9.737674e-007
𝑛=500	42/67/9.154342e-007	37/58/8.340650e-007	49/74/8.277585e-007	37/60/6.328648e-007
𝑛=1000	43/62/7.874632e-007	40/61/8.997602e-007	55/76/8.830280e-007	41/68/8.430165e-007

(c) (Small-scales). Test results for Algorithm 2.2.

x0	(0.5,…,0.5)	(1,…,1)	(3,…,3)	(−0.75,…,−0.75)	(−2,…,−2)	(−3,…,−3)
Dim	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG
𝑛=10	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000
𝑛=50	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000
𝑛=99	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000
x0	(0.5,0,0.5,0,…)	(1,0,1,0,…)	(3,0,3,0,…)	(−0.75,0,−0.75,0,…)	(−2,0,−2,0,…)	(−3,0,−3,0,…)
Dim	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG
𝑛=10	196/391/9.528159e-007	204/407/9.939962e-007	213/425/9.975163e-007	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000
𝑛=50	199/397/9.791882e-007	208/415/9.592104e-007	217/433/9.634499e-007	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000
𝑛=99	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000

(d) (Large-scales). Test results for Algorithm 2.2.

x0	(0.5,…,0.5)	(1,…,1)	(3,…,3)	(−0.75,…,−0.75)
Dim	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG
𝑛=200	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000
𝑛=500	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000
𝑛=1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000	𝑁𝐼>1000
x0	(0.5,0,0.5,0,…)	(1,0,1,0,…)	(3,0,3,0,…)	(−0.75,0,−0.75,0,…)
Dim	NI/NG/EG	NI/NG/EG	NI/NG/EG	NI/NG/EG
𝑛=200	200/399/9.537272e-007	208/415/9.804205e-007	217/433/9.775482e-007	𝑁𝐼>1000
𝑛=500	201/401/9.425775e-007	209/417/9.430954e-007	217/433/9.908579e-007	𝑁𝐼>1000
𝑛=1000	202/403/9.503816e-007	209/417/9.824140e-007	218/435/9.470469e-007	𝑁𝐼>1000