强Wolfe线搜索下的修正PRP和HS共轭梯度法

摘要/Abstract

摘要：

共轭梯度法是求解大规模无约束优化问题最有效的方法之一.结合强Wolfe线搜索的第二个不等式，该文提出了修正的PRP和HS公式.在常规假设下，由修正的PRP方法和HS方法所产生的搜索方向均满足充分下降条件，且得到了较大的参数σ取值范围，并证明了两个修正的方法是全局收敛的.最后，对新算法进行数值试验，并与其它同类算法进行比对，其结果验证了该文所提出算法的有效性.

关键词: 无约束优化, 共轭梯度法, 强Wolfe线搜索, 全局收敛性

Abstract:

The conjugate gradient method is one of the most effective methods for solving large-scale unconstrained optimization. Combining the second inequality of the strong Wolfe line search, two new conjugate parameters are constructed. Under usual assumptions, it is proved that the improved PRP and HS conjugate gradient methods satisfy sufficient descent condition with the greater range of parameter in the strong Wolfe line search and converge globally for unconstrained optimization. Finally, two group numerical experiments for the proposed methods and their comparisons are tested, the numerical results and their corresponding performance files are reported, which show that the proposed methods are promising.

Key words: Unconstrained optimization, Conjugate gradient method, Strong Wolfe line search, Global convergence

中图分类号:

O221

马国栋. 强Wolfe线搜索下的修正PRP和HS共轭梯度法[J]. 数学物理学报, 2021, 41(3): 837-847.

Guodong Ma. Improved PRP and HS Conjugate Gradient Methods with the Strong Wolfe Line Search[J]. Acta mathematica scientia,Series A, 2021, 41(3): 837-847.

图/表 10

表 1

第一组方法数值试验报告"

序号	算例	IPRP	VPRP	WYL	IFR	FR
序号	算例名/维数	Itr/NF/NG/Tcpu/\|\|g_k\|\|	Itr/NF/NG/Tcpu/\|\|g_k\|\|	Itr/NF/NG/Tcpu/\|\|g_k\|\|	Itr/NF/NG/Tcpu/\|\|g_k\|\|	Itr/NF/NG/Tcpu/\|\|g_k\|\|
1	bdexp 10	3/1/3/0.005/2.27e-48	3/1/3/0.001/6.25e-49	3/1/3/0.000/6.25e-49	3/1/3/0.000/1.06e-48	F/F/F/F/F
2	bdexp 100	3/2/3/0.000/1.33e-82	3/2/3/0.000/1.24e-82	3/2/3/0.000/1.24e-82	3/2/3/0.000/1.23e-82	4/4/4/0.001/8.22e-10
3	bdexp 1000	3/2/3/0.001/4.45e-107	3/2/3/0.001/4.45e-107	3/2/3/0.002/4.45e-107	3/2/3/0.001/4.41e-107	3/2/3/0.001/3.25e-65
4	bdexp 10000	3/2/3/0.009/1.14e-109	3/2/3/0.009/1.14e-109	3/2/3/0.009/1.14e-109	3/2/3/0.009/1.13e-109	3/2/3/0.007/5.74e-104
5	bdexp 20000	3/2/3/0.019/1.07e-109	3/2/3/0.019/1.07e-109	3/2/3/0.018/1.07e-109	3/2/3/0.019/1.07e-109	3/2/3/0.022/8.65e-107
6	exdenschnb 6	18/333/142/0.027/5.87e-06	21/424/199/0.031/2.28e-06	F/F/F/F/F	92/2469/1261/0.120/7.46e-06	F/F/F/F/F
7	exdenschnb 8	18/333/142/0.016/6.77e-06	21/424/198/0.020/2.63e-06	F/F/F/F/F	81/2181/1132/0.103/9.46e-06	F/F/F/F/F
8	himmelbg 200	3/6/7/0.001/7.14e-29	3/6/7/0.001/7.12e-29	3/6/7/0.001/7.12e-29	3/6/7/0.001/7.13e-29	3/6/7/0.001/2.78e-27
9	himmelbg 1000	3/6/7/0.002/1.60e-28	3/6/7/0.001/1.59e-28	3/6/7/0.001/1.59e-28	3/6/7/0.001/1.59e-28	3/6/7/0.001/6.23e-27
10	himmelbg 2000	3/6/7/0.002/2.26e-28	3/6/7/0.002/2.25e-28	3/6/7/0.002/2.25e-28	3/6/7/0.002/2.26e-28	3/6/7/0.002/8.81e-27
11	himmelbg 5000	3/6/7/0.006/3.57e-28	3/6/7/0.004/3.56e-28	3/6/7/0.004/3.56e-28	3/6/7/0.005/3.57e-28	3/6/7/0.003/1.39e-26
12	genquartic 1000	26/527/272/0.051/8.90e-06	49/1193/591/0.087/1.72e-06	F/F/F/F/F	135/3820/1894/0.262/7.72e-06	F/F/F/F/F
13	genquartic 2000	24/465/195/0.041/3.68e-07	40/829/402/0.071/3.32e-07	F/F/F/F/F	63/1527/759/0.152/4.42e-06	80/1963/985/0.166/8.99e-06
14	genquartic 3000	28/533/256/0.074/9.56e-07	39/819/394/0.110/4.38e-06	130/3558/1746/0.438/2.90e-06	109/2817/1389/0.343/3.04e-06	60/14 36/699/0.198/1.29e-06
15	biggsb1 5	47/1099/508/0.060/3.03e-06	51/1260/580/0.058/3.30e-06	83/2167/1067/0.096/9.24e-06	101/2486/1235/0.114/3.20e-06	116/3006/1485/0.151/1.63e-06
16	biggsb1 10	72/1771/868/0.097/3.99e-06	110/3065/1354/0.176/6.47e-06	321/9146/4502/0.464/6.68e-06	165/4546/2275/0.249/7.11e-06	260/73 83/3699/0.394/7.49e-06
17	sinqua d 3	211/6309/2454/0.327/5.45e-06	141/3934/1661/0.167/8.82e-07	F/F/F/F/F	209/5635/2820/0.292/1.67e-06	F/F/F/F/F
18	fletcbv3 10	1/1/1/0.000/5.97e-06	1/1/1/0.000/5.97e-06	1/1/1/0.000/5.97e-06	1/1/1/0.000/5.97e-06	1/1/1/0.000/5.97e-06
19	fletcbv3 20	123/2561/1320/0.158/5.54e-06	207/3854/1963/0.230/9.67e-06	144/3777/1938/0.208/9.27e-06	144/3549/1786/0.173/9.18e-06	235/6092/3110/0.287/8.70e-06
20	nonscomp 50	69/1550/742/0.090/5.32e-06	350/10319/4581/0.511/8.98e-06	332/9490/4661/0.474/8.39e-06	F/F/F/F/F	F/F/F/F/F
21	dixmaana 1500	15/159/64/0.217/9.53e-07	16/263/115/0.249/2.16e-06	56/1370/656/1.247/7.33e-06	54/1387/654/1.258/6.30e-06	63/1542/738/1.411/8.05e-06
22	dixmaan b 1500	11/140/50/0.145/2.91e-06	11/140/50/0.120/2.90e-06	65/1613/757/1.566/6.91e-06	50/1278/623/1.139/5.80e-06	F/F/F/F/F
23	dixmaanc 1500	60/1667/793/1.578/7.39e-06	21/430/199/0.406/3.25e-06	F/F/F/F/F	141/4104/2008/3.680/1.69e-06	49/1260/611/1.156/8.49e-06
24	dixmaand 1500	63/1757/855/1.670/6.73e-06	F/F/F/F/F	59/1419/655/1.251/8.17e-06	75/1811/888/1.692/2.60e-06	54/1303/644/1.186/8.75e-06
25	dixon3dq 20	521/14809/6503/0.737/4.61e-06	403/11099/5010/0.493/9.85e-06	F/F/F/F/F	411/11043/5496/0.490/8.31e-06	F/F/F/F/F
26	dqdrtic 1000	89/2094/957/0.135/7.15e-06	123/3254/1469/0.195/7.20e-06	337/8603/4223/0.506/8.01e-06	145/3860/1933/0.236/7.33e-06	F/F/F/F/F
27	dqdrtic 3000	115/2569/1149/0.318/2.11e-06	130/3633/1604/0.394/8.77e-06	F/F/F/F/F	267/7418/3623/0.882/4.49e-06	348/9565/4759/1.167/5.93e-06
28	dqrtic 50	17/226/94/0.019/2.87e-06	23/340/152/0.025/9.97e-07	30/571/257/0.039/3.97e-06	37/718/334/0.047/3.17e-06	42/847/422/0.058/3.39e-06
29	dqrtic 100	23/292/117/0.024/9.22e-07	37/740/350/0.059/4.05e-06	39/834/412/0.069/4.43e-06	F/F/F/F/F	56/1250/607/0.100/2.99e-06
30	dqrtic 150	23/345/158/0.035/5.20e-06	27/473/221/0.047/7.25e-06	41/890/411/0.087/7.32e-06	F/F/F/F/F	55/1239/607/0.124/9.64e-06
31	edensch 100	38/805/372/0.075/6.94e-06	37/747/358/0.066/4.94e-06	F/F/F/F/F	F/F/F/F/F	F/F/F/F/F
32	edensch 200	39/837/402/0.099/7.72e-06	41/894/409/0.103/3.57e-06	586/17937/8632/2.164/6.27e-06	90/2221/1066/0.257/2.92e-06	203/5577/2766/0.698/6.38e-06
33	edensch 1000	53/1251/618/0.559/8.40e-06	57/1412/678/0.604/6.60e-06	212/6028/2941/2.500/9.89e-06	F/F/F/F/F	121/3187/1554/1.336/9.97e-06
34	fletchcr 10	92/2330/1063/0.122/4.66e-06	137/3824/1720/0.198/5.62e-06	181/4817/2375/0.239/9.63e-06	130/3342/1649/0.161/8.36e-06	F/F/F/F/F
35	fletchcr 100	63/1570/739/0.072/4.99e-06	126/3389/1561/0.153/6.47e-06	200/5116/2552/0.272/2.95e-06	136/3723/1839/0.194/8.70e-06	F/F/F/F/F
36	liarwhd 10	85/2053/931/0.126/4.38e-06	118/3128/1461/0.153/6.45e-06	67/1637/822/0.079/7.80e-06	143/3622/1726/0.174/1.86e-06	F/F/F/F/F
37	liarwhd 10	85/2053/931/0.102/4.38e-06	118/3128/1461/0.152/6.45e-06	67/1637/822/0.079/7.80e-06	143/3622/1726/0.175/1.86e-06	F/F/F/F/F
38	liarwhd 20	54/1201/548/0.059/2.51e-06	83/2005/946/0.096/3.67e-06	F/F/F/F/F	112/2678/1341/0.132/6.28e-06	F/F/F/F/F
39	penalt y1 1000	14/243/93/0.755/2.34e-06	14/243/93/0.753/2.34e-06	20/440/186/1.402/8.67e-07	14/243/93/0.702/2.34e-06	F/F/F/F/F
40	penalt y1 2000	10/129/39/1.282/5.53e-06	10/129/39/1.295/5.53e-06	19/421/183/4.409/1.67e-07	13/220/81/2.262/7.43e-07	14/233/82/2.331/1.04e-06
41	power1 30	390/11133/4828/0.479/7.51e-06	F/F/F/F/F	F/F/F/F/F	554/14994/7484/0.661/7.38e-06	F/F/F/F/F
42	power1 50	974/27796/12081/1.271/7.98e-06	F/F/F/F/F	F/F/F/F/F	844/22960/11507/1.063/7.28e-06	F/F/F/F/F
43	quartc 20	17/218/97/0.015/1.93e-06	25/436/214/0.024/3.39e-06	32/624/296/0.034/5.70e-06	21/369/172/0.022/9.11e-06	21/348/170/0.023/5.01e-06
44	quartc 100	23/292/117/0.034/9.22e-07	37/740/350/0.075/4.05e-06	39/834/412/0.068/4.43e-06	F/F/F/F/F	56/1250/607/0.099/2.99e-06
45	tridia 5	90/2208/1021/0.131/7.47e-06	91/2347/1098/0.111/5.04e-06	233/6288/3116/0.288/9.52e-06	132/3694/1822/0.167/9.45e-06	178/4808 /2347/0.224/8.08e-06
46	raydan2 1000	11/173/64/0.025/6.93e-06	11/173/59/0.015/6.97e-06	13/235/98/0.019/5.88e-06	11/173/64/0.015/6.93e-06	F/F/F/F/F
47	raydan2 5000	15/270/119/0.068/8.63e-07	12/204/71/0.052/8.98e-06	F/F/F/F/F	13/235/81/0.071/7.52e-06	F/F/F/F/F
48	raydan2 9000	13/235/89/0.119/3.57e-06	14/236/80/0.122/8.17e-07	F/F/F/F/F	12/204/73/0.105/9.68e-06	F/F/F/F/F
49	diagonal 1 12	105/3134/1528/0.152/6.47e-06	105/3134/1528/0.138/6.46e-06	178/5045/2412/0.254/8.59e-06	123/3267/1606/0.175/2.63e-06	154/4290/2074/0.188/9.47e-06
50	diagonal 2 20	110/3291/1615/0.167/7.74e-07	112/3352/1629/0.152/1.21e-06	177/4865/2381/0.234/8.10e-06	91/2436/1199/0.109/7.14e-06	158/4381/2169/0.195/6.73e-06
51	diagonal2 100	86/2332/1156/0.125/9.74e-06	138/3993/1881/0.226/5.03e-06	258/7007/3440/0.367/5.18e-06	142/3889/1927/0.228/4.61e-06	314/8540/4265/0.457/3.56e-06
52	diagonal 3 20	94/2642/1291/0.153/4.23e-06	92/2518/1239/0.173/6.97e-06	198/5426/2744/0.254/7.35e-06	136/3671/1808/0.169/8.58e-06	297/8635/4233/0.389/8.19e-06
53	diagonal 3 40	78/1848/864/0.088/8.04e-06	70/1662/798/0.082/2.75e-06	330/9184/4561/0.436/8.35e-06	F/F/F/F/F	166/4212/2153/0.221/7.77e-06
54	bv 1000	1/1/1/0.000/4.99e-06	1/1/1/0.000/4.99e-06	1/1/1/0.000/4.99e-06	1/1/1/0.000/4.99e-06	1/1/1/0.000/4.99e-06
55	bv 10000	1/1/1/0.000/5.00e-08	1/1/1/0.000/5.00e-08	1/1/1/0.000/5.00e-08	1/1/1/0.000/5.00e-08	1/1/1/0.000/5.00e-08
56	ie 50	12/193/77/0.255/6.88e-06	11/166/63/0.178/4.87e-06	42/1006/516/1.155/2.93e-06	30/598/294/0.676/7.83e-06	49/1249/599/1.258/9.32e-06
57	ie 200	14/225/98/3.642/3.11e-06	11/166/64/2.631/9.62e-06	58/1475/739/24.086/9.66e-06	42/961/446/15.469/6.32e-06	37/834/424/13.787/4.12e-06
58	gauss 3	8/142/65/0.018/3.85e-06	8/142/64/0.014/3.78e-06	35/912/446/0.082/2.70e-06	5/51/16/0.004/2.43e-06	27/699/363/0.055/3.86e-06
59	kowosb 4	403/11443/5023/0.678/5.35e-06	501/13830/6067/0.789/6.93e-06	F/F/F/F/F	241/6791/3346/0.387/8.22e-06	F/F/F/F/F
60	lin 500	2/2/2/0.021/9.93e-14	2/2/2/0.018/9.93e-14	2/2/2/0.020/9.93e-14	2/2/2/0.016/9.93e-14	2/2/2/0.016/9.93e-14
61	rosex 50	454/13379/5609/0.780/9.65e-06	518/14946/6508/1.015/8.36e-06	F/F/F/F/F	F/F/F/F/F	F/F/F/F/F
62	trid 20	101/2542/1164/0.206/4.07e-06	111/2785/1293/0.253/4.46e-06	717/20092/10002/1.691/9.73e-06	173/4652/2343/0.360/9.95e-06	449/12935/6383/1.034/9.91e-06
63	vardim 5	10/137/43/0.011/5.41e-07	10/137/43/0.008/5.41e-07	13/228/96/0.013/9.27e-07	10/137/43/0.008/5.41e-07	14/228/88/0.017/5.31e-07
64	watson 4	106/2780/1261/0.345/9.27e-06	190/5428/2430/0.585/9.92e-06	F/F/F/F/F	138/3688/1822/0.368/9.18e-06	F/F/F/F/F

表 1

表 2

第二组方法数值试验报告"

序号	算例	IHS	VHS	YWH	IDY	DY
序号	算例名/维数	Itr/NF/NG/Tcpu/\|\|g_k\|\|	Itr/NF/NG/Tcpu/\|\|g_k\|\|	Itr/NF/NG/Tcpu/\|\|g_k\|\|	Itr/NF/NG/Tcpu/\|\|g_k\|\|	Itr/NF/NG/Tcpu/\|\|g_k\|\|
1	bdexp 10	3/1/3/0.001/2.25e-48	3/1/3/0.001/5.63e-49	3/1/3/0.001/5.63e-49	3/1/3/0.001/9.95e-49	3/1/3/0.000/8.27e-54
2	bdexp 100	3/2/3/0.001/1.33e-82	3/2/3/0.001/1.24e-82	3/2/3/0.001/1.24e-82	3/2/3/0.001/1.22e-82	3/2/3/0.001/2.02e-83
3	bdexp 1000	3/2/3/0.002/4.45e-107	3/2/3/0.001/4.45e-107	3/2/3/0.001/4.45e-107	3/2/3/0.001/4.40e-107	3/2/3/0.001/3.49e-107
4	bdexp 10000	3/2/3/0.012/1.14e-109	3/2/3/0.009/1.14e-109	3/2/3/0.006/1.14e-109	3/2/3/0.005/1.13e-109	3/2/3/0.005/1.11e-109
5	bdexp 20000	3/2/3/0.015/1.07e-109	3/2/3/0.017/1.07e-109	3/2/3/0.019/1.07e-109	3/2/3/0.020/1.07e-109	3/2/3/0.018/1.06e-109
6	exdenschnb 6	27/575/270/0.031/6.48e-07	46/1205/572/0.060/5.91e-06	85/2160/1080/0.102/9.61e-06	F/F/F/F/F	45/1082/500/0.049/2.80e-06
7	exdenschnb 8	27/575/270/0.035/7.48e-07	46/1205/581/0.070/6.83e-06	82/2086/1040/0.099/9.94e-06	F/F/F/F/F	45/1082/521/0.050/3.23e-06
8	himmelbg 200	3/6/7/0.001/7.14e-29	3/6/7/0.001/7.12e-29	3/6/7/0.001/7.12e-29	3/6/7/0.001/7.13e-29	3/6/7/0.001/6.93e-29
9	himmelbg 1000	3/6/7/0.001/1.60e-28	3/6/7/0.001/1.59e-28	3/6/7/0.001/1.59e-28	3/6/7/0.001/1.59e-28	3/6/7/0.001/1.55e-28
10	himmelbg 2000	3/6/7/0.001/2.26e-28	3/6/7/0.001/2.25e-28	3/6/7/0.001/2.25e-28	3/6/7/0.001/2.25e-28	3/6/7/0.001/2.19e-28
11	himmelbg 5000	3/6/7/0.003/3.57e-28	3/6/7/0.003/3.56e-28	3/6/7/0.003/3.56e-28	3/6/7/0.004/3.57e-28	3/6/7/0.004/3.47e-28
12	genquartic 200	21/337/151/0.021/2.29e-06	38/872/413/0.058/5.45e-06	92/2398/1194/0.124/6.62e-06	63/1571/737/0.087/3.97e-06	821/19378 /9592/1.049/9.80e-06
13	genquartic 500	30/599/289/0.033/5.23e-06	36/712/351/0.047/4.93e-06	65/1501/760/0.112/1.08e-06	78/1989/962/0.114/9.02e-06	645/17776/8774/1.104/9.29e-06
14	genquartic 1000	26/497/224/0.034/2.70e-06	32/582/258/0.038/4.41e-06	149/4040/1993/0.282/6.24e-06	F/F/F/F/F	918/25177/12373/1.652/9.83e-06
15	biggsb1 10	92/2354/1072/0.137/8.90e-06	99/2590/1194/0.130/4.88e-06	93/2353/1157/0.118/6.10e-06	F/F/F/F/F	300/7576/3703/0.363/9.64e-06
16	biggsb1 20	223/6267/2797/0.328/4.70e-06	231/6441/2952/0.310/9.38e-06	141/3413/1740/0.158/9.96e-06	F/F/F/F/F	585/15571/7647/0.817/6.94e-06
17	fletcbv3 10	1/1/1/0.000/5.97e-06	1/1/1/0.000/5.97e-06	1/1/1/0.000/5.97e-06	1/1/1/0.000/5.97e-06	1/1/1/0.000/5.97e-06
18	nonscomp 50	71/1796/850/0.118/6.23e-06	49/988/455/0.047/4.92e-06	F/F/F/F/F	F/F/F/F/F	F/F/F/F/F
19	dixmaana 1500	15/177/73/0.187/8.55e-06	13/184/61/0.156/5.51e-06	F/F/F/F/F	18/267/110/0.237/2.08e-06	25/423/187/0.394/2.57e-06
20	dixmaan b 1500	11/140/44/0.147/2.91e-06	11/140/53/0.128/2.90e-06	18/316/132/0.297/9.49e-06	10/117/35/0.097/7.97e-06	13/172/69/0.155/5.31e-07
21	dixmaanc 1500	20/324/142/0.330/1.63e-06	24/400/169/0.369/4.89e-07	F/F/F/F/F	29/579/278/0.526/2.10e-06	F/F/F/F/F
22	dixmaand 1500	20/303/137/0.316/9.11e-06	18/297/129/0.272/7.43e-06	59/1312/656/1.246/7.34e-06	F/F/F/F/F	F/F/F/F/F
23	dixmaanf 1500	384/10668/4805/9.600/6.61e-06	328/8716/3884/7.877/6.86e-06	368/9331/4633/8.518/9.40e-06	F/F/F/F/F	569/13296/6208/12.037/9.45e-06
24	dixmaang 1500	462/12937/5721/11.405/6.91e-06	349/9644/4160/8.895/9.67e-06	251/6303/3170/5.904/7.43e-06	F/F/F/F/F	F/F/F/F/F
25	dixmaanh 1500	340/8992/4020/8.074/9.24e-06	443/12355/5798/11.331/6.07e-06	253/6592/3315/6.061/9.63e-06	F/F/F/F/F	545/12688/6055/11.547/9.72e-06
26	dqdrtic 500	112/2969/1315/0.169/4.71e-06	808/24158/10585/1.257/6.11e-06	234/5935/2961/0.308/8.73e-06	F/F/F/F/F	332/7838/3899/0.402/8.00e-06
27	dqdrtic 1000	448/12853/5663/0.829/6.15e-06	677/20184/8818/1.191/7.38e-06	195/5120/2524/0.303/6.75e-06	F/F/F/F/F	356/9315/4555/0.572/9.28e-06
28	dqdrtic 5000	211/5452/2496/0.871/9.23e-06	786/23433/10290/3.794/7.65e-06	329/9479/4559/1.520/7.36e-06	F/F/F/F/F	355/9103/4407/1.442/9.96e-06
29	edensch 200	40/923/459/0.120/3.76e-06	37/773/344/0.090/3.17e-06	99/2586/1330/0.303/2.67e-06	F/F/F/F/F	F/F/F/F/F
30	edensch 500	48/1121/538/0.261/5.01e-06	48/1121/507/0.250/9.24e-06	F/F/F/F/F	F/F/F/F/F	646/14842/7011/3.463/9.95e-06
31	fletchcr 20	96/2510/1184/0.133/3.12e-06	122/3213/1517/0.168/4.57e-06	100/2485/1271/0.143/5.84e-06	F/F/F/F/F	720/19851/9712/0.967/9.81e-06
32	fletchcr 50	108/2811/1292/0.143/9.45e-06	111/2937/1382/0.131/8.33e-06	70/1702/811/0.104/1.88e-06	F/F/F/F/F	353/9180/4634/0.447/8.43e-06
33	genrose 40000	394/11155/4844/14.998/1.15e-06	484/13780/6083/18.593/9.07e-06	617/16168/8104/22.158/6.56e-06	F/F/F/F/F	F/F/F/F/F
34	genrose 50000	239/6311/2783/10.298/6.13e-06	307/8594/3772/13.857/3.30e-06	726/18954/9375/33.155/4.74e-06	F/F/F/F/F	F/F/F/F/F
35	liarwhd 10	75/1884/846/0.104/7.13e-06	92/2380/1068/0.121/9.74e-06	81/1932/948/0.098/1.58e-06	F/F/F/F/F	169/4333/2143/0.236/7.07e-06
36	liarwhd 20	91/2350/1048/0.123/1.68e-06	147/3927/1783/0.204/4.36e-06	71/1791/870/0.101/1.15e-06	F/F/F/F/F	158/3925/1941/0.244/2.74e-06
37	penalt y1 1000	14/243/93/0.791/2.34e-06	14/243/93/0.758/2.34e-06	19/406/179/1.333/9.37e-07	19/406/177/1.367/9.37e-07	14/243/93/0.750/2.34e-06
38	penalt y1 2000	10/129/39/1.354/5.53e-06	10/129/39/1.301/5.53e-06	18/385/157/4.370/3.61e-08	18/385/157/4.268/3.61e-08	10/129/39/1.302/5.53e-06
39	penalt y1 5000	11/152/50/11.744/3.01e-06	11/152/50/11.720/3.01e-06	10/156/50/12.004/4.03e-06	10/156/50/11.701/4.03e-06	11/152/51/11.610/3.01e-06
40	quartc 20	12/131/50/0.012/1.48e-06	15/193/77/0.013/6.78e-07	25/465/214/0.027/7.65e-06	27/554/243/0.031/4.83e-06	42/907/438/0.049/8.77e-06
41	quartc 100	21/268/112/0.034/2.55e-06	22/298/107/0.042/5.43e-06	F/F/F/F/F	40/835/412/0.077/7.91e-06	F/F/F/F/F
42	tridia 10	93/2351/1061/0.138/6.02e-06	160/4450/1973/0.215/5.81e-06	143/3824/1885/0.193/6.63e-06	F/F/F/F/F	509/13202/6489/0.659/6.52e-06
43	tridia 30	243/6517/2942/0.339/7.01e-06	244/6428/2942/0.362/8.75e-06	229/6007/2989/0.310/9.35e-06	F/F/F/F/F	F/F/F/F/F
44	raydan 1 50	54/1229/560/0.067/4.85e-06	65/1361/679/0.071/1.69e-06	91/2358/1158/0.114/5.16e-06	F/F/F/F/F	685/15740/7582/0.776/9.90e-06
45	raydan 1 80	69/1547/712/0.072/8.33e-06	98/2546/1222/0.123/8.71e-06	107/2819/1366/0.139/2.74e-06	F/F/F/F/F	602/13450/6380/0.789/9.59e-06
46	raydan2 1000	11/173/64/0.018/6.93e-06	11/173/66/0.014/6.95e-06	11/173/65/0.014/6.93e-06	11/173/65/0.014/6.95e-06	11/173/65/0.027/6.95e-06
47	raydan2 4000	12/204/79/0.055/9.01e-06	13/235/109/0.062/4.97e-06	15/268/96/0.063/4.95e-07	16/301/98/0.067/1.75e-06	15/271/92/0.079/7.57e-07
48	raydan2 10000	13/208/79/0.135/6.37e-09	14/236/90/0.139/2.02e-06	14/238/92/0.155/5.95e-07	15/272/115/0.157/1.62e-06	12/204/72/0.126/8.83e-06
49	diagonal 1 50	156/4632/2239/0.252/5.98e-06	168/5006/2447/0.314/9.21e-06	106/2632/1279/0.134/5.76e-06	F/F/F/F/F	F/F/F/F/F
50	diagonal2 200	168/4488/2112/0.294/6.79e-06	232/6717/3155/0.423/5.53e-06	156/3915/1904/0.264/3.42e-06	F/F/F/F/F	901/23693/11745/1.642/9.54e-06
51	diagonal2 800	468/13834/6665/1.469/9.51e-06	460/13514/6589/1.395/4.10e-06	297/8009/3930/0.846/7.15e-06	F/F/F/F/F	F/F/F/F/F
52	diagonal 3 5	82/2438/1160/0.146/4.57e-06	82/2438/1151/0.130/4.58e-06	90/2387/1194/0.115/7.49e-06	74/1924/927/0.106/6.19e-06	322/8250 /4083/0.460/9.78e-06
53	diagonal 3 20	107/3017/1401/0.150/8.29e-06	117/3398/1613/0.174/7.60e-06	91/2344/1142/0.131/8.12e-06	F/F/F/F/F	542/13718/6653/0.701/9.80e-06
54	bv 1000	1/1/1/0.000/4.99e-06	1/1/1/0.000/4.99e-06	1/1/1/0.000/4.99e-06	1/1/1/0.000/4.99e-06	1/1/1/0.000/4.99e-06
55	bv 10000	1/1/1/0.000/5.00e-08	1/1/1/0.000/5.00e-08	1/1/1/0.000/5.00e-08	1/1/1/0.000/5.00e-08	1/1/1/0.000/5.00e-08
56	ie 100	12/193/85/0.894/9.38e-06	11/166/75/0.729/2.85e-06	70/1873/917/8.274/4.81e-06	F/F/F/F/F	71/1838/904/7.827/9.92e-06
57	ie 200	14/225/89/3.598/2.89e-06	11/166/67/2.651/4.02e-06	102/2876/1443/48.512/3.88e-06	15/207/104/3.495/2.82e-06	62/1551/711/25.058/5.32e-06
58	singx 10	174/4588/2004/0.307/2.52e-06	618/18316/7284/1.186/8.99e-06	334/8361/4166/0.620/7.59e-06	F/F/F/F/F	731/19522/9716/1.219/9.55e-06
59	singx 150	175/4802/2117/0.758/6.53e-06	720/21252/8241/3.513/9.69e-06	601/15530/7775/2.721/3.25e-06	F/F/F/F/F	228/5911/2866/1.009/4.27e-06
60	beale 2	134/3736/1642/0.201/7.28e-06	203/5839/2717/0.305/3.43e-06	F/F/F/F/F	F/F/F/F/F	F/F/F/F/F
61	froth 2	933/26993/11996/1.634/9.76e-06	F/F/F/F/F	F/F/F/F/F	F/F/F/F/F	210/6330/3026/0.396/9.06e-06
62	lin 100	2/2/2/0.007/3.13e-14	2/2/2/0.004/3.13e-14	2/2/2/0.003/3.13e-14	2/2/2/0.002/3.13e-14	2/2/2/0.002/3.13e-14
63	lin 500	2/2/2/0.027/9.93e-14	2/2/2/0.019/9.93e-14	2/2/2/0.019/9.93e-14	2/2/2/0.016/9.93e-14	2/2/2/0.024/9.93e-14
64	trid 120	963/28247/12176/6.211/6.81e-06	F/F/F/F/F	461/11845/5899/2.515/6.45e-06	F/F/F/F/F	F/F/F/F/F

表 2

图 1

图 2

图 3

图 4

图 5

图 6

图 7

图 8

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