1 Introduction
2 The generalized Newton method
3 The analysis of convergence
4 Numerical experiments
The performance of numerical results
|
The solution pairs
\(\boldsymbol {(u^{*},w^{*})}\)
| ||
---|---|---|---|
n = 3 | It | 3 |
u∗ = (0.3571,0.4286,0.3571) |
CPU | 0.0068 | ||
RES | 0 |
w∗ = (0,0,0) | |
n = 5 | It | 3 |
u∗ = (0.3654,0.4615,0.4808,0.4615,0.3654) |
CPU | 0.0073 | ||
RES | 0 |
w∗ = 1.0e − 015∗(0.4441,0,0,0,0) | |
n = 8 | It | 3 |
\(\begin{array}{lcl}u*&=&(0.3660,0.4641,0.4902,0.4967,\\ &&{}0.4967,0.4902,0.4641,0.3660) \end{array}\)
|
CPU | 0.0074 | ||
RES | 0 |
w∗ = (0,0,0,0,0,0,0,0) | |
n = 10 | It | 3 |
\(\begin{array}{lcl}u*&=&(0.3660,0.4641,0.4904,0.4974,0.4991,\\ &&{}0.4991,0.4974,0.4904,0.4641,0.3660) \end{array}\)
|
CPU | 0.0075 | ||
RES | 0 |
\(\begin{array}{lcl} w*&=&1.0e\!-\!015*(-0.1110,0.4441,-0.4441,\\ &&{}0,0,0,0.2220,0,0,0) \end{array}\)
|
The performance of numerical results
|
The solution pairs
\(\boldsymbol {(u^{*},w^{*})}\)
| ||
---|---|---|---|
n = 3 | It | 4 |
u∗ = (0.3571,0.4286,0.3571) |
CPU | 0.0104 | ||
RES | 0 |
w∗ = 1.0e − 011∗(0.8880,−0.5869,0.8880) | |
n = 5 | It | 4 |
u∗ = (0.3654,0.4615,0.4808,0.4615,0.3654) |
CPU | 0.0106 | ||
RES | 6.1063e − 016 |
w∗ = 1.0e − 015∗(0,0,0,0,−0.1110) | |
n = 8 | It | 4 |
\(\begin{array}{lcl} u*&=&(0.3660,0.4641,0.4902,0.4967,0.4967,\\ &&{}0.4902,0.4641,0.3660) \end{array}\)
|
CPU | 0.0113 | ||
RES | 6.2042e − 016 |
\(\begin{array}{lcl} w*&=&1.0e\!-\!011*(0.4839,-0.1609,-0.0001,0,0,\\ &&{}-0.0001,-0.1608,0.4839) \end{array}\)
| |
n = 10 | It | 3 |
\(\begin{array}{lcl} u*&=&(0.3660,0.4641,0.4904,0.4974, 0.4991,\\ &&{}0.4991,0.4974,0.4904,0.4641,0.3660) \end{array}\)
|
CPU | 0.0115 | ||
RES | 6.2232e − 016 |
\(\begin{array}{lcl} w*&=&1.0e\!-\!011*(0.4835,-0.1607,-0.0001,0,0,\\ &&{}0,0,-0.0001,-0.1607,0.4835) \end{array}\)
|
Initials
|
\(\boldsymbol {z^{0}_{(1)}}\)
|
\(\boldsymbol {z^{0}_{(1)}}\)
|
\(\boldsymbol {z^{0}_{(2)}}\)
|
\(\boldsymbol {z^{0}_{(2)}}\)
| |
---|---|---|---|---|---|
Methods
|
GNIM
|
CBSN
|
GNIM
|
CBSN
| |
n = 50 | It | 2 | 3 | 3 | 4 |
CPU | 0.0084 | 0.0250 | 0.0119 | 0.0447 | |
RES | 5.7787e − 016 | 1.7554e − 016 | 5.3787e − 016 | 1.8619e − 016 | |
n = 100 | It | 2 | 3 | 3 | 4 |
CPU | 0.0135 | 0.0230 | 0.0142 | 0.0364 | |
RES | 6.1815e − 016 | 2.1678e − 016 | 5.6610e − 016 | 2.7616e − 016 | |
n = 300 | It | 2 | 3 | 3 | 4 |
CPU | 0.0376 | 0.0547 | 0.0562 | 0.0905 | |
RES | 6.2315e − 016 | 2.2648e − 016 | 5.7610e − 016 | 2.9616e − 016 |
Initials
|
\(\boldsymbol {z^{0}_{(1)}}\)
|
\(\boldsymbol {z^{0}_{(1)}}\)
|
\(\boldsymbol {z^{0}_{(2)}}\)
|
\(\boldsymbol {z^{0}_{(2)}}\)
| |
---|---|---|---|---|---|
Methods
|
GNIM
|
FBSN
|
GNIM
|
FBSN
| |
n = 500 | It | 2 | 11 | 3 | 11 |
CPU | 0.0886 | 0.3229 | 0.1366 | 0.3831 | |
RES | 6.1815e − 016 | 6.6613e − 016 | 5.6610e − 016 | 1.1047e − 015 | |
n = 1000 | It | 2 | 11 | 3 | 11 |
CPU | 0.5114 | 1.0758 | 0.7728 | 1.3069 | |
RES | 6.2113e − 016 | 6.6723e − 016 | 5.6720e − 016 | 1.3027e − 015 | |
n = 3000 | It | 2 | 11 | 3 | 11 |
CPU | 9.1418 | 26.6155 | 11.1484 | 11.5429 | |
RES | 6.2812e − 016 | 6.6921e − 016 | 5.5610e − 016 | 1.2123e − 015 | |
n = 5000 | It | 2 | 11 | 3 | 11 |
CPU | 42.9507 | 71.3675 | 66.0831 | 79.3341 | |
RES | 6.4615e − 016 | 6.6990e − 016 | 5.7810e − 016 | 1.3056e − 015 |
The performance of numerical results
|
The solution pairs
\(\boldsymbol {(u^{*},w^{*})}\)
| ||
---|---|---|---|
n = 3 | It | 2 |
u∗ = (3,1.5,1) |
CPU | 0.0067 | ||
RES | 0 |
w∗ = (0,0,0) | |
n = 5 | It | 2 |
u∗ = (5,2.5,1.6667,1.25,1) |
CPU | 0.0070 | ||
RES | 0 |
w∗ = 1.0e − 015∗(0.2220,0,0,0,0) | |
n = 8 | It | 2 |
u∗ = (8,4,2.6667,2,1.6,1.3333,1.1429,1) |
CPU | 0.0072 | ||
RES | 0 |
w∗ = (0,0,0,0,0,0,0,0) | |
n = 10 | It | 2 |
\(\begin{array}{lcl} u*&=&(10,5,3.3333,2.5,2,1.6667,1.4286,\\ &&{}1.25,1.1111,1) \end{array}\)
|
CPU | 0.0075 | ||
RES | 0 |
\(\begin{array}{lcl}w*&=&1.0e\!-\!015*(0,0.2220,-0.1110,0,0,\\ &&{}0,0.2220,0,0,0) \end{array}\)
|
The performance of numerical results
|
The solution pairs
\(\boldsymbol {(u^{*},w^{*})}\)
| ||
---|---|---|---|
n = 3 | It | 4 |
u∗ = (3,1.5,1) |
CPU | 0.0155 | ||
RES | 0 |
w∗ = (0,0,0) | |
n = 5 | It | 4 |
u∗ = (5,2.5,1.6667,1.25,1) |
CPU | 0.0161 | ||
RES | 0 |
w∗ = 1.0e − 015∗(0.4441,0,0,0,0) | |
n = 8 | It | 4 |
u∗ = (8,4,2.6667,2,1.6,1.3333,1.1429,1) |
CPU | 0.0163 | ||
RES | 2.2204e − 016 |
\(\begin{array}{lcl} w*&=&1.0e\!-\!011*(-0.1110,0.2220,0,0,\\ &&{}0.2220,0,0,0) \end{array}\)
| |
n = 10 | It | 4 |
\(\begin{array}{lcl} u*&=&(10,5,3.3333,2.5,2,1.6667,\\ &&{}1.4286,1.25,1.1111,1) \end{array}\)
|
CPU | 0.0147 | ||
RES | 8.8818e − 016 |
\(\begin{array}{lcl} w*&=&1.0e\!-\!014*(-0.1221,0.0444,-0.0111,\\ &&{}0.0222,0,-0.0111,0.0222,0,0,0) \end{array}\)
|
Initials
|
\(\boldsymbol {z^{0}_{(1)}}\)
|
\(\boldsymbol {z^{0}_{(1)}}\)
|
\(\boldsymbol {z^{0}_{(2)}}\)
|
\(\boldsymbol {z^{0}_{(2)}}\)
| |
---|---|---|---|---|---|
Methods
|
GNIM
|
FBSN
|
GNIM
|
FBSN
| |
n = 500 | It | 2 | 11 | 3 | 11 |
CPU | 0.0957 | 0.3025 | 0.1421 | 0.3308 | |
RES | 2.3182e − 015 | 1.7073e + 001 | 2.1006e − 015 | 1.8073e + 001 | |
n = 1000 | It | 2 | 11 | 3 | 11 |
CPU | 0.5016 | 0.9461 | 0.6327 | 1.1754 | |
RES | 6.6438e − 014 | 2.4129e + 001 | 6.6351e − 014 | 2.6249e + 001 |