1 Introduction and preliminaries
2 Strong convergence theorems
2.1 Ergodic convergence of the first iterative algorithm
2.2 Ergodic convergence of the second iterative algorithm
3 Examples and numerical experiments
n
|
\(\boldsymbol {y_{n}}\)
|
\(\boldsymbol {u_{n}}\)
|
\(\boldsymbol {x_{n}}\)
|
\(\boldsymbol {z_{n}}\)
|
---|---|---|---|---|
0 | −4.0000000000 | −1.333333333333333 | −4.0000000000000 | |
1 | −0.8477427 | −0.635807052254677 | −1.0952380952381 | −1.0952380952381 |
2 | −0.408616 | −0.348080327113469 | −0.564535296490403 | −0.829886695864249 |
3 | −0.2269676 | −0.203325144325693 | −0.321650478060882 | −0.660474623263127 |
4 | −0.1335324 | −0.122849839925766 | −0.191673798999402 | −0.543274417197196 |
5 | −0.08100538 | −0.075755030882579 | −0.117198947637147 | −0.458059323285186 |
6 | −0.05007082 | −0.047345876009489 | −0.072845261243154 | −0.393856979611514 |
7 | −0.03134004 | −0.029870972328354 | −0.045785050770546 | −0.344132418348518 |
8 | −0.01979206 | −0.018977571393789 | −0.029008549436138 | −0.304741934734471 |
9 | −0.01258285 | −0.012121482143799 | −0.018490222101272 | −0.272936188886338 |
10 | −0.008041086 | −0.007775264661297 | −0.011841190590053 | −0.246826689056709 |
11 | −0.005159962 | −0.005004685808672 | −0.007611755159024 | −0.225079876884192 |
12 | −0.003322376 | −0.003230633801933 | −0.004908180655962 | −0.20673223553184 |
13 | −0.002145251 | −0.002090525234888 | −0.003173114874022 | −0.191073841635085 |
14 | −0.001388506 | −0.001355593524252 | −0.002055956135385 | −0.177572564099392 |
15 | −0.000900557 | −0.000880623111319 | −0.001334667828875 | −0.165823371014691 |
16 | −0.000585129 | −0.000572980669566 | −0.000867876867878 | −0.155513652630515 |
17 | −0.000380779 | −0.000373335337785 | −0.000565174013077 | −0.146399036241254 |
18 | −0.000248140 | −0.000243557044559 | −0.000368530757053 | −0.13828623038102 |
19 | −0.000619034 | −0.000159070048612 | −0.000240587259597 | −0.131020670216735 |
20 | −0.000105754 | −0.000103995450936 | −0.000157227239453 | −0.124477498067871 |
21 | −0.0000691467 | −0.000068051443395 | −0.000102847940106 | −0.118554895680834 |
22 | −0.0000452522 | −0.000044567863803 | −0.000067334619358 | −0.113169097450767 |
23 | −0.0000296392 | −0.0000292103517604 | −0.000044118782606 | −0.108250620117369 |
24 | −0.0000194276 | −0.000019158216414 | −0.000028928262428 | −0.103741382956746 |
25 | −0.0000127431 | −0.000012573398811 | −0.000018980591458 | −0.099592486862135 |
26 | −0.0000083638 | −0.000008256737423 | −0.000012461301835 | −0.095762485879046 |
27 | −0.0000054928 | −0.000005425047229 | −0.000008185847743 | −0.092216030322332 |
28 | −0.0000036092 | −0.000003566325208 | −0.000005380130737 | −0.088922792815489 |
29 | −0.0000023728 | −0.000002345554771 | −0.000003537814776 | −0.085856611608568 |
30 | −0.0000015607 | −0.000001543347773 | −0.000002327427839 | −0.082994802135877 |
31 | 0.0000000000 | 0.000000000000000 | −0.000001531804738 | −0.080317599867130 |
n
|
\(\boldsymbol {x_{n}}\)
|
\(\boldsymbol {u_{n}}\)
|
\(\boldsymbol {z_{n}}\)
|
---|---|---|---|
0 | 8.0000000000000 | 2.6666666666667 | |
1 | 0.0907029478458 | 0.0680272108844 | 0.09070295 |
2 | 0.0199727386316 | 0.0170138143899 | 0.05533785 |
3 | 0.0068807569080 | 0.0061640113968 | 0.03918548 |
4 | 0.0028754047605 | 0.0026453723797 | 0.03010796 |
5 | 0.0013373257810 | 0.0012506472582 | 0.02435383 |
6 | 0.0006653502921 | 0.0006291407524 | 0.02040576 |
7 | 0.0003466462534 | 0.0003303972103 | 0.01754017 |
8 | 0.0001867447646 | 0.0001790597949 | 0.01537099 |
9 | 0.0001031871958 | 0.0000994036653 | 0.01367457 |
10 | 0.0000581631893 | 0.0000562404392 | 0.01231293 |
11 | 0.0000333151231 | 0.0000323125847 | 0.01119660 |
12 | 0.0000193365400 | 0.0000188025922 | 0.01026516 |
13 | 0.0000113483809 | 0.0000110588814 | 0.00947641 |
14 | 0.0000067234088 | 0.0000065640391 | 0.00879999 |
15 | 0.0000040158582 | 0.0000039269655 | 0.00821360 |
16 | 0.0000024157047 | 0.0000023655516 | 0.00770040 |
17 | 0.0000014622202 | 0.0000014336377 | 0.00724752 |
18 | 0.0000008899717 | 0.0000008735364 | 0.00684493 |
19 | 0.0000005443463 | 0.0000005348202 | 0.00648470 |
20 | 0.0000003344191 | 0.0000003288581 | 0.00616048 |