So I solved a question using Newton's Method for systems of equations. Then they asked:
How can you ensure that Newton's method converges as it should? What convergence rate do you observe?
My idea was to check $|x_{n+1} - x_{n}|$ at every iteration and see that the error is decreasing quadratically. I am using the same equations for 3 different scenarios.
First one it seems like it is decreasing by half until around iteration 4, and after that it kind of looks like it is decreasing quadratically.
Second one, decreases by half until around iteration 8 and after it doesn't seem to have any pattern.
Third one decreases by half until aorund iteration 10 then decreases kinda quadratically.
My question is why? Why is the second and third not decreasing quadratically from the get go? Also why is the second not behaving like Newton's method. Is it because ther might be double roots?
The print is as follows:
First
Iter__________x___________y_____________$|\vec{x}_{n+1} - \vec{x}_{n}|$
1 -93.2691075514873 925.137299771167 262.652751611591
2 32.2982202134813 957.6116086759 129.698629680021
3 92.0628220739525 973.067971226022 61.7309223875802
4 116.816890313347 979.469885425866 25.5685040591496
5 123.281762471589 981.141835121963 6.67757349706366
6 123.792362353483 981.273886815556 0.527399174413201
7 123.795587686334 981.274720953362 0.00333144981658602
8 123.79558781504 981.274720986648 1.32939971766592e-07
9 123.79558781504 981.274720986648 4.98866793196928e-14
Second
1 -811.324797843665 2089.42580862534 1257.96088578934
2 -207.971102392269 2264.39838030624 628.212609436874
3 92.2371291895562 2351.45876746497 312.577179783993
4 239.453474253177 2394.15150753342 153.281839460659
5 307.65763474734 2413.93071407673 71.0142557533089
6 333.308186188558 2421.36937399468 26.7073856941016
7 338.367321823104 2422.8365233287 5.26757824213792
8 338.580730096101 2422.89841172787 0.222200956198675
9 338.581111187739 2422.89852224444 0.000396793081132822
10 338.581111188954 2422.8985222448 1.26530035414541e-09
11 338.581111188954 2422.8985222448 1.62967534042146e-13
Third
1 -12738.1250000001 -6768.71250000003 15878.8624410971
2 -6030.76538636056 -2520.71807802813 7939.40360454024
3 -2677.1322400338 -396.750418688086 3969.64657092593
4 -1000.40898331252 665.174310568713 1984.71282815351
5 -162.233951488883 1196.01849739038 992.135542481428
6 256.480661911184 1461.20441921042 495.62637198429
7 465.094481230214 1593.32650477914 246.933130838899
8 567.932740426527 1658.45740227013 121.728145317798
9 616.554533055372 1689.2512042684 57.552905747243
10 636.211610445708 1701.70068661561 23.2677953885447
11 640.984825100492 1704.72372256364 5.64998446756525
12 641.303898483309 1704.92580237276 0.377682502739795
13 641.305337111327 1704.92671350384 0.00170288297146311
14 641.305337140574 1704.92671352236 3.4619367114589e-08
15 641.305337140574 1704.92671352236 3.29609026243582e-14