| ODE Taylor methods | |||||||||||||
| Example: y' = cos(t) - sin(y) + t^2, with y(-1) = 3, t in [-1,1] | |||||||||||||
| Try various degree methods, with h = | 0.1 | ||||||||||||
| n=1 (Euler) | n=2 | n=3 | |||||||||||
| i | ti | wi | y'(ti,wi) | w(i+1) | wi | y'(ti,wi) | y''(ti,wi) | w(i+1) | wi | y'(ti,wi) | y''(ti,wi) | y'''(ti,wi) | w(i+1) |
| 0 | -1 | 3.00000000000000 | 1.39918229780827 | 3.13991822978083 | 3.0000000000 | 1.39918229780827 | 0.226650961 | 3.141051485 | 3 | 1.39918229780827 | 0.226650961 | 1.96035265 | 3.14137821 |
| 1 | -0.9 | 3.13991822978083 | 1.42993554524413 | 3.28291178430524 | 3.141051485 | 1.43106879929318 | 0.414395499 | 3.286230342 | 3.14137821 | 1.43139552471032 | 0.414722401 | 1.7935518 | 3.2868903 |
| 2 | -0.8 | 3.28291178430524 | 1.47755592653823 | 3.43066737695906 | 3.286230342 | 1.48084062013585 | 0.582734058 | 3.437228074 | 3.2868903 | 1.48149365534085 | 0.583239032 | 1.56260486 | 3.43821629 |
| 3 | -0.7 | 3.43066737695906 | 1.53990764944418 | 3.58465814190348 | 3.437228074 | 1.54618995058404 | 0.723329599 | 3.595463717 | 3.438216295 | 1.54713515673784 | 0.723787633 | 1.22769639 | 3.59675336 |
| 4 | -0.6 | 3.58465814190348 | 1.61404657996063 | 3.74606279989954 | 3.595463717 | 1.62378356947493 | 0.824028642 | 3.761962218 | 3.596753365 | 1.62494228342678 | 0.824150017 | 0.75415411 | 3.76349404 |
| 5 | -0.5 | 3.74606279989954 | 1.69590875255946 | 3.91565367515549 | 3.761962218 | 1.70891846286969 | 0.869910409 | 3.937203616 | 3.763494035 | 1.71016416483201 | 0.869399457 | 0.12518719 | 3.93887831 |
| 6 | -0.4 | 3.91565367515549 | 1.78010593504109 | 4.09366426865960 | 3.937203616 | 1.79535231324858 | 0.845892951 | 4.120968312 | 3.938878314 | 1.79652334575084 | 0.844561693 | -0.6401589 | 4.12264676 |
| 7 | -0.3 | 4.09366426865960 | 1.85995527108516 | 4.27965979576811 | 4.120968312 | 1.87548592539886 | 0.741180415 | 4.312222807 | 4.122646764 | 1.87642056110664 | 0.739085501 | -1.4705053 | 4.31373916 |
| 8 | -0.2 | 4.27965979576811 | 1.92789122635714 | 4.47244891840383 | 4.312222807 | 1.94106284799067 | 0.554851889 | 4.509103351 | 4.313739163 | 1.94165251798375 | 0.552369107 | -2.2400434 | 4.51029292 |
| 9 | -0.1 | 4.47244891840383 | 1.97635638580982 | 4.67008455698481 | 4.509103351 | 1.98441270038079 | 0.300463299 | 4.709046937 | 4.51029292 | 1.98465216750156 | 0.298199095 | -2.7938296 | 4.70978349 |
| 10 | -0 | 4.67008455698481 | 1.99910530132653 | 4.86999508711746 | 4.709046937 | 1.99999441537931 | 0.006684055 | 4.909079799 | 4.709783494 | 1.99999660572135 | 0.005210959 | -2.9999593 | 4.90930922 |
| 11 | 0.1 | 4.86999508711746 | 1.99261001030887 | 5.06925608814835 | 4.909079799 | 1.98572280854905 | -0.287893358 | 5.106212613 | 4.909309216 | 1.98567794898342 | -0.288331345 | -2.8053074 | 5.1059678 |
| 12 | 0.2 | 5.06925608814835 | 1.95706244400795 | 5.26496233254914 | 5.106212613 | 1.94351517918313 | -0.544439134 | 5.297841935 | 5.105967803 | 1.94360909070565 | -0.544035756 | -2.2602203 | 5.29723183 |
| 13 | 0.3 | 5.26496233254914 | 1.89651313159035 | 5.45461364570818 | 5.297841935 | 1.87879836798594 | -0.733700897 | 5.482053268 | 5.29723183 | 1.87913534325364 | -0.732931368 | -1.4949768 | 5.48123154 |
| 14 | 0.4 | 5.45461364570818 | 1.81802766164493 | 5.63641641187267 | 5.482053268 | 1.79920532453641 | -0.84147487 | 5.657766426 | 5.481231544 | 1.79977691457423 | -0.840810136 | -0.6645049 | 5.65689443 |
| 15 | 0.5 | 5.63641641187267 | 1.73019358292288 | 5.80943577016496 | 5.657766426 | 1.71301948078258 | -0.868201201 | 5.824727368 | 5.656894434 | 1.71372619734908 | -0.86789877 | 0.10417491 | 5.82394492 |
| 16 | 0.6 | 5.80943577016496 | 1.64156167718655 | 5.97359193788361 | 5.824727368 | 1.62790142718001 | -0.824440331 | 5.983395309 | 5.823944923 | 1.62860293850239 | -0.824504996 | 0.73804299 | 5.9828057 |
| 17 | 0.7 | 5.97359193788361 | 1.55951355044811 | 6.12954329292843 | 5.983395309 | 1.55016176502362 | -0.725239955 | 6.134785286 | 5.982805699 | 1.55072502647523 | -0.725507819 | 1.21665677 | 6.13445344 |
| 18 | 0.8 | 6.12954329292843 | 1.48974496112411 | 6.27851778904084 | 6.134785286 | 1.48456263870329 | -0.585601768 | 6.280313541 | 6.134453438 | 1.48489083054314 | -0.585853415 | 1.55594748 | 6.28027258 |
| 19 | 0.9 | 6.27851778904084 | 1.43627746946189 | 6.42214553598702 | 6.280313541 | 1.43448173074232 | -0.417802725 | 6.4216727 | 6.280272579 | 1.43452269239237 | -0.417843517 | 1.79023781 | 6.421934 |
| 20 | 1 | 6.42214553598702 | 6.4216727 | 6.421934004 | |||||||||