| 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 = | 0.05 | ||||||||||||
| 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 | 3.06995911489041 | 3.0000000000 | 1.39918229780827 | 0.226650961 | 3.070242429 | 3 | 1.39918229780827 | 0.226650961 | 1.96035265 | 3.07028327 |
| 1 | -0.95 | 3.06995911489041 | 1.41261079800698 | 3.14058965479076 | 3.070242429 | 1.41289338799484 | 0.322714001 | 3.14129049 | 3.070283269 | 1.41293412482163 | 0.322758747 | 1.8824956 | 3.14137264 |
| 2 | -0.9 | 3.14058965479076 | 1.43060696963980 | 3.21212000327275 | 3.14129049 | 1.43130780517796 | 0.414634649 | 3.213374174 | 3.141372643 | 1.43138995728722 | 0.414716832 | 1.79355763 | 3.2134979 |
| 3 | -0.85 | 3.21212000327275 | 1.45295204167638 | 3.28476760535657 | 3.213374174 | 1.45420303881989 | 0.501738599 | 3.286711499 | 3.213497902 | 1.45432644787723 | 0.501848774 | 1.68861553 | 3.28687672 |
| 4 | -0.8 | 3.28476760535657 | 1.47939300341723 | 3.35873725552743 | 3.286711499 | 1.48131673652226 | 0.583102302 | 3.361506214 | 3.286876715 | 1.48148021381006 | 0.583228646 | 1.562631 | 3.36171232 |
| 5 | -0.75 | 3.35873725552743 | 1.50963103041046 | 3.43421880704796 | 3.361506214 | 1.51233413492185 | 0.657550308 | 3.437944859 | 3.361712316 | 1.51253526907642 | 0.657678562 | 1.41059598 | 3.43819057 |
| 6 | -0.7 | 3.43421880704796 | 1.54330991890397 | 3.51138430299316 | 3.437944859 | 1.54687556384921 | 0.723662049 | 3.516193214 | 3.438190566 | 1.54711055114642 | 0.723775736 | 1.2277716 | 3.51647639 |
| 7 | -0.65 | 3.51138430299316 | 1.58000497170785 | 3.59038455157855 | 3.516193214 | 1.58448461715978 | 0.779793117 | 3.596392187 | 3.516476391 | 1.58474814224082 | 0.779874105 | 1.01003331 | 3.59670968 |
| 8 | -0.6 | 3.59038455157855 | 1.61921299991182 | 3.67134520157414 | 3.596392187 | 1.62461784819777 | 0.824116468 | 3.678653225 | 3.596709683 | 1.62490304905406 | 0.824145978 | 0.75432595 | 3.67900073 |
| 9 | -0.55 | 3.67134520157414 | 1.66034434444028 | 3.75436241879615 | 3.678653225 | 1.66663712244350 | 0.854688911 | 3.763053442 | 3.679000733 | 1.66693567683014 | 0.854648983 | 0.45921727 | 3.7634254 |
| 10 | -0.5 | 3.75436241879615 | 1.70271805424603 | 3.83949832150845 | 3.763053442 | 1.70980600616403 | 0.869547097 | 3.849630676 | 3.763425396 | 1.71010837498281 | 0.869422494 | 0.125515 | 3.85002021 |
| 11 | -0.45 | 3.83949832150845 | 1.74556154438943 | 3.92677639872792 | 3.849630676 | 1.75329173077619 | 0.866834835 | 3.938378806 | 3.850020207 | 1.75358758472145 | 0.866615241 | -0.2431163 | 3.93877779 |
| 12 | -0.4 | 3.92677639872792 | 1.78801614233074 | 4.01617720584446 | 3.938378806 | 1.79617427494061 | 0.844959572 | 4.029243719 | 3.938777791 | 1.79645311166768 | 0.844641819 | -0.6396349 | 4.02964292 |
| 13 | -0.35 | 4.01617720584446 | 1.82914984815379 | 4.10763469825215 | 4.029243719 | 1.83746387051526 | 0.802770378 | 4.122120376 | 4.029642923 | 1.83771580016668 | 0.802360321 | -1.0533196 | 4.12250972 |
| 14 | -0.3 | 4.10763469825215 | 1.86797831693084 | 4.20103361409869 | 4.122120376 | 1.87612769714324 | 0.739743235 | 4.21685144 | 4.122509719 | 1.87634433647654 | 0.739256806 | -1.4698198 | 4.21722039 |
| 15 | -0.25 | 4.20103361409869 | 1.90349446895350 | 4.29620833754637 | 4.21685144 | 1.91112565559789 | 0.656153003 | 4.313227914 | 4.217220386 | 1.91130103121642 | 0.655615988 | -1.8716729 | 4.31356596 |
| 16 | -0.2 | 4.29620833754637 | 1.93470623734441 | 4.39294364941359 | 4.313227914 | 1.94145394372337 | 0.553206663 | 4.410992119 | 4.313565964 | 1.94158527279083 | 0.552652904 | -2.2393509 | 4.41128939 |
| 17 | -0.15 | 4.39294364941359 | 1.96068083031843 | 4.49097769092951 | 4.410992119 | 1.96619383447495 | 0.433111404 | 4.5098432 | 4.411289391 | 1.96628203859674 | 0.432579392 | -2.5528063 | 4.51009103 |
| 18 | -0.1 | 4.49097769092951 | 1.98059265767919 | 4.59000732381347 | 4.5098432 | 1.98456179932775 | 0.299055247 | 4.609445109 | 4.510091033 | 1.98461162427495 | 0.29858346 | -2.793376 | 4.60963665 |
| 19 | -0.05 | 4.59000732381347 | 1.99377096740629 | 4.68969587218378 | 4.609445109 | 1.99595621784907 | 0.155087909 | 4.70943678 | 4.609636649 | 1.99597588259073 | 0.154709642 | -2.9457885 | 4.70956746 |
| 20 | 3E-16 | 4.68969587218378 | 1.99974252246996 | 4.78968299830728 | 4.70943678 | 1.99999564225977 | 0.005904379 | 4.809443943 | 4.709567459 | 1.99999601951163 | 0.005643024 | -2.9999522 | 4.80951181 |
| 21 | 0.05 | 4.78968299830728 | 1.99826456470550 | 4.88959622654256 | 4.809443943 | 1.99654412347245 | -0.143449612 | 4.909091837 | 4.809511815 | 1.99653754417662 | -0.143583846 | -2.9522034 | 4.90909771 |
| 22 | 0.1 | 4.88959622654256 | 1.98934400621608 | 4.98906342685336 | 4.909091837 | 1.98572045601824 | -0.287916341 | 5.008017964 | 4.909097708 | 1.98571930853864 | -0.28792755 | -2.8057703 | 5.00796531 |
| 23 | 0.15 | 4.98906342685336 | 1.97324023575749 | 5.08772543864124 | 5.008017964 | 1.96789016035049 | -0.422766411 | 5.105884014 | 5.007965311 | 1.96790549922685 | -0.422671757 | -2.5703628 | 5.1057787 |
| 24 | 0.2 | 5.08772543864124 | 1.95045091167066 | 5.18524798422477 | 5.105884014 | 1.94364122004738 | -0.543897672 | 5.202386203 | 5.105778697 | 1.94368159566922 | -0.543724089 | -2.260969 | 5.20223602 |
| 25 | 0.25 | 5.18524798422477 | 1.92168226424704 | 5.28133209743712 | 5.202386203 | 1.91374658738320 | -0.648057957 | 5.29726346 | 5.202236018 | 1.91381725819423 | -0.647837599 | -1.8960918 | 5.29707758 |
| 26 | 0.3 | 5.28133209743712 | 1.88780732188966 | 5.37572246353160 | 5.29726346 | 1.87911788071543 | -0.732971286 | 5.39030314 | 5.297077582 | 1.87922048874630 | -0.732736672 | -1.495746 | 5.39009152 |
| 27 | 0.35 | 5.37572246353160 | 1.84981674420060 | 5.46821330074163 | 5.39030314 | 1.84075530106205 | -0.797363877 | 5.4813442 | 5.390091524 | 1.84088800278893 | -0.797143655 | -1.0795402 | 5.481117 |
| 28 | 0.4 | 5.46821330074163 | 1.80876739241787 | 5.55865167036253 | 5.4813442 | 1.79969857999570 | -0.840901379 | 5.570278002 | 5.481117004 | 1.79985655014350 | -0.840717332 | -0.6651026 | 5.57004508 |
| 29 | 0.45 | 5.55865167036253 | 1.76573340034163 | 5.64693834037961 | 5.570278002 | 1.75698290457024 | -0.864059137 | 5.657047074 | 5.570045079 | 1.75715908521069 | -0.863924689 | -0.2670013 | 5.65681756 |
| 30 | 0.5 | 5.64693834037961 | 1.72176351248956 | 5.73302651600409 | 5.657047074 | 1.71360252095897 | -0.867951864 | 5.74164226 | 5.656817564 | 1.71378847608990 | -0.867872007 | 0.10380609 | 5.74142431 |
| 31 | 0.55 | 5.73302651600409 | 1.67784711777091 | 5.81691887189263 | 5.74164226 | 1.67048338611367 | -0.854148371 | 5.824098744 | 5.741424311 | 1.67067013752612 | -0.854120678 | 0.44004931 | 5.82389933 |
| 32 | 0.6 | 5.81691887189263 | 1.63489002362038 | 5.89866337307365 | 5.824098744 | 1.62846504962287 | -0.824492412 | 5.904491381 | 5.823899335 | 1.62864380267778 | -0.824508714 | 0.73786227 | 5.90431626 |
| 33 | 0.65 | 5.89866337307365 | 1.59369982167564 | 5.97834836415743 | 5.904491381 | 1.58829104759920 | -0.780944321 | 5.982929753 | 5.904316261 | 1.58845375415106 | -0.780992636 | 0.99627315 | 5.98278346 |
| 34 | 0.7 | 5.97834836415743 | 1.55497982710099 | 6.05609735551248 | 5.982929753 | 1.55060652477686 | -0.725451528 | 6.059553265 | 5.982783464 | 1.55074626590300 | -0.725517905 | 1.21659106 | 6.05943923 |
| 35 | 0.75 | 6.05609735551248 | 1.51933006618704 | 6.13206385882183 | 6.059553265 | 1.51596154587216 | -0.65985043 | 6.134526529 | 6.059439225 | 1.51607274406378 | -0.659920507 | 1.40178698 | 6.13444717 |
| 36 | 0.8 | 6.13206385882183 | 1.48725360331973 | 6.20642653898782 | 6.134526529 | 1.48481854663898 | -0.585798007 | 6.208035209 | 6.134447166 | 1.48489703403466 | -0.58585817 | 1.55593523 | 6.20799211 |
| 37 | 0.85 | 6.20642653898782 | 1.45916656033932 | 6.27938486700479 | 6.208035209 | 1.45756252889307 | -0.504729051 | 6.280282424 | 6.20799211 | 1.45760550588530 | -0.504767189 | 1.68375173 | 6.2802765 |
| 38 | 0.9 | 6.27938486700479 | 1.43541039929696 | 6.35115538696964 | 6.280282424 | 1.43451284759743 | -0.417833713 | 6.351485774 | 6.280276504 | 1.43451876693949 | -0.417839608 | 1.79024202 | 6.35151744 |
| 39 | 0.95 | 6.35115538696964 | 1.41626533377393 | 6.42196865365833 | 6.351485774 | 1.41593571333103 | -0.326049864 | 6.421874997 | 6.35151744 | 1.41590412125617 | -0.326015285 | 1.88045598 | 6.4219443 |
| 40 | 1 | 6.42196865365833 | 6.421874997 | 6.421944303 | |||||||||