 | ||
| 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 | 1.40196404398432 | 6.49206685585755 | 6.421874997 | 1.40205680039718 | -0.230065223 | 6.491690256 | 6.421944303 | 1.40198816047671 | -0.229983806 | 1.95933724 | 6.49179705 | |||||
| 41 | 1.05 | 6.49206685585755 | 1.39270515817184 | 6.56170211376614 | 6.491690256 | 1.39307358690118 | -0.1303249 | 6.561181029 | 6.491797051 | 1.39296910607644 | -0.130191881 | 2.03165112 | 6.56132509 | |||||
| 42 | 1.1 | 6.56170211376614 | 1.38866620740709 | 6.63113542413649 | 6.561181029 | 1.38916724903538 | -0.027040868 | 6.63060559 | 6.561325092 | 1.38902871952382 | -0.026852728 | 2.10197516 | 6.63078675 | |||||
| 43 | 1.15 | 6.63113542413649 | 1.39001595670690 | 6.70063622197184 | 6.63060559 | 1.39051408724554 | 0.079799397 | 6.700231044 | 6.630786754 | 1.39034375340449 | 0.080045334 | 2.17473778 | 6.7004493 | |||||
| 44 | 1.2 | 6.70063622197184 | 1.39692616518698 | 6.77048253023119 | 6.700231044 | 1.39729658189399 | 0.190427201 | 6.770333907 | 6.700449305 | 1.39709703783923 | 0.190733189 | 2.25429409 | 6.77058954 | |||||
| 45 | 1.25 | 6.77048253023119 | 1.40958293919252 | 6.84096167719081 | 6.770333907 | 1.40971426797560 | 0.305291464 | 6.841201235 | 6.770589538 | 1.40948838979826 | 0.30565977 | 2.34503069 | 6.84149489 | |||||
| 46 | 1.3 | 6.84096167719081 | 1.42819792425672 | 6.91237157340365 | 6.841201235 | 1.42799469053106 | 0.425063416 | 6.913132299 | 6.841494887 | 1.42774560624594 | 0.425496769 | 2.45149475 | 6.91346511 | |||||
| 47 | 1.35 | 6.91237157340365 | 1.45301964340843 | 6.98502255557407 | 6.913132299 | 1.45240476232150 | 0.550648285 | 6.986440847 | 6.913465111 | 1.45213586310986 | 0.551150342 | 2.57854496 | 6.98681456 | |||||
| 48 | 1.4 | 6.98502255557407 | 1.48434533864466 | 7.05923982250630 | 6.986440847 | 1.48326289964466 | 0.683205048 | 7.061457998 | 6.986814562 | 1.48297789745906 | 0.68378092 | 2.73152442 | 7.06187509 | |||||
| 49 | 1.45 | 7.05923982250630 | 1.52253371527325 | 7.13536650826997 | 7.061457998 | 1.52095236588622 | 0.824175372 | 7.138535836 | 7.061875089 | 1.52065540506783 | 0.824832217 | 2.91645634 | 7.13899966 | |||||
| 50 | 1.5 | 7.13536650826997 | 1.56801902887092 | 7.21376745971351 | 7.138535836 | 1.56593630843961 | 0.975322874 | 7.218051805 | 7.138999659 | 1.56563214273933 | 0.976070622 | 3.14026373 | 7.21856678 | |||||
| 51 | 1.55 | 7.21376745971351 | 1.62132699220645 | 7.29483380932384 | 7.218051805 | 1.61877503294698 | 1.13878392 | 7.300414036 | 7.218566777 | 1.61846928439352 | 1.139636179 | 3.41101254 | 7.30098585 | |||||
| 52 | 1.6 | 7.29483380932384 | 1.68309301059581 | 7.37898845985363 | 7.300414036 | 1.68014611467936 | 1.317130975 | 7.386067756 | 7.300985849 | 1.67984563728812 | 1.31810619 | 3.73817254 | 7.38670364 | |||||
| 53 | 1.65 | 7.37898845985363 | 1.75408326785385 | 7.46669262324632 | 7.386067756 | 1.75086798766202 | 1.51344907 | 7.475502967 | 7.386703642 | 1.75058136716647 | 1.51457202 | 4.13288022 | 7.47621203 | |||||
| 54 | 1.7 | 7.46669262324632 | 1.83521915918529 | 7.55845358120558 | 7.475502967 | 1.83192765158433 | 1.731424672 | 7.56926363 | 7.476212028 | 1.83166588208553 | 1.732728414 | 4.60816823 | 7.57005724 | |||||
| 55 | 1.75 | 7.55845358120558 | 1.92760546140247 | 7.65483385427571 | 7.56926363 | 1.92451304454299 | 1.975443705 | 7.667958587 | 7.570057236 | 1.92429043345206 | 1.976972051 | 5.17908942 | 7.66885087 | |||||
| 56 | 1.8 | 7.65483385427571 | 2.03256237335032 | 7.75646197294322 | 7.667958587 | 2.03005035498135 | 2.250690441 | 7.772274468 | 7.66885087 | 2.02988571664565 | 2.252501025 | 5.86259961 | 7.77328292 | |||||
| 57 | 1.85 | 7.75646197294322 | 2.15166102810941 | 7.86404502434869 | 7.772274468 | 2.15024592702066 | 2.563229717 | 7.882990801 | 7.77328292 | 2.15016412772714 | 2.565397585 | 6.67695072 | 7.88413698 | |||||
| 58 | 1.9 | 7.86404502434869 | 2.28676106862208 | 7.97838307777980 | 7.882990801 | 2.28713116952282 | 2.920038378 | 8.000997408 | 7.884136976 | 2.28716507101519 | 2.922659705 | 7.64015581 | 8.00230772 | |||||
| 59 | 1.95 | 7.97838307777980 | 2.44004705432694 | 8.10038543049614 | 8.000997408 | 2.44310653697179 | 3.328923033 | 8.127313888 | 8.002307724 | 2.44329933014681 | 3.332117927 | 8.7667738 | 8.12882048 | |||||
| 60 | 2 | 8.10038543049614 | 2.61405729399126 | 8.23108829519571 | 8.127313888 | 2.62097643324625 | 3.79821283 | 8.263110476 | 8.128820479 | 2.62138421727050 | 3.802124844 | 10.0617672 | 8.26485197 | |||||
| 61 | 2.05 | 8.23108829519571 | 2.81169336307932 | 8.37167296334967 | 8.263110476 | 2.82395958217707 | 4.336037613 | 8.409728502 | 8.264851966 | 2.82465375573282 | 4.340825179 | 11.5094629 | 8.41175047 | |||||
| 62 | 2.1 | 8.37167296334967 | 3.03618989688299 | 8.52348245819382 | 8.409728502 | 3.05564724932070 | 4.948882644 | 8.568696968 | 8.411750466 | 3.05671572866888 | 4.954693484 | 13.0547295 | 8.57105159 | |||||
| 63 | 2.15 | 8.52348245819382 | 3.29101078514185 | 8.68803299745092 | 8.568696968 | 3.31986249188637 | 5.638947189 | 8.741738777 | 8.571051593 | 3.32140781117972 | 5.64586075 | 14.5727096 | 8.74448291 | |||||
| 64 | 2.2 | 8.68803299745092 | 3.57961828124462 | 8.86701391151315 | 8.741738777 | 3.62034557956059 | 6.399654016 | 8.930755623 | 8.744482908 | 3.62247646077578 | 6.407570275 | 15.8239073 | 8.93394586 | |||||
| 65 | 2.25 | 8.86701391151315 | 3.90503592629919 | 9.06226570782811 | 8.930755623 | 3.96015526506924 | 7.208577034 | 9.137774108 | 8.933945858 | 3.96296646066574 | 7.217029185 | 16.3959233 | 9.14145705 | |||||
| 66 | 2.3 | 9.06226570782811 | 4.26909965004414 | 9.27572069033031 | 9.137774108 | 4.34064407210489 | 8.017390579 | 9.364828049 | 9.141457049 | 4.34417828000735 | 8.025281053 | 15.6471625 | 9.36902355 | |||||
| 67 | 2.35 | 9.27572069033031 | 4.67128100074469 | 9.50928474036755 | 9.364828049 | 4.75987291530081 | 8.739848652 | 9.613746506 | 9.369023547 | 4.76406139120939 | 8.745185284 | 12.6990718 | 9.61842266 | |||||
| 68 | 2.4 | 9.50928474036755 | 5.10701251723908 | 9.76463536622950 | 9.613746506 | 5.21045218616040 | 9.242235204 | 9.885821909 | 9.618422662 | 5.21504302962400 | 9.242107461 | 6.57312107 | 9.89086439 | |||||
| 69 | 2.45 | 9.76463536622950 | 5.56562140303578 | 10.04291643638130 | 9.885821909 | 5.67715204349892 | 9.346627667 | 10.1813628 | 9.890864388 | 5.68166235491430 | 9.337856657 | -3.3949493 | 10.1865491 | |||||
| 70 | 2.5 | 10.04291643638130 | 6.02837548454692 | 10.34433521060860 | 10.1813628 | 6.13529838242899 | 8.863022107 | 10.49920649 | 10.1865491 | 6.13906053293017 | 8.843842299 | -16.81178 | 10.5042067 | |||||
| 71 | 2.55 | 10.34433521060860 | 6.46777977985819 | 10.66772419960150 | 10.49920649 | 6.55176460584167 | 7.662494741 | 10.83637284 | 10.50420668 | 6.55413486825682 | 7.6347677 | -31.442028 | 10.8408018 | |||||
| 72 | 2.6 | 10.66772419960150 | 6.84984806726949 | 11.01021660296500 | 10.83637284 | 6.89046543940320 | 5.776842748 | 11.18811717 | 10.84080184 | 6.89115788195332 | 5.746806913 | -43.177626 | 11.1916437 | |||||
| 73 | 2.65 | 11.01021660296500 | 7.14081060733681 | 11.36725713333190 | 11.18811717 | 7.12243861943976 | 3.465052368 | 11.54857041 | 11.19164371 | 7.12175769394780 | 3.440540097 | -47.536331 | 11.5510419 | |||||
| 74 | 2.7 | 11.36725713333190 | 7.31764534071585 | 11.73313940036770 | 11.54857041 | 7.23688251925007 | 1.171526098 | 11.91187895 | 11.55104193 | 7.23558178317881 | 1.157003402 | -42.188542 | 11.9133883 | |||||
| 75 | 2.75 | 11.73313940036770 | 7.37830580440848 | 12.10205469058810 | 11.91187895 | 7.24695365392067 | -0.631085889 | 12.27343777 | 11.91338835 | 7.24575546663872 | -0.636786558 | -28.467291 | 12.2742871 | |||||
| 76 | 2.8 | 12.10205469058810 | 7.34558891381672 | 12.46933413627890 | 12.27343777 | 7.18653902856740 | -1.615388541 | 12.63074549 | 12.27428707 | 7.18572580677971 | -1.616369738 | -10.377971 | 12.6313367 | |||||
| 77 | 2.85 | 12.46933413627890 | 7.26159702836458 | 12.83241398769710 | 12.63074549 | 7.10038234249385 | -1.673153001 | 12.98367316 | 12.63133669 | 7.09979237681556 | -1.672292998 | 7.89899769 | 12.9844005 | |||||
| 78 | 2.9 | 12.83241398769710 | 7.17612575766034 | 13.19122027558020 | 12.98367316 | 7.03374587447756 | -0.869398034 | 13.33427371 | 12.9844005 | 7.03308105738914 | -0.866715298 | 23.843578 | 13.3354679 | |||||
| 79 | 2.95 | 13.19122027558020 | 7.13582244987794 | 13.54801139807410 | 13.33427371 | 7.02616947769010 | 0.655170112 | 13.68640115 | 13.3354679 | 7.02531090692364 | 0.661618988 | 36.8320541 | 13.6883278 | |||||
| 80 | 3 | 13.54801139807410 | 13.68640115 | 13.68832781 | ||||||||||||||
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