12c01ce7-ae94-47a1-8924-a852e5cbcb09 38 3 5 decimal

&[DATE] &[TIME] - &[FILENAME]

A difficult/uncertain case

Whether the integration is right/wrong ? Good question, apparently YES from lab test. This example from Yuk [Ru.] Mathsoft Collaboratoty, Mathcad 11 [2004].

t y f 3250341 y * - 9 t * + (* + 3250341 y * 9 t * + (* - / :

init 0 :

t Y D t Y 1 el f :

N 328649129513 mat ≡

t y f 0 ≡ y solve maple 325027 t * + 123 /

rkfixed  ... N = 328, 649, 1295 S init 030649 t Y D rkfixed :

x datum 26.423 :

15.521

rkfix S 1 col S 2 col augment :

Useless solvers. # # # # # al_rkckadapt NON-stop compute ≡ # # # # # dn_AdamsMoulton NON-stop compute ≡ # # # # # dn_ExplicitRK45 NFG ≡ # # # # # dn_GearsBDF crashes Smath ≡ # # # # # dn_ImplicitRK5 NON-stop compute ≡ 51 line

r 3250123 / : 26.422764

r r f 1 -

y 23 :

ONE  possible limited solver. # # # # # mwode113 NON-stop compute ≡ # # # # # mwode15s upper limit 15.475 ≡ # # # # # mwode23 NON-stop compute ≡ # # # # # mwode23s NON-stop compute ≡ # # # # # mwode45 NON-stop compute ≡ 51 line

3250 3 41 y * - 9 x * + ( * + 3250 3 41 y * 9 x * + ( * - / 15.521 100 15.521 100 - 2 2 mat 2 1 sys

# # # # # Rkadapt result above max ≡ # # # # # rkfixed OK N: 1295 3 sec ≡ # # # # # rkm9mka NON-stop compute ≡ # # # # # rkm9mkn NON-stop compute ≡ # # # # # rkm9st NON-stop compute ≡ 51 line

# # # # # mk52lfa NON-stop compute ≡ # # # # # mk52lfn NON-stop compute ≡ 21 line

# # log ln

y0 x0 x1 n f rkf4 X 1 el x0 : Y 1 el y0 : Δ x1 x0 - n / : 13 mat i 2 n range k1 X i 1 - el Y i 1 - el f eval : k2 X i 1 - el Δ 2 / + Y i 1 - el k1 2 / Δ * + f eval : k3 X i 1 - el Δ 2 / + Y i 1 - el k2 2 / Δ * + f eval : k4 X i 1 - el Δ + Y i 1 - el k3 Δ * + f eval : Y i el Y i 1 - el k1 2 k2 * + 2 k3 * + k4 + 6 / Δ * + : X i el X i 1 - el Δ + : 61 line for X Y augment 31 line :

N 500999 # 13 mat ≡

rkf 0030500 t Y D rkf4 :

Unknown reaction: Yuk [Ru.] 2004

rkf rkfix x datum 15.521 100 15.521 0 2 2 mat 5 1 sys

al_rkckadapt("1:vector", "2:number", "3:number", "4:number", "5:function") - (init, x1, x2, intvls, D) Uses the 4rth-order Runge-Kutta-Cash-Karp adaptive ODE solver.

We define the function f(x) to satisfy a Lipschitz condition on the interval [a, b] if there exists a constant K (dependent on both f and the interval) such that. |f(x1) − f(x2)| < K|x1 − x2| ================================================= All solvers fail: around 15.2 the derivative reaches infinity, thus the Lipschitz condition fails. At sufficiently high mesh, "rkfixed" jumps to the next existence of the derivative ... 1295 was found manually [different than Mathcad 11].