78f6b0f7-ebcb-45fc-9242-7ce7f20987ad 20 3 5 decimal

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

Under "while", the module outputs the last RK4 value. 
Computes on-line integral value(s) @ user 'u' ... computes for a given vector. init t1 t2 n f RK4 vx init : t t1 : h t2 t1 - n / : 13 mat t t2 ≤ k1 t vx f h * eval : k2 t h 2 / + vx k1 2 / + f h * eval : 12 mat k3 t h 2 / + vx k2 2 / + f h * eval : k4 t h + vx k3 + f h * eval : 12 mat vx vx 16 / k1 2 k2 * + 2 k3 * + k4 + (* + eval : t t h + : 12 mat 31 line while vx 31 line :

init t1 t2 N f YUK X 1 el t1 : Y 1 el init : Δ t2 t1 - 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 :

ε 1015 -^:

x t f x ε + ln t 0 + (+ :

init 0 : start 0 : end 2 : N 100 : 14 mat

RK4 init start end N f RK4 : yuk init start end N f YUK : fixed init start end N x t f rkfixed : adapt init start end N x t f Rkadapt : 41 sys

Rkadapt max iterations 5000

=============================================================================

N 100 6.101538 - 5.889469 - 9.907851 - 5.31995 - 41 mat

N 1000 5.390762 - 5.372009 - 5.765418 - 5.319613 - 41 mat

N 5000 5.332982 - 5.329271 - 5.406418 - 5.319562 - 41 mat

RK4 yuk yuk rows 2 el fixed fixed rows 2 el adapt adapt rows 2 el 41 mat 6.101538 - 5.889469 - 9.907851 - 5.31995 - 41 mat

adapt yuk 2 1 sys

yuk 11 mat 00 0.02 0.193 - 0.04 0.268 - 0.06 0.334 - 0.08 0.394 - 0.1 0.451 - 0.12 0.505 - 0.14 0.556 - 0.16 0.606 - 0.18 0.654 - 0.2 0.7 - 0.22 0.746 - 0.24 0.791 - 0.26 0.835 - 0.28 0.878 - 0.3 0.921 - 0.32 0.963 - 0.34 1.005 - 0.36 1.047 - 0.38 1.088 - 0.4 1.129 - 0.42 1.17 - 0.44 1.21 - 0.46 1.251 - 0.48 1.292 - 0.5 1.332 - 0.52 1.373 - 0.54 1.413 - 0.56 1.454 - 0.58 1.494 - 0.6 1.535 - 0.62 1.576 - 0.64 1.617 - 0.66 1.659 - 0.68 1.7 - 0.7 1.742 - 0.72 1.784 - 0.74 1.827 - 0.76 1.87 - 0.78 1.913 - 0.8 1.956 - 0.82 2 - 0.84 2.044 - 0.86 2.089 - 0.88 2.134 - 0.9 2.179 - 0.92 2.225 - 0.94 2.271 - 0.96 2.318 - 0.98 2.366 - 1 2.414 - 1.02 2.462 - 1.04 2.511 - 1.06 2.561 - 1.08 2.612 - 1.1 2.663 - 1.12 2.714 - 1.14 2.767 - 1.16 2.82 - 1.18 2.873 - 1.2 2.928 - 1.22 2.983 - 1.24 3.039 - 1.26 3.096 - 1.28 3.154 - 1.3 3.213 - 1.32 3.272 - 1.34 3.332 - 1.36 3.394 - 1.38 3.456 - 1.4 3.519 - 1.42 3.583 - 1.44 3.648 - 1.46 3.714 - 1.48 3.782 - 1.5 3.85 - 1.52 3.919 - 1.54 3.99 - 1.56 4.062 - 1.58 4.135 - 1.6 4.209 - 1.62 4.284 - 1.64 4.361 - 1.66 4.439 - 1.68 4.518 - 1.7 4.599 - 1.72 4.681 - 1.74 4.765 - 1.76 4.849 - 1.78 4.936 - 1.8 5.024 - 1.82 5.113 - 1.84 5.204 - 1.86 5.297 - 1.88 5.392 - 1.9 5.488 - 1.92 5.585 - 1.94 5.685 - 1.96 5.786 - 1.98 5.889 - 1002 mat 11 mat

adapt 11 mat 00 0.02 0.099 - 0.04 0.173 - 0.06 0.237 - 0.08 0.295 - 0.1 0.35 - 0.12 0.402 - 0.14 0.451 - 0.16 0.499 - 0.18 0.544 - 0.2 0.589 - 0.22 0.632 - 0.24 0.675 - 0.26 0.717 - 0.28 0.757 - 0.3 0.798 - 0.32 0.838 - 0.34 0.877 - 0.36 0.916 - 0.38 0.954 - 0.4 0.993 - 0.42 1.031 - 0.44 1.069 - 0.46 1.106 - 0.48 1.144 - 0.5 1.182 - 0.52 1.219 - 0.54 1.256 - 0.56 1.294 - 0.58 1.331 - 0.6 1.369 - 0.62 1.407 - 0.64 1.444 - 0.66 1.482 - 0.68 1.52 - 0.7 1.558 - 0.72 1.597 - 0.74 1.636 - 0.76 1.674 - 0.78 1.713 - 0.8 1.753 - 0.82 1.793 - 0.84 1.832 - 0.86 1.873 - 0.88 1.913 - 0.9 1.954 - 0.92 1.996 - 0.94 2.038 - 0.96 2.08 - 0.98 2.122 - 1 2.166 - 1.02 2.209 - 1.04 2.253 - 1.06 2.298 - 1.08 2.343 - 1.1 2.388 - 1.12 2.434 - 1.14 2.481 - 1.16 2.528 - 1.18 2.576 - 1.2 2.625 - 1.22 2.674 - 1.24 2.724 - 1.26 2.774 - 1.28 2.826 - 1.3 2.878 - 1.32 2.93 - 1.34 2.984 - 1.36 3.038 - 1.38 3.093 - 1.4 3.149 - 1.42 3.205 - 1.44 3.263 - 1.46 3.321 - 1.48 3.381 - 1.5 3.441 - 1.52 3.502 - 1.54 3.564 - 1.56 3.627 - 1.58 3.692 - 1.6 3.757 - 1.62 3.823 - 1.64 3.89 - 1.66 3.959 - 1.68 4.028 - 1.7 4.099 - 1.72 4.171 - 1.74 4.244 - 1.76 4.319 - 1.78 4.395 - 1.8 4.472 - 1.82 4.55 - 1.84 4.63 - 1.86 4.711 - 1.88 4.793 - 1.9 4.877 - 1.92 4.963 - 1.94 5.05 - 1.96 5.138 - 1.98 5.228 - 2 5.32 - 1012 mat 11 mat

u atUser init start u N f RK4 start u ≤ (u end ≤ (& out range cases :

u 3 sqrt : 1.7321

on-line evaluate @ user 1.23456789 atUser u atUser 2.5 atUser 31 mat 2.9005 - 4.6597 - out range 31 mat

Smath ODE librairies

Matlab solvers

Intel solvers

Useless solvers. # # # # # al_rkckadapt # # # # # dn_AdamsMoulton # # # # # dn_ExplicitRK45 # # # # # dn_GearsBDF # # # # # dn_ImplicitRK5 51 line

ONE  possible limited solver. # # # # # mwode113 # # # # # mwode15s # # # # # mwode23 # # # # # mwode23s # # # # # mwode45 51 line

# # # # # Rkadapt # # # # # rkfixed # # # # # rkm9mka # # # # # rkm9mkn # # # # # rkm9st 51 line

# # # # # mk52lfa # # # # # mk52lfn 21 line

Intel init start end N x t f Rkadapt init start end N x t f rkfixed init start end N x t f rkm9mka init start end N x t f rkm9mkn init start end N x t f rkm9st 51 mat :

Matlab init start end N x t f mwode113 init start end N x t f mwode15s init start end N x t f mwode23 init start end N x t f mwode23s init start end N x t f mwode45 51 mat :

DotNumeric init start end N x t f al_rkckadapt init start end N x t f dn_AdamsMoulton init start end N x t f dn_ExplicitRK45 init start end N x t f dn_GearsBDF init start end N x t f dn_ImplicitRK5 51 mat :

Intel rows Matlab rows DotNumeric rows 31 mat 55531 mat

A difficult/uncertain case

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

init 0 :

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 :

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

rkf4 0030500 t Y D rkf4 :

Unknown reaction: Yuk [Ru.] 2004

rkf4 15.521 100 15.521 0 2 2 mat 2 1 sys