<p> plotG(vx,vy,a,b,c) </p>

89e13956-5d5e-439c-9e0f-0356f3b1ba30 15 4 5 decimal

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

This ODE solver is coded to take the function name 'f' ... as ccded NOT  ► commutative. 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 :

Cauchy ODE solver rkf4  ... as  coded  ►  NOT comutative y0 x0 x1 n f rkf4 X 1 el x0 : Y 1 el y0 : Δ x1 x0 - n 1 - / : 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 :

both sanity checked
Mathcad, Mathematica 2 x * - y - # 1 x y 3 nthroot * - 13 mat

y0 1 : x0 0 : x1 5 : n 400 : 14 mat

x y f 1 x y 3 nthroot * - :

rkf y0 x0 x1 n f rkf4 :

integral curve of

rkf 1 1 sys

1 x y 3 nthroot * -

rkf4 interpolation

vx rkf 1 col : vy rkf 2 col : 12 mat x Cauchy vx vy x ainterp 0 x ≤ (x 5 ≤ (& out range cases : 21 sys

u 2 :

Ω u Cauchy : 0.8596

track 6242 u Ω 32 mat :

x Cauchy track 2 1 sys

x y f 1 x y 3 nthroot * - :

y0 1 : x0 0 : x1 2 : N 400 : 14 mat

rkf y0 x0 x1 N f rkf4 :

u SOL 10 u N f RK4 0 u ≤ (u 5 ≤ (& out range cases : X 05 0.25 range : Y X SOL vectorize eval : XY X Y augment : x sol X Y x ainterp 0 x ≤ (x 5 ≤ (& out range cases : 51 sys

rkf x sol 2 1 sys

===== More companionship =====

x t f x t sqrt 0 + (+ :

mesh fine enough for smooth plot. init 0 : start 0 : end 5 : N 100 : 14 mat

last value @ 'end' ... sanity MCD 11 RK4 init start end N f RK4 : 24.9957

<p> plotG(vx,vy,a,b,c) </p>

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

t0 1 time :

u SOL init start u N f RK4 start u ≤ (u end ≤ (& out range cases : X 05 0.25 range : Y X SOL vectorize eval : XY X Y augment : x sol X Y x ainterp 0 x ≤ (x end ≤ (& out range cases : 51 sys

Notes :

1. Takes no time to interpolate @ user point(s), plot canvas.

2. This solver is identical to Mathsoft Odesolve, which by

same token does not rescue to spline interpolation. Rather it

is based on ¬ if IsScalar. The Odesolve block is a wraper of

fixed steps rk, adaptive, stiff from menu option.

Observe ========================

RK4(init,t1,t2,n,f) ... is sensibly more accurate than Smath

built-in rkfixed. On the other hand, Rkadapt is generally

not equally spaced for export. The proposal will adequately

interpolate from in-situ command.

]]>

U 0.625 :

Ω U sol : 0.3906

track 0.5 - 313 U Ω 32 mat :

track x sol U U sol + 15 blue 1 5 mat X Y . 12 black plotG 4 1 sys

1 time t0 - 11.6 s *

x t q x t sqrt 0 + (+ : t ω D t ω 1 el q : rkf 002 N t ω D rkfixed : rka 002 N t ω D Rkadapt : 41 sys

U Ω 12 mat 0.625 0.3906 12 mat

Compare the 3 ODE solvers RK4 rkf rkf rows row rka rka rows row 31 mat 24.996 2 3.959 12 mat 2412 mat 31 mat

U 5 :

Ω U sol : 24.996

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

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