rkf45 Compactum

93dfa631-ef45-410a-a224-36b4893caabb 15 4 5 decimal

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

Fehlberg Integral Curve @ point(s)

y x diff x y f ≡

x y f 2 x * - y - :

x exact 3 x - exp * - 2 x * - 2 + :

n 25 :

xstart 0 : xend 1 : y0 1 - : 31 line

h xend xstart - n / : 0.04

j 1 :

h j h * : 0.04

X xstart : Y y0 : 21 line

X 0

Y 1 -

rkf45 Compactum

fhelberg companion in compactum format given as point compute/integrate X Y h fehlberg k1 h X Y f (* eval : k2 h X 14 / h * + Y 14 / k1 * + f * eval : k3 h X 38 / h * + Y 332 / k1 * + 932 / k2 * + f * eval : k4 h X 1213 / h * + Y 19322197 / k1 * + 72002197 / k2 * - 72962197 / k3 * + f * eval : k5 h X h + Y 439216 / k1 * + 8 k2 * - 3680513 / k3 * + 8454104 / k4 * - f * eval : k6 h X h 2 / + Y 827 / k1 * - 2 k2 * + 35442565 / k3 * - 18594104 / k4 * + 1140 / k5 * - f * eval : out X h + Y 25216 / k1 * 14082565 / k3 * + 21974104 / k4 * + k5 5 / - (+ Y 16135 / k1 * 665612825 / k3 * + 2856156430 / k4 * + 950 / k5 * - 255 / k6 * + (+ k1 360 / 1284275 / k3 * - 219775240 / k4 * - k5 50 / + 255 / k6 * + 41 mat : 71 line :

k1 h X Y f (* eval : k2 h X 14 / h * + Y 14 / k1 * + f * eval : k3 h X 38 / h * + Y 332 / k1 * + 932 / k2 * + f * eval : k4 h X 1213 / h * + Y 19322197 / k1 * + 72002197 / k2 * - 72962197 / k3 * + f * eval : k5 h X h + Y 439216 / k1 * + 8 k2 * - 3680513 / k3 * + 8454104 / k4 * - f * eval : k6 h X h 2 / + Y 827 / k1 * - 2 k2 * + 35442565 / k3 * - 18594104 / k4 * + 1140 / k5 * - f * eval : 61 line

out X h + Y 25216 / k1 * 14082565 / k3 * + 21974104 / k4 * + k5 5 / - (+ eval Y 16135 / k1 * 665612825 / k3 * + 2856156430 / k4 * + 950 / k5 * - 255 / k6 * + (+ eval k1 360 / 1284275 / k3 * - 219775240 / k4 * - k5 50 / + 255 / k6 * + eval 41 mat :

out 0.04 0.962368317046 - 0.962368317446 - 3.99754 1010 -^* - 41 mat

k1 k2 k3 k4 k5 k6 61 mat X Y h fehlberg 12 mat 0.04 0.0388 0.038214 0.035649 0.035293 0.037624 61 mat 0.04 0.962368 - 0.962368 - 3.997539 1010 -^* - 41 mat 12 mat

Fehlberg components RKF45 X+h Global estimate RK order 4 Global estimate RK order 5 Error estimate 41 mat ≡

X Y h fehlberg 0.04 0.9624 - 0.9624 - 3.9975 1010 -^* - 41 mat

n 25 :

collect/populate the 'n' integrate values 
RKF45 is only for plotting. RKF45 j 1 n 1 + (range h j el 1 - j + n / : for sol will be nested colwise j 1 h rows range sol 1 j el X Y h j el fehlberg : for Unest sol U sol 1 el : j 2 sol cols range U U sol j el augment : for U transpose 71 line :

interpolate @ 'h' h 0.5678 :

Integrate/Interpolate @ 'h' explorer X Y h fehlberg 0.5678 0.835557 - 0.835832 - 0.000275 - 41 mat

Global estimate Fehlberg order 5 X Y h fehlberg 3 el 0.835832 -

plotG(vx,vy,a,b,c)

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 :

vx RKF45 1 col : (collect the variate 'x' ≡ vy RKF45 3 col : (collect Fehlberg solutions ≡ XY vx vy augment : (export data solution ≡ 31 line

The code sources agree: Mathematica & Mathcad 11 Smath as well .

RKF45 rows 26

vx vy . 12 black plotG x exact 2 1 sys

RKF45 11 mat 01 - 1 - 0 0.04 0.9624 - 0.9624 - 0 0.08 0.9293 - 0.9293 - 0 0.12 0.9008 - 0.9008 - 0 0.16 0.8764 - 0.8764 - 0 0.2 0.8562 - 0.8562 - 0 0.24 0.8399 - 0.8399 - 0 0.28 0.8273 - 0.8273 - 0 0.32 0.8184 - 0.8184 - 0 0.36 0.813 - 0.813 - 0 0.4 0.8109 - 0.8109 - 0 0.44 0.812 - 0.8121 - 0.0001 - 0.48 0.8162 - 0.8163 - 0.0001 - 0.52 0.8233 - 0.8235 - 0.0002 - 0.56 0.8333 - 0.8336 - 0.0003 - 0.6 0.846 - 0.8463 - 0.0004 - 0.64 0.8612 - 0.8617 - 0.0005 - 0.68 0.8789 - 0.8796 - 0.0007 - 0.72 0.899 - 0.8999 - 0.0009 - 0.76 0.9213 - 0.9226 - 0.0013 - 0.8 0.9457 - 0.9474 - 0.0016 - 0.84 0.9722 - 0.9743 - 0.0021 - 0.88 1.0006 - 1.0033 - 0.0027 - 0.92 1.0308 - 1.0342 - 0.0034 - 0.96 1.0627 - 1.0669 - 0.0043 - 1 1.0962 - 1.1014 - 0.0053 - 264 mat 11 mat

x 0.6 :

Balistic accuracy test X Y x fehlberg 3 el x exact 21 mat 0.846323 - 0.846435 - 21 mat

Δ x exact X Y x fehlberg 3 el - : 0.000112 -

x canvas X Y x fehlberg 3 el 0 x ≤ (x 1 ≤ (& cases :

x canvas Y vx vy . 12 black plotG 3 1 sys x exact x canvas -

t Y D 2 t * - Y 1 el - :

can't use D(t,Y) 'Y' is an existing vector
just renamed arguments of   D(t,dummy) adapt 1 - 01100 t dummy D Rkadapt : Gears 1 - 01100 t dummy D dn_GearsBDF : fixed 1 - 01100 t dummy D rkfixed : 31 line

1 exact adapt adapt rows row Gears Gears rows row fixed fixed rows row X Y 1 fehlberg 3 el 51 mat 1.10363832351433 - 1 1.10363832351431 - 12 mat 1 1.10381207518406 - 12 mat 1 1.09730658258777 - 12 mat 1.10144230769231 - 51 mat

1 exact 1 exact 1.10363832351431 - (- 1 exact 1.10381207518406 - (- 1 exact 1.09730658258777 - (- 1 exact 1.10144230769231 - (- 51 mat 1.1036 - 1.7097 1014 -^* - 0.0002 0.0063 - 0.0022 - 51 mat

0.5 exact 0.8196 -

y0 x0 x1 n f Euler X 1 el x0 : Y 1 el y0 : Δ x1 x0 - n 1 - / : 13 mat i 2 n range Y i el Y i 1 - el X i 1 - el Y i 1 - el f Δ * + eval : X i el X i 1 - el Δ + : 21 line for X Y augment 31 line :

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 :

x0 0 : x1 1 : y0 1 - : n 1000 : 41 line

x y f 2 x * - y - :

euler y0 x0 x1 n f Euler :

rkf y0 x0 x1 n f rkf4 :

Euler & rkf4 2 variables integrators. "Integrators" in the sense of 'Integral Curve'

euler rkf 2 1 sys

rkf rkf rows row 1 1.10363832351437 - 12 mat

1 exact 1.10363832351433 -