<p> rkf45 Compactum </p>

7656b0b9-3499-43bf-af48-aa518c7081b7 44 6 9 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 :

j 13 :

xstart 0 : xend 1 : 21 line

y0 1 - : yalt y0 1 - : 21 line

h xend xstart - n / : 0.04

h j h * : 0.52

x xstart 0 (+ : y y0 0 (+ : 21 line

<p> rkf45 Compactum </p>

rkf45 companion in compactum format given as point compute/integrate x y h rkf 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 0.52 0.8233387776 - 0.823513525222 - 0.000174747622 - 41 mat

k1 k2 k3 k4 k5 k6 61 mat x y h rkf 12 mat 0.52 0.3172 0.24546 0.100306 - 0.177167 - 0.174478 61 mat 0.52 0.823339 - 0.823514 - 0.000175 - 41 mat 12 mat

0.16 0.1408 0.132064 0.093883736440601 0.088509676307692 0.123214481880615 61 mat 0.16 0.876430895262 - 0.876431322758 - 0.000000427496 - 41 mat 12 mat

0.52 0.3172 0.2454595 0.10030592 - 0.177166912 - 0.174478001984 61 mat 0.52 0.8233387776 - 0.823513525222 - 0.000174747622 - 41 mat 12 mat

1 0.25 0.0859375 1.19162494310423 - 1.89423076923077 - 0.066105769230771 - 61 mat 1 1.096153846154 - 1.101442307692 - 0.005288461538 - 41 mat 12 mat

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 x+h Global estimate RK order 4 Global estimate RK order 5 Error estimate 41 mat ≡

Integrate and collect

j 126 range h j el 1 - j + 25 / : for

sol will be nested colwise j 1 h rows range sol 1 j el x y h j el rkf : for 21 line

data UnestCol v data 1 el : j 2 data cols range v v data j el augment : for v 31 line :

U sol UnestCol :

U 0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 0.44 0.48 0.52 0.56 0.6 0.64 0.68 0.72 0.76 0.8 0.84 0.88 0.92 0.96 11 - 0.962368 - 0.929349 - 0.900761 - 0.876431 - 0.856191 - 0.83988 - 0.827343 - 0.81843 - 0.812997 - 0.810905 - 0.812017 - 0.816205 - 0.823339 - 0.833296 - 0.845957 - 0.861202 - 0.878917 - 0.898987 - 0.921301 - 0.945748 - 0.972218 - 1.000603 - 1.030793 - 1.06268 - 1.096154 - 1 - 0.962368 - 0.929349 - 0.900761 - 0.876431 - 0.856192 - 0.839883 - 0.82735 - 0.818444 - 0.813023 - 0.81095 - 0.812091 - 0.81632 - 0.823514 - 0.833553 - 0.846323 - 0.861714 - 0.879618 - 0.899932 - 0.922554 - 0.947386 - 0.974333 - 1.003302 - 1.034203 - 1.066945 - 1.101442 - 0 3.997539 1010 -^* - 000 0.000001 - 0.000003 - 0.000007 - 0.000014 - 0.000026 - 0.000045 - 0.000074 - 0.000116 - 0.000175 - 0.000256 - 0.000366 - 0.000512 - 0.000702 - 0.000945 - 0.001253 - 0.001638 - 0.002115 - 0.0027 - 0.003409 - 0.004265 - 0.005288 - 426 mat

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 :

X U 1 row transpose : collect the variate 'x' ≡ Y U 3 row transpose : Fehleberg collect solution ≡ XY X Y augment : export data solution ≡ 31 line

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

X Y . 12 black plotG x exact 2 1 sys

Balistic accuracy test x y 1 rkf 3 el 1 exact 21 mat 1.101442 - 1.103638 - 21 mat

Uses the Runge-Kutta-Fehlberg Method (RKF45). # # # # # Rkadapt