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Working an 'RK' example

RK4 and the likes, output the cumulative area under a given function of one independent variable 'x'. An RK project goes in two steps: 1. Plot the function 2. Construct RK ,i.e: the cumulative area under the function. Let's follow the Smath forum example [this document Smath 5346].

rc 281 : rw 262 : Bc 36180 / π * : 31 line

RK's can integrate many rows in D(x,y). The setup is most likely universal

0. The given function

r F r 2^rw 2^- (rc 2^r 2^Bc sin (2^* - sqrt rc 2^rw 2^- (2^rc 2^* Bc cos (2^* r 2^rw 2^- (2^rc 2^r 2^Bc sin (2^* - (* - sqrt / * :

1. Express the function in term of its variable of integration

x Integral x 2^rw 2^- (rc 2^x 2^Bc sin (2^* - sqrt rc 2^rw 2^- (2^rc 2^* Bc cos (2^* x 2^rw 2^- (2^rc 2^x 2^Bc sin (2^* - (* - sqrt / * :

2. Express the D(x,y) RK integrand argument

x y D x 2^rw 2^- (rc 2^x 2^Bc sin (2^* - sqrt rc 2^rw 2^- (2^rc 2^* Bc cos (2^* x 2^rw 2^- (2^rc 2^x 2^Bc sin (2^* - (* - sqrt / * :

sol 011 mat 255280100 x y D rkfixed :

sol45 011 mat 255280100 x y D Rkadapt :

rkfixed (init, x1, x2, intvls, D) Uses the fourth-order Runge-Kutta fixed-step method

Rkadapt (init, x1, x2, intvls, D) Uses the Runge-Kutta-Fehlberg Method (RKF45).

unary matrix 11 mat start end Npoints x y D RK cumulative integrator ≡ rk

As the rk will proceed, the unary matrix will fillup with the discrete variate 'x'

The given function

255 x ≤ ( x 280 ≤ ( & x Integral if 1 1 sys

RK cumulative area under Integral(x)

sol sol45 2 1 sys

f a b Int Romberg core algorithm x f x a b int 32 / b a - (* w G w 01 int * ≡ memo ≡ w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 51 line :

x f x Integral :

f 255280 Int 11.583599

sol 101 row 280 11.172309 12 mat

sol45 101 row 280 11.583539 12 mat

Romberg x 255280256 range : i 1 x rows range y i el f 255 x i el Int eval : for x y augment 31 line :

Romberg 2550256 0.362444 - 257 0.663675 - 258 0.906607 - 259 1.093452 - 260 1.225827 - 261 1.304833 - 262 1.331103 - 263 1.304824 - 264 1.22575 - 265 1.093191 - 266 0.905978 - 267 0.662417 - 268 0.360206 - 269 0.003682 270 0.43317 271 0.933323 272 1.510717 273 2.173998 274 2.934775 275 3.809109 276 4.820176 277 6.00349 278 7.41855 279 9.180502 280 11.583599 262 mat

Lorenz attractor Rkadapt project

u Y D 8 Y 1 el * - 8 Y 2 el * + 30 Y 1 el * Y 1 el + Y 1 el Y 3 el * - Y 1 el Y 2 el * 83 / Y 3 el * - 31 mat :

t0 0 : t1 10 : N 1000 : 31 line

vinit 1 - 0131 mat :

s vinit t0 t1 N u Y D Rkadapt :

y0 s 1 col s 2 col augment : y1 s 1 col s 3 col augment : y2 s 1 col s 4 col augment : 31 line

Lorenz triplet solutions

y0 y1 y2 3 1 sys

Lorenz s 2 col s 3 col augment : θ rot θ cos θ sin - θ sin θ cos 22 mat : θ π - 4 / : Rot Lorenz θ rot * : 41 line

s 2 col s 3 col augment 0 Rot 2 1 sys