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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 :

rkfixed("1:vector", "2:number", "3:number", "4:number", "5:function") - (init, x1, x2, intvls, D) Uses the fourth-order Runge-Kutta fixed-step method

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

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

RK4 sol 1 col sol 2 col augment : collect for 2D plot/export ≡

The given function

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

RK cumulative area under Integral(x)

RK4 0 2 1 sys

x f x Integral :

f a b Int 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 * 41 line :

f 255280 Int 11.583599

RK4 101 row 280 11.172309 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 #