# Verfahren von Heun zur numerischen Lösung von Anfangswertaufgaben.

def heun(xk,yk,h,ys):
    xk1 = xk + h
    yk1p = yk + h * ys(xk,yk)
    yk1 = 0.5*(yk+yk1p+h*ys(xk1,yk1p))
    return (xk1,yk1p)

def run_heun(K,h,x0,y0,ys):
    if not isinstance(ys,list): ys = [ys]
    if not isinstance(y0,list): y0 = [y0]
    xk = x0
    yk = y0
    xks = x0
    yks = [ [v] for v in y0 ]

    for i in range(K):
        for j in range(len(ys)):
            xk,yk[j] = heun(xk,yk[j],h,ys[i])
            xks.append(xk)
            yks[j].append(yk[j])