636.sagews

#simple Runga-Kutta Order 2 Slash 3

var('t','y')
f(t,y) = y
start = 0.0
end = 1.0
solution(t) = float(e)^t

h = 0.5 #initial step size

#let z00 represent higher-order solution

z00 = y00 = 1.0 #initial value

t00 = start;
soln = [[t00,y00,z00]]

while t00 < end:
    s1 = f(t00,y00)
    s2 = f(t00+h,y00+h*s1) 
    s3 = f(t00+h/2,y00+h/2*((s1+s2)/2))
    y01 = z00+h*((s1+s2)/2.0);
    z01 = z00+h*((s1+4*s3+s2)/6.0);
    err = abs(z01-y01)
    if err < .025:
        t00=t00+h
        y00 = y01
        z00 = z01    
        soln.append([t00,y00,z00])
    else:
        h = h/2.0
print soln

[[0.000000000000000, 1.00000000000000, 1.00000000000000], [0.500000000000000, 1.62500000000000, 1.64583333333333], [0.750000000000000, 2.10286458333333, 2.10709635416667], [1.00000000000000, 2.69852701822917, 2.70400322808160]]

abs(3.2-3)

0.200000000000000