634.sagews

#simple example of Trapezoid Method

var('t','y')
f(t,y) = y
start = 0.0
end = 1.0
solution(t) = float(e)^t
numOfIterations = 10
h = (end-start)/numOfIterations

y00 = 1.0 #initial value
t00 = start;
soln = [[t00,y00]]

for i in range(numOfIterations):
    y00 = y00+h/2*(f(t00,y00) + f(t00+h,y00 + h*f(t00,y00)))
    t00=t00+h
    soln.append([t00,y00])
    print "%10r %20r %20r"%(t00,y00,solution(t00))

0.100000000000000 1.10500000000000 1.10517091807565 0.200000000000000 1.22102500000000 1.22140275816017 0.300000000000000 1.34923262500000 1.34985880757600 0.400000000000000 1.49090205062500 1.49182469764127 0.500000000000000 1.64744676594062 1.64872127070013 0.600000000000000 1.82042867636439 1.82211880039051 0.700000000000000 2.01157368738265 2.01375270747048 0.800000000000000 2.22278892455783 2.22554092849247 0.900000000000000 2.45618176163640 2.45960311115695 1.00000000000000 2.71408084660822 2.71828182845904

#simple example of Trapezoid Method

var('t','y')
f(t,y) = t*y+t^3
start = 0.0
end = 1.0
solution(t) = 3*float(e)^(t^2/2)-t^2-2
numOfIterations = 2
h = (end-start)/numOfIterations

y00 = 1.0 #initial value
t00 = start;
soln = [[t00,y00]]

for i in range(numOfIterations):
    y00 = y00+h/2*(f(t00,y00) + f(t00+h,y00 + h*f(t00,y00)))
    t00=t00+h
    soln.append([t00,y00])
    print "%10r %20r %20r"%(t00,y00,solution(t00))

0.500000000000000 1.15625000000000 1.14944535920048 1.00000000000000 1.95898437500000 1.94616381210038