I'm trying to implement the trapezoidal rule in Python 2.7.2. I've written the following function:
def trapezoidal(f, a, b, n):
h = float(b - a) / n
s = 0.0
s += h * f(a)
for i in range(1, n):
s += 2.0 * h * f(a + i*h)
s += h * f(b)
return s
However, f(lambda x:x**2, 5, 10, 100) returns 583.333 (it's supposed to return 291.667), so clearly there is something wrong with my script. I can't spot it though.
You are off by a factor of two. Indeed, the Trapezoidal Rule as taught in math class would use an increment like
s += h * (f(a + i*h) + f(a + (i-1)*h))/2.0
(f(a + i*h) + f(a + (i-1)*h))/2.0
is averaging the height of the function at two adjacent points on the grid.
Since every two adjacent trapezoids have a common edge, the formula above requires evaluating the function twice as often as necessary.
A more efficient implementation (closer to what you posted), would combine common terms from adjacent iterations of the for-loop
:
f(a + i*h)/2.0 + f(a + i*h)/2.0 = f(a + i*h)
to arrive at:
def trapezoidal(f, a, b, n):
h = float(b - a) / n
s = 0.0
s += f(a)/2.0
for i in range(1, n):
s += f(a + i*h)
s += f(b)/2.0
return s * h
print( trapezoidal(lambda x:x**2, 5, 10, 100))
which yields
291.66875