Numerical Integration

Unbloated Reveal.js Example

The Midpoint Rule

Approximate an integral by sampling the midpoint of each subinterval.

$M_n(f) = h \sum_{k=0}^{n-1} f\left(a + \left(k+\frac{1}{2}\right)h\right), \qquad h = \frac{b-a}{n}.$

This is the same example used in the Unbloated technical book.

Reference Implementation

def midpoint(f, a, b, n):
    if n < 1:
        raise ValueError("n must be positive")
    h = (b - a) / n
    return h * sum(f(a + (k + 0.5) * h) for k in range(n))

Checking the Result

For the integral of $x^2$ on $[0,1]$ :

$n$	Estimate	Absolute error
1	0.250000	0.083333
2	0.312500	0.020833
4	0.328125	0.005208

Note

These values check one integrand. They do not establish an error bound for arbitrary continuous functions.

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