Many numerical integration formulas over a finite interval have the form

That is, the integral on the left can be approximated by evaluating the integrand *f* at particular nodes and taking the weighted sum, and the error is some multiple of a derivative of *f* evaluated at a point in the interval [*a*, *b*].

This post will give several examples, showing how they all fit into to framework above, then discuss the impossibility of extending this framework to infinite integrals.

## Simpson’s 3/8 rule

Simpson’s rule says

In this case the *x*‘s are evenly spaced a distance of *h* apart. The weights are 3*h*/8, 9*h*/8, 9*h*/8, and 3*h*/8.

We have *n* = 4, *k* = 4, and *C* = −3*h*^{5/80. }

We don’t know *a priori* what the value of ξ will be, only that there will be some value of ξ between *x*_{0} and *x*_{3} that makes the equation hold. If we have an explicit bound on the 4th derivative of *f* then we have an explicit bound on the error in approximating the integral by the weighted sum.

## Bode’s rules

A sequence of quadrature rules go by the name “Bode’s rule.” Here is one example, also known as Weddle’s rule.

As with Simpson’s 3/8 rule, you could map the formula for Bode’s rule(s) to the template at the top of the post.

## Gauss quadrature

Gauss’ formula says

for some ξ in [−1, 1]. Here the limits of integration are fixed, though you could use a change of variables to integrals over other finite integrals into this form.

Unlike Bode’s rule and Simpson’s rule, the *x*‘s are not evenly spaced but are the zeros of *P*_{n}, the Legendre polynomial of degree *n*. The weights are related to the *x*‘s and the derivative *P*′_{n} evaluated at the *x*‘s. The constant *C* is a complicated function of n but is independent of *f*.

Note that the error term involves the (2*n*)th derivative of *f*. This explains why Gaussian integration can be more accurate than other methods using the same number of function evaluations. The non-uniform spacing of the integration nodes enables higher-order error terms.

## Non-existence theorem

Although many integration rules over a finite interval have the form

Davis and Rabinowitz [1] proved that there cannot be an integration rule of the form

The proof, given in [1], takes only about one page. The entire article is a little more than a page, and about half the article is preamble to the proof.

## Related posts

- Three surprises with the trapezoid rule
- Romberg integration
- Lobatto quadrature
- Orthogonal polynomials

[1] P. J. Davis and P. Rabinowitz. On the Nonexistence of Simplex Integration Rules for Infinite Integrals. Mathematics of Computation, Vol. 26, No. 119 (July, 1972), pp. 687–688