Rectangle rule

A formula for calculating an integral over a finite interval :

(*)

where and . Its algebraic degree of accuracy is 1 if and 0 otherwise.

The quadrature formula (*) is exact for the trigonometric functions

If , then (*) is exact for all trigonometric polynomials of order at most ; moreover, its trigonometric degree of accuracy is . No other quadrature formula with real nodes can have trigonometric degree of accuracy larger than , so that the rectangle rule with has the highest trigonometric degree of accuracy.

Let be the error of the rectangle rule, i.e. the difference between the left- and right-hand sides of (*). If the integrand is twice continuously differentiable on , then for one has

for some . If is a periodic function with period and has a continuous derivative of order (where is a natural number) on the entire real axis, then for any ,

for some , where is the Bernoulli number (cf. Bernoulli numbers).

References

[a1]	D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , Dover, reprint (1988) pp. 362ff