Quadrature rule

Remarks

A quadrature rule is a numerical integration method, and $Q$ is a linear functional on the continuous functions. More generally, a rule for a weight function $\omega \ge 0$ approximates $\int_a^b f(x)\, \omega(x)\, dx$ by the same kind of sum. Example: the midpoint rule has $n = 1$, $x_1 = (a + b)/2$, $w_1 = b - a$; for $f = 1$ it gives $b - a = I(1)$. Non-example: $f \mapsto (b - a) \max_{[a, b]} f$ is not a quadrature rule, because it is not of the form $\sum w_i f(x_i)$ with fixed nodes; it is not even linear. A rule is called interpolatory if $Q(f)$ is the integral of the polynomial that interpolates $f$ at the nodes. A rule with nonnegative weights is stable, since $\abs{Q(f)} \le \big(\sum_i w_i\big) \max \abs{f}$.