---
tag: 4T1MNB8Z11
kind: definition
name: Quadrature rule
msc: [65D32, 41A55]
defines: ADSVMCY77A
uses: [021WEM680W, 7HYF6YG0RS, 0X3WGX95FE]
sources:
  - url: https://en.wikipedia.org/wiki/Quadrature_(mathematics)
---

## Definition

Let $a < b$ be real numbers and let $n \ge 1$.
A *quadrature rule* on $[a, b]$ with $n$ nodes consists of distinct nodes $x_1, \dots, x_n \in [a, b]$ and real weights $w_1, \dots, w_n$.
It assigns to each [[continuous function|0X3WGX95FE]] $f \colon [a, b] \to \R$ the number
$$Q(f) = \sum_{i=1}^{n} w_i\, f(x_i),$$
which is used as an approximation of the [[definite integral|021WEM680W]] $I(f) = \int_a^b f(x)\, dx$.

## Remarks

A quadrature rule is a [[numerical integration|7HYF6YG0RS]] 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 $\abs{Q(f)} \le \big(\sum_i w_i\big) \max \abs{f}$.