Editing Gaussian quadrature (section)

== Gauss–Legendre quadrature == <!-- The section "Other forms of Gaussian quadrature" below links to this section -->
{{Further|Gauss–Legendre quadrature}}
[[File:Legendrepolynomials6.svg|thumb|Graphs of Legendre polynomials (up to {{math|1=''n'' = 5)}}]]

For the simplest integration problem stated above, i.e., {{math|''f''(''x'')}} is well-approximated by polynomials on <math>[-1, 1]</math>, the associated orthogonal polynomials are [[Legendre polynomials]], denoted by {{math|''P''<sub>''n''</sub>(''x'')}}. With the {{mvar|n}}-th polynomial normalized to give {{math|1=''P''<sub>''n''</sub>(1) = 1}}, the {{mvar|i}}-th Gauss node, {{mvar|x<sub>i</sub>}}, is the {{mvar|i}}-th root of {{mvar|P<sub>n</sub>}} and the weights are given by the formula<ref>{{harvnb|Abramowitz|Stegun|1983|p=887}}</ref>
<math display="block"> w_i = \frac{2}{\left( 1 - x_i^2 \right) \left[P'_n(x_i)\right]^2}.</math>

Some low-order quadrature rules are tabulated below (over interval {{math|[−1, 1]}}, see the section below for other intervals).

{| class="wikitable" style="margin:auto; background:white; text-align:center;"
! Number of points, {{mvar|n}}
! colspan="2" | Points, {{mvar|x<sub>i</sub>}}
! colspan="2" | Weights, {{mvar|w<sub>i</sub>}}
|-
| 1
| colspan="2" | 0
| colspan="2" | 2
|-
| 2
| <math>\pm\frac{1}{\sqrt{3}}</math> || ±0.57735...
| colspan="2" | 1
|-
| rowspan="2" | 3
| colspan="2" | 0
| <math>\frac{8}{9}</math> || 0.888889...
|-
| <math>\pm\sqrt{\frac{3}{5}}</math> || ±0.774597...
| <math>\frac{5}{9}</math> || 0.555556...
|-
| rowspan="2" | 4
| <math>\pm\sqrt{\frac{3}{7} - \frac{2}{7}\sqrt{\frac{6}{5}}}</math> || ±0.339981...
| <math>\frac{18 + \sqrt{30}}{36}</math> || 0.652145...
|-
| <math>\pm\sqrt{\frac{3}{7} + \frac{2}{7}\sqrt{\frac{6}{5}}}</math> || ±0.861136...
| <math>\frac{18 - \sqrt{30}}{36}</math> || 0.347855...
|-
| rowspan="3" | 5
| colspan="2" | 0
| <math>\frac{128}{225}</math> || 0.568889...
|-
| <math>\pm\frac{1}{3}\sqrt{5 - 2\sqrt{\frac{10}{7}}}</math> || ±0.538469...
| <math>\frac{322 + 13\sqrt{70}}{900}</math> || 0.478629...
|-
| <math>\pm\frac{1}{3}\sqrt{5 + 2\sqrt{\frac{10}{7}}}</math> || ±0.90618...
| <math>\frac{322 - 13\sqrt{70}}{900}</math> || 0.236927...
|}