Editing Gaussian quadrature (section)

=== Gauss–Lobatto rules ===
Also known as '''Lobatto quadrature''',<ref>{{harvnb|Abramowitz|Stegun|1983|p=888}}</ref> named after Dutch mathematician [[Rehuel Lobatto]]. It is similar to Gaussian quadrature with the following differences:
# The integration points include the end points of the integration interval.
# It is accurate for polynomials up to degree {{math|2''n'' − 3}}, where {{mvar|n}} is the number of integration points.<ref>{{harvnb|Quarteroni|Sacco|Saleri|2000}}</ref>

Lobatto quadrature of function {{math|''f''(''x'')}} on interval {{math|[−1, 1]}}:
<math display="block">\int_{-1}^1 {f(x) \, dx} = \frac {2} {n(n-1)}[f(1) + f(-1)] + \sum_{i = 2}^{n-1} {w_i f(x_i)} + R_n.</math>

Abscissas: {{mvar|x<sub>i</sub>}} is the <math>(i - 1)</math>st zero of <math>P'_{n-1}(x)</math>, here <math>P_m(x)</math> denotes the standard Legendre polynomial of {{mvar|m}}-th degree and the dash denotes the derivative.

Weights:
<math display="block">w_i = \frac{2}{n(n - 1)\left[P_{n-1}\left(x_i\right)\right]^2}, \qquad x_i \ne \pm 1.</math>

Remainder:
<math display="block">R_n = \frac{-n\left(n - 1\right)^3 2^{2n-1} \left[\left(n - 2\right)!\right]^4}{(2n-1) \left[\left(2n - 2\right)!\right]^3} f^{(2n-2)}(\xi), \qquad -1 < \xi < 1.</math>

Some of the weights are:

{| class="wikitable" style="margin:auto; background:white; text-align:center;"
! Number of points, ''n''
! Points, {{mvar|x<sub>i</sub>}}
! Weights, {{mvar|w<sub>i</sub>}}
|-
| rowspan="2" | <math>3</math>
| <math>0</math> || <math>\frac{4}{3}</math>
|-
| <math>\pm 1</math> || <math>\frac{1}{3}</math>
|-
| rowspan="2" | <math>4</math>
| <math>\pm \sqrt{\frac{1}{5}}</math> || <math>\frac{5}{6}</math>
|-
| <math>\pm 1</math> || <math>\frac{1}{6}</math>
|-
| rowspan="3" | <math>5</math>
| <math>0</math> || <math>\frac{32}{45}</math>
|-
| <math>\pm\sqrt{\frac{3}{7}}</math> || <math>\frac{49}{90}</math>
|-
| <math>\pm 1</math> || <math>\frac{1}{10}</math>
|-
| rowspan="3" | <math>6</math>
| <math>\pm\sqrt{\frac{1}{3}-\frac{2\sqrt{7}}{21}}</math> || <math>\frac{14+\sqrt{7}}{30}</math>
|-
| <math>\pm\sqrt{\frac{1}{3} + \frac{2\sqrt{7}}{21}}</math> || <math>\frac{14 - \sqrt{7}}{30}</math>
|-
| <math>\pm 1</math> || <math>\frac{1}{15}</math>
|-
| rowspan="4" | <math>7</math>
| <math>0</math> || <math>\frac{256}{525}</math>
|-
| <math>\pm\sqrt{\frac{5}{11}-\frac{2}{11}\sqrt{\frac{5}{3}}}</math> || <math>\frac{124 + 7\sqrt{15}}{350}</math>
|-
| <math>\pm\sqrt{\frac{5}{11} + \frac{2}{11}\sqrt{\frac{5}{3}}}</math> || <math>\frac{124 - 7\sqrt{15}}{350}</math>
|-
| <math>\pm 1</math> || <math>\frac{1}{21}</math>
|}

An adaptive variant of this algorithm with 2 interior nodes<ref>{{harvnb|Gander|Gautschi|2000}}</ref> is found in [[GNU Octave]] and [[MATLAB]] as <code>quadl</code> and <code>integrate</code>.<ref>{{harvnb|MathWorks|2012}}</ref><ref>{{harvnb|Eaton|Bateman|Hauberg|Wehbring|2018}}</ref>