"Gaussian integration" redirects here. For the integral of a Gaussian function, see
Gaussian integral.
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration.
(See numerical integration for more on quadrature rules.)
An npoint Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points x_{i} and weights w_{i} for i = 1,...,n.
The domain of integration for such a rule is conventionally taken as [−1, 1],
so the rule is stated as
 $\backslash int\_\{1\}^1\; f(x)\backslash ,dx\; \backslash approx\; \backslash sum\_\{i=1\}^n\; w\_i\; f(x\_i).$
Gaussian quadrature as above will only produce accurate results if the function f(x) is well approximated by a polynomial function within the range [1,1]. The method is not, for example, suitable for functions with singularities. However, if the integrated function can be written as
$f(x)\; =\; \backslash omega(x)\; g(x)\backslash ,$,
where g(x) is approximately polynomial, and ω(x) is known, then there are alternative weights $w\_i\text{'}$ such that
 $\backslash int\_\{1\}^1\; f(x)\backslash ,dx\; =\; \backslash int\_\{1\}^1\; \backslash omega(x)\; g(x)\backslash ,dx\; \backslash approx\; \backslash sum\_\{i=1\}^n\; w\_i\text{'}\; g(x\_i).$
Common weighting functions include
$\backslash omega(x)=(1x^2)^\{1/2\}\backslash ,$
(Chebyshev–Gauss) and $\backslash omega(x)=e^\{x^2\}$ (Gauss–Hermite).
It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the roots of a polynomial belonging to a class of orthogonal polynomials.
Gauss–Legendre quadrature
For the simplest integration problem stated above, i.e. with $\backslash omega(x)=1$,
the associated polynomials are Legendre polynomials, P_{n}(x), and the method is usually known as Gauss–Legendre quadrature. With the n^{th} polynomial normalized to give P_{n}(1) = 1, the i^{th} Gauss node, x_{i}, is the i^{th} root of P_{n}; its weight is given by (Abramowitz & Stegun 1972, p. 887)
 $w\_i\; =\; \backslash frac\{2\}\{\backslash left(\; 1x\_i^2\; \backslash right)\; [P\text{'}\_n(x\_i)]^2\}.\; \backslash ,\backslash !$
Some loworder rules for solving the integration problem are listed below.
Number of points, n 
Points, x_{i } 
Weights, w_{i}

1 
0 
2

2 
$\backslash pm\; \backslash sqrt\{3\}/3$ 
1

3 
0 
^{8}⁄_{9}

$\backslash pm\backslash sqrt\{3/5\}$ 
^{5}⁄_{9}

4 
$\backslash pm\backslash sqrt\{\backslash Big(\; 3\; \; 2\backslash sqrt\{6/5\}\; \backslash Big)/7\}$ 
$\backslash tfrac\{18+\backslash sqrt\{30\}\}\{36\}$

$\backslash pm\backslash sqrt\{\backslash Big(\; 3\; +\; 2\backslash sqrt\{6/5\}\; \backslash Big)/7\}$ 
$\backslash tfrac\{18\backslash sqrt\{30\}\}\{36\}$ 

5 
0 
^{128}⁄_{225}

$\backslash pm\backslash tfrac13\backslash sqrt\{52\backslash sqrt\{10/7\}\}$ 
$\backslash tfrac\{322+13\backslash sqrt\{70\}\}\{900\}$

$\backslash pm\backslash tfrac13\backslash sqrt\{5+2\backslash sqrt\{10/7\}\}$ 
$\backslash tfrac\{32213\backslash sqrt\{70\}\}\{900\}$

Change of interval
An integral over [a, b] must be changed into an integral over [−1, 1] before applying the Gaussian quadrature rule. This change of interval can be done in the following way:
 $$
\int_a^b f(x)\,dx = \frac{ba}{2} \int_{1}^1 f\left(\frac{ba}{2}z
+ \frac{a+b}{2}\right)\,dz.
After applying the Gaussian quadrature rule, the following approximation is:
 $$
\int_a^b f(x)\,dx \approx \frac{ba}{2} \sum_{i=1}^n w_i f\left(\frac{ba}{2}z_i + \frac{a+b}{2}\right).
Other forms
The integration problem can be expressed in a slightly more general way by introducing a positive weight function ω into the integrand,
and allowing an interval other than [−1, 1].
That is, the problem is to calculate
 $\backslash int\_a^b\; \backslash omega(x)\backslash ,f(x)\backslash ,dx$
for some choices of a, b, and ω.
For a = −1, b = 1, and ω(x) = 1,
the problem is the same as that considered above.
Other choices lead to other integration rules.
Some of these are tabulated below.
Equation numbers are given for Abramowitz and Stegun (A & S).
Interval 
ω(x) 
Orthogonal polynomials 
A & S 
For more information, see ...

[−1, 1] 
$1\backslash ,$ 
Legendre polynomials 
25.4.29 
Section Gauss–Legendre quadrature, above

(−1, 1) 
$(1x)^\backslash alpha\; (1+x)^\backslash beta,\backslash quad\; \backslash alpha,\; \backslash beta\; >\; 1\backslash ,$ 
Jacobi polynomials 
25.4.33 ($\backslash beta=0$) 
Gauss–Jacobi quadrature

(−1, 1) 
$\backslash frac\{1\}\{\backslash sqrt\{1\; \; x^2\}\}$ 
Chebyshev polynomials (first kind) 
25.4.38 
Chebyshev–Gauss quadrature

[−1, 1] 
$\backslash sqrt\{1\; \; x^2\}$ 
Chebyshev polynomials (second kind) 
25.4.40 
Chebyshev–Gauss quadrature

[0, ∞) 
$e^\{x\}\backslash ,$ 
Laguerre polynomials 
25.4.45 
Gauss–Laguerre quadrature

[0, ∞) 
$x^\backslash alpha\; e^\{x\}\backslash ,$ 
Generalized Laguerre polynomials 

Gauss–Laguerre quadrature

(−∞, ∞) 
$e^\{x^2\}$ 
Hermite polynomials 
25.4.46 
Gauss–Hermite quadrature

Fundamental theorem
Let $p\_n$ be a nontrivial polynomial of degree n such that
 $$
\int_a^b \omega(x) \, x^k p_n(x) \, dx = 0, \quad \text{for all }k=0,1,\ldots,n1.
If we pick the n nodes x_{i} to be the zeros of p_{n}, then there exist n weights w_{i} which make the Gaussquadrature computed integral exact for all polynomials $h(x)$ of degree 2n − 1 or less. Furthermore, all these nodes x_{i} will lie in the open interval (a, b) (Stoer & Bulirsch 2002, pp. 172–175).
The polynomial $p\_n$ is said to be an orthogonal polynomial of degree n associated to the weight function $\backslash omega\; (x)$. It is unique up to a constant normalization factor. The idea underlying the proof is that, because of its sufficiently low degree, $h(x)$ can be divided by $p\_n(x)$ to produce a quotient $q(x)$ of degree strictly lower than n, and a remainder $r(x)$ of still lower degree, so that both will be orthogonal to $p\_n(x)$, by the defining property of $p\_n(x)$. Thus
 $\backslash int\_a^b\; \backslash omega(x)\backslash ,h(x)\backslash ,dx\; =\; \backslash int\_a^b\; \backslash omega(x)\backslash ,r(x)\backslash ,dx.$
Because of the choice of nodes x_{i}, the corresponding relation
 $\backslash sum\_\{i=1\}^n\; w\_i\; h(x\_i)\; =\; \backslash sum\_\{i=1\}^n\; w\_i\; r(x\_i)$
holds also. The exactness of the computed integral for $h(x)$ then follows from corresponding exactness for polynomials of degree only n or less (as is $r(x)$).
General formula for the weights
The weights can be expressed as
 $w\_\{i\}\; =\; \backslash frac\{a\_\{n\}\}\{a\_\{n1\}\}\backslash frac\{\backslash int\_\{a\}^\{b\}\backslash omega(x)p\_\{n1\}(x)^\{2\}dx\}\{p\text{'}\_\{n\}(x\_\{i\})p\_\{n1\}(x\_\{i\})\}$ (1)
where $a\_\{k\}$ is the coefficient of $x^\{k\}$ in $p\_\{k\}(x)$. To prove this, note that using Lagrange interpolation one can express $r(x)$ in terms of $r(x\_\{i\})$ as
 $r(x)\; =\; \backslash sum\_\{i=1\}^\{n\}r(x\_\{i\})\backslash prod\_\{\backslash begin\{smallmatrix\}1\backslash leq\; j\backslash leq\; n\backslash \backslash j\backslash neq\; i\backslash end\{smallmatrix\}\}\backslash frac\{xx\_\{j\}\}\{x\_\{i\}x\_\{j\}\}$
because r(x) has degree less than n and is thus fixed by the values it attains at n different points. Multiplying both sides by $\backslash omega(x)$ and integrating from a to b yields
 $\backslash int\_\{a\}^\{b\}\backslash omega(x)r(x)dx=\; \backslash sum\_\{i=1\}^\{n\}r(x\_\{i\})\backslash int\_\{a\}^\{b\}\backslash omega(x)\backslash prod\_\{\backslash begin\{smallmatrix\}1\backslash leq\; j\backslash leq\; n\backslash \backslash j\backslash neq\; i\backslash end\{smallmatrix\}\}\backslash frac\{xx\_\{j\}\}\{x\_\{i\}x\_\{j\}\}dx$
The weights $w\_\{i\}$ are thus given by
 $w\_\{i\}\; =\; \backslash int\_\{a\}^\{b\}\backslash omega(x)\backslash prod\_\{\backslash begin\{smallmatrix\}1\backslash leq\; j\backslash leq\; n\backslash \backslash j\backslash neq\; i\backslash end\{smallmatrix\}\}\backslash frac\{xx\_\{j\}\}\{x\_\{i\}x\_\{j\}\}dx$
This integral expression for $w\_\{i\}$ can be expressed in terms of the orthogonal polynomials $p\_\{n\}(x)$ and $p\_\{n+1\}(x)$ as follows.
We can write
 $\backslash prod\_\{\backslash begin\{smallmatrix\}1\backslash leq\; j\backslash leq\; n\backslash \backslash j\backslash neq\; i\backslash end\{smallmatrix\}\}\backslash left(xx\_\{j\}\backslash right)\; =\; \backslash frac\{\backslash prod\_\{1\backslash leq\; j\backslash leq\; n\}\; \backslash left(x\; \; x\_\{j\}\backslash right)\}\{xx\_\{i\}\}\; =\; \backslash frac\{p\_\{n\}(x)\}\{a\_\{n\}\backslash left(xx\_\{i\}\backslash right)\}$
where $a\_\{n\}$ is the coefficient of $x^\{n\}$ in $p\_\{n\}(x)$. Taking the limit of x to $x\_\{i\}$ yields using L'Hôpital's rule
 $\backslash prod\_\{\backslash begin\{smallmatrix\}1\backslash leq\; j\backslash leq\; n\backslash \backslash j\backslash neq\; i\backslash end\{smallmatrix\}\}\backslash left(x\_\{i\}x\_\{j\}\backslash right)\; =\; \backslash frac\{p\text{'}\_\{n\}(x\_\{i\})\}\{a\_\{n\}\}$
We can thus write the integral expression for the weights as
 $w\_\{i\}\; =\; \backslash frac\{1\}\{p\text{'}\_\{n\}(x\_\{i\})\}\backslash int\_\{a\}^\{b\}\backslash omega(x)\backslash frac\{p\_\{n\}(x)\}\{xx\_\{i\}\}dx$ (2)
In the integrand, writing
 $\backslash frac\{1\}\{xx\_\{i\}\}\; =\; \backslash frac\{1\backslash left(\backslash frac\{x\}\{x\_\{i\}\}\backslash right)^\{k\}\}\{xx\_\{i\}\}\; +\; \backslash left(\backslash frac\{x\}\{x\_\{i\}\}\backslash right)^\{k\}\; \backslash frac\{1\}\{xx\_\{i\}\}$
yields
 $\backslash int\_\{a\}^\{b\}\backslash omega(x)\backslash frac\{x^\{k\}p\_\{n\}(x)\}\{xx\_\{i\}\}dx=\; x\_\{i\}^\{k\}\backslash int\_\{a\}^\{b\}\backslash omega(x)\backslash frac\{p\_\{n\}(x)\}\{xx\_\{i\}\}dx$
provided $k\backslash leq\; n$, because $\backslash frac\{1\backslash left(\backslash frac\{x\}\{x\_\{i\}\}\backslash right)^\{k\}\}\{xx\_\{i\}\}$ is a polynomial of degree k1 which is then orthogonal to $p\_\{n\}(x)$. So, if q(x) is a polynomial of at most nth degree we have
 $\backslash int\_\{a\}^\{b\}\backslash omega(x)\backslash frac\{p\_\{n\}(x)\}\{xx\_\{i\}\}dx=\backslash frac\{1\}\{q(x\_\{i\})\}\backslash int\_\{a\}^\{b\}\backslash omega(x)\backslash frac\{q(x)p\_\{n\}(x)\}\{xx\_\{i\}\}dx$
We can evaluate the integral on the right hand side for $q(x)\; =\; p\_\{n1\}(x)$ as follows. Because $\backslash frac\{p\_\{n\}(x)\}\{xx\_\{i\}\}$ is a polynomial of degree n1, we have
 $\backslash frac\{p\_\{n\}(x)\}\{xx\_\{i\}\}\; =\; a\_\{n\}x^\{n1\}\; +\; s(x)$
where s(x) is a polynomial of degree n2. Since s(x) is orthogonal to $p\_\{n1\}(x)$ we have
 $\backslash int\_\{a\}^\{b\}\backslash omega(x)\backslash frac\{p\_\{n\}(x)\}\{xx\_\{i\}\}dx=\backslash frac\{a\_\{n\}\}\{p\_\{n1\}(x\_\{i\})\}\backslash int\_\{a\}^\{b\}\backslash omega(x)p\_\{n1\}(x)x^\{n1\}dx$
We can then write
 $x^\{n1\}\; =\; \backslash left(x^\{n1\}\; \; \backslash frac\{p\_\{n1\}(x)\}\{a\_\{n1\}\}\backslash right)\; +\; \backslash frac\{p\_\{n1\}(x)\}\{a\_\{n1\}\}$
The term in the brackets is a polynomial of degree n2, which is therefore orthogonal to $p\_\{n1\}(x)$. The integral can thus be written as
 $\backslash int\_\{a\}^\{b\}\backslash omega(x)\backslash frac\{p\_\{n\}(x)\}\{xx\_\{i\}\}dx=\backslash frac\{a\_\{n\}\}\{a\_\{n1\}p\_\{n1\}(x\_\{i\})\}\backslash int\_\{a\}^\{b\}\backslash omega(x)p\_\{n1\}(x)^\{2\}dx$
According to Eq. (2), the weights are obtained by dividing this by $p\text{'}\_\{n\}(x\_\{i\})$ and that yields the expression in Eq. (1).
Proof that the weights are positive
Consider the following polynomial of degree 2n2
 $f(x)\; =\; \backslash prod\_\{\backslash begin\{smallmatrix\}1\backslash leq\; j\backslash leq\; n\backslash \backslash j\backslash neq\; i\backslash end\{smallmatrix\}\}(xx\_\{j\})^\{2\}$
where as above the $x\_\{j\}$ are the roots of the polynomial $p\_\{n\}(x)$. Since the degree of f(x) is less than 2n1, the Gaussian quadrature formula involving the weights and nodes obtained from $p\_\{n\}(x)$ applies. Since $f(x\_\{j\})=0$ for j not equal to i, we have
 $\backslash int\_\{a\}^\{b\}\backslash omega(x)f(x)=\backslash sum\_\{j=1\}^\{N\}w\_\{j\}f(x\_\{j\})\; =\; w\_\{i\}\; f(x\_\{i\}).$
Since both $\backslash omega(x)$ and f(x) are nonnegative functions, it follows that $w\_\{i\}>0$.
Computation of Gaussian quadrature rules
For computing the nodes $x\_i$ and weights $w\_i$ of Gaussian quadrature rules, the fundamental tool is the threeterm recurrence relation satisfied by the set of orthogonal polynomials associated to the corresponding weight function. For n points, these nodes and weights can be computed in O(n^{2}) operations by an algorithm derived by Gautschi (1968).
Gautschi's theorem
Gautschi's theorem (Gautschi, 1968) states that orthogonal polynomials $p\_r$ with $(p\_r,p\_s)=0$ for $r\backslash ne\; s$ for a scalar product $(\; ,\; )$ to be specified later, degree$(p\_r)=r$ and leading coefficient one (i.e. monic orthogonal polynomials) satisfy the recurrence relation
 $p\_\{r+1\}(x)=(xa\_\{r,r\})p\_r(x)a\_\{r,r1\}p\_\{r1\}(x)\backslash ldotsa\_\{r,0\}p\_0(x)$
for $r=0,1,\backslash ldots,n1$ where $n$ is the maximal degree which can be taken to
be infinity, and where $a\_\{r,s\}=(xp\_r,p\_s)/(p\_s,p\_s)$. First of all, it is obvious that the polynomials defined by the recurrence relation starting with $p\_0(x)=1$ have leading coefficient one and correct degree. Given the starting point by $p\_0$, the orthogonality of $p\_r$ can be shown by induction. For $r=s=0$ one has
 $(p\_1,p\_0)=((xa\_\{0,0\}p\_0,p\_0)=(xp\_0,p\_0)a\_\{0,0\}(p\_0,p\_0)=(xp\_0,p\_0)(xp\_0,p\_0)=0.$
Now if $p\_0,p\_1,\backslash ldots,p\_r$ are orthogonal, then also $p\_\{r+1\}$, because in
 $(p\_\{r+1\},p\_s)=(xp\_r,p\_s)a\_\{r,r\}(p\_r,p\_s)a\_\{r,r1\}(p\_\{r1\},p\_s)\backslash ldotsa\_\{r,0\}(p\_0,p\_s)$
all scalar products vanish except for the first one and the one where $p\_s$ meets the same
orthogonal polynom. Therefore,
 $(p\_\{r+1\},p\_s)=(xp\_r,p\_s)a\_\{r,s\}(p\_s,p\_s)=(xp\_r,p\_s)(xp\_r,p\_s)=0.$
However, if the scalar product satisfies $(xf,g)=(f,xg)$ (which is the case for Gaussian quadrature), the recurrence relation reduces to a threeterm recurrence relation: For $s\backslash le\; r1$, $xp\_s$ is a polynomial of degree less or equal to $r1$. On the other hand, $p\_r$ is orthogonal to every polynomial of degree less or equal to $r1$. Therefore, one has $(xp\_r,p\_s)=(p\_r,xp\_s)=0$ and $a\_\{r,s\}=0$ for $smath>.\; The\; recurrence\; relation\; then\; simplifies\; to$ p\_\{r+1\}(x)=(xa\_\{r,r\})p\_r(x)a\_\{r,r1\}p\_\{r1\}(x)$or$
 $p\_\{r+1\}(x)=(xa\_r)p\_r(x)b\_rp\_\{r1\}(x)$
(with the convention $p\_\{1\}(x)\backslash equiv\; 0$) where
 $a\_r:=\backslash frac\{(xp\_r,p\_r)\}\{(p\_r,p\_r)\},\backslash qquad$
b_r:=\frac{(xp_r,p_{r1})}{(p_{r1},p_{r1})}=\frac{(p_r,p_r)}{(p_{r1},p_{r1})}
(the last because of $(xp\_r,p\_\{r1\})=(p\_r,xp\_\{r1\})=(p\_r,p\_r)$, since $xp\_\{r1\}$ differs from $p\_r$ by a degree less than $r$).
The GolubWelsch algorithm
The threeterm recurrence relation can be written in the matrix form $J\backslash tilde\{P\}=x\backslash tilde\{P\}$
where $\backslash tilde\{P\}=[p\_0(x),p\_1(x),...,p\_\{n1\}(x)]^\{T\}$ and $J$ is the socalled Jacobi matrix:
 $$
\mathbf{J}=\left(
\begin{array}{llllll}
a_0 & 1 & 0 & \ldots & \ldots & \ldots\\
b_1 & a_1 & 1 & 0 & \ldots & \ldots \\
0 & b_2 & a_2 & 1 & 0 & \ldots \\
0 & \ldots & \ldots & \ldots & \ldots & 0 \\
\ldots & \ldots & 0 & b_{n2} & a_{n2} & 1 \\
\ldots & \ldots & \ldots & 0 & b_{n1} & a_{n1}
\end{array}
\right).
The zeros $x\_j$ of the polynomials up to degree $n$ which are used as nodes for the Gaussian quadrature can be found by computing the eigenvalues of this tridiagonal matrix. This procedure is known as Golub–Welsch algorithm.
For computing the weights and nodes, it is preferable to consider the symmetric tridiagonal matrix $\backslash mathcal\{J\}$ with elements $\backslash mathcal\{J\}\_\{i,i\}=J\_\{i,i\}=a\_\{i1\},\backslash ,\; i=1,\backslash ldots,n$ and $\backslash mathcal\{J\}\_\{i1,i\}=\backslash mathcal\{J\}\_\{i,i1\}=\backslash sqrt\{J\_\{i,i1\}J\_\{i1,i\}\}=\backslash sqrt\{b\_\{i1\}\},\backslash ,\; i=2,\backslash ldots,n.$
$\backslash mathbf\{J\}$ and $\backslash mathcal\{J\}$ are similar matrices and therefore have the same eigenvalues (the nodes). The weights can be computed from the corresponding eigenvectors: If $\backslash phi^\{(j)\}$ is a normalized eigenvector (i.e., an eigenvector with euclidean norm equal to one) associated to the eigenvalue $x\_j$, the corresponding weight can be computed from
the first component of this eigenvector, namely:
 $$
w_j=\mu_0 \left(\phi_1^{(j)}\right)^2
where $\backslash mu\_0$ is the integral of the weight function
 $$
\mu_0=\int_a^b \omega(x) dx.
See, for instance, (Gil, Segura & Temme 2007) for further details.
There are alternative methods for obtaining the same weights and nodes in O(n) operations using the Prüfer Transform.
Error estimates
The error of a Gaussian quadrature rule can be stated as follows (Stoer & Bulirsch 2002, Thm 3.6.24).
For an integrand which has 2n continuous derivatives,
 $\backslash int\_a^b\; \backslash omega(x)\backslash ,f(x)\backslash ,dx\; \; \backslash sum\_\{i=1\}^n\; w\_i\backslash ,f(x\_i)$
= \frac{f^{(2n)}(\xi)}{(2n)!} \, (p_n,p_n)
for some ξ in (a, b), where p_{n} is the monic (i.e. the leading coefficient is 1) orthogonal polynomial of degree n and where
 $(f,g)\; =\; \backslash int\_a^b\; \backslash omega(x)\; f(x)\; g(x)\; \backslash ,\; dx\; .\; \backslash ,\backslash !$
In the important special case of ω(x) = 1, we have the error estimate (Kahaner, Moler & Nash 1989, §5.2)
 $\backslash frac\{(ba)^\{2n+1\}\; (n!)^4\}\{(2n+1)[(2n)!]^3\}\; f^\{(2n)\}\; (\backslash xi)\; ,\; \backslash qquad\; a\; <\; \backslash xi\; <\; b\; .\; \backslash ,\backslash !$
Stoer and Bulirsch remark that this error estimate is inconvenient in practice,
since it may be difficult to estimate the order 2n derivative,
and furthermore the actual error may be much less than a bound established by the derivative.
Another approach is to use two Gaussian quadrature rules of different orders,
and to estimate the error as the difference between the two results.
For this purpose, Gauss–Kronrod quadrature rules can be useful.
Important consequence of the above equation is that Gaussian quadrature of order n is accurate for all polynomials up to degree 2n–1.
Gauss–Kronrod rules
If the interval [a, b] is subdivided,
the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at zero for odd numbers),
and thus the integrand must be evaluated at every point.
Gauss–Kronrod rules are extensions of Gauss quadrature rules generated by adding $n+1$ points to an $n$point rule in such a way that the resulting rule is of order $3n+1$.
This allows for computing higherorder estimates while reusing the function values of a lowerorder estimate.
The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error.
Gauss–Lobatto rules
Also known as Lobatto quadrature (Abramowitz & Stegun 1972, p. 888), 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 2n–3, where n is the number of integration points.
Lobatto quadrature of function f(x) on interval [–1, +1]:
 $$
\int_{1}^1 {f(x) \, dx} =
\frac {2} {n(n1)}[f(1) + f(1)] +
\sum_{i = 2} ^{n1} {w_i f(x_i)} + R_n.
Abscissas: $x\_i$ is the $(i1)$^{st} zero of $P\text{'}\_\{n1\}(x)$.
Weights:
 $$
w_i = \frac{2}{n(n1)[P_{n1}(x_i)]^2} \quad (x_i \ne \pm 1).
Remainder:
$$
R_n = \frac
{ n (n1)^3 2^{2n1} [(n2)!]^4}
{(2n1) [(2n2)!]^3}
f^{(2n2)}(\xi), \quad (1 < \xi < 1)
Some of the weights are:
Number of points, n 
Points, x_{i } 
Weights, w_{i}

$3$ 
$0$ 
$\backslash frac\{4\}\{3\}$

$\backslash pm\; 1$ 
$\backslash frac\{1\}\{3\}$

$4$ 
$\backslash pm\; \backslash sqrt\{\backslash frac\; \{1\}\; \{5\}\}$ 
$\backslash frac\{5\}\{6\}$

$\backslash pm\; 1$ 
$\backslash frac\{1\}\{6\}$

$5$ 
$0$ 
$\backslash frac\{32\}\{45\}$

$\backslash pm\backslash sqrt\{\backslash frac\; \{3\}\; \{7\}\}$ 
$\backslash frac\{49\}\{90\}$

$\backslash pm\; 1$ 
$\backslash frac\{1\}\{10\}$

See also
References
External links

 ALGLIB contains a collection of algorithms for numerical integration (in C# / C++ / Delphi / Visual Basic / etc.)
 GNU Scientific Library)
 From Lobatto Quadrature to the Euler constant e
 Gaussian Quadrature Rule of Integration  Notes, PPT, Matlab, Mathematica, Maple, Mathcad at Holistic Numerical Methods Institute
 MathWorld.
 Wolfram Demonstrations Project.
 Tabulated weights and abscissae with Mathematica source code, high precision (16 and 256 decimal places) LegendreGaussian quadrature weights and abscissas, for n=2 through n=64, with Mathematica source code.
 Mathematica source code distributed under the GNU LGPL for abscissas and weights generation for arbitrary weighting functions W(x), integration domains and precisions.
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