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Gaussian quadrature

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 n-point 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

int_{-1}^1 f(x),dx approx 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, for example, not suitable for functions with singularities. However, if the integrated function can be written as $f(x) = W(x) g(x),$ , where g(x) is approximately polynomial, and W(x) is known, then there are alternative weights $w_i$ such that

int_{-1}^1 f(x),dx = int_{-1}^1 W(x) g(x),dx approx sum_{i=1}^n w_i g(x_i).

Common weighting functions include $W(x)=(1-x^2)^{-1/2},$ (Gauss-Chebyshev) and $W(x)=e^{-x^2}$ (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.

Rules for the basic problem

For the integration problem stated above, the associated polynomials are Legendre polynomials, P_n(x). 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

w_i = frac{2}{left(1-x_i^2 right) (P'_n(x_i))^2} ,!

Some low-order rules for solving the integration problem are listed below.

Number of points, n	Points, x_i	Weights, wi
1	0	2
2	$pmsqrt{1/3}$	1
3	0	⁸⁄₉
3	$pmsqrt{3/5}$	⁵⁄₉
4	$pmsqrt{Big(3 - 2sqrt{6/5} Big)/7}$	$tfrac{18+sqrt{30}}{36}$
4	$pmsqrt{Big(3 + 2sqrt{6/5} Big)/7}$	$tfrac{18-sqrt{30}}{36}$
5	0	¹²⁸⁄₂₂₅
	$pmtfrac13sqrt{5-2sqrt{10/7}}$	$tfrac{322+13sqrt{70}}{900}$
	$pmtfrac13sqrt{5+2sqrt{10/7}}$	$tfrac{322-13sqrt{70}}{900}$

Change of interval for Gaussian quadrature

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{b-a}{2} int_{-1}^1 fleft(frac{b-a}{2}x + frac{a+b}{2}right),dx

After applying the Gaussian quadrature rule, the following approximation is obtained:

frac{b-a}{2} sum_{i=1}^n w_i fleft(frac{b-a}{2}x_i + frac{a+b}{2}right)

Other forms of Gaussian quadrature

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

int_a^b omega(x),f(x),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]	$1,$	Legendre polynomials	25.4.29	Section Rules for the basic problem, above
(−1, 1)	$(1-x)^alpha (1+x)^beta,quad alpha, beta > -1,$	Jacobi polynomials	25.4.33 ( $beta=0$ )
(−1, 1)	$frac{1}{sqrt{1 - x^2}}$	Chebyshev polynomials (first kind)	25.4.38	Chebyshev–Gauss quadrature
[−1, 1]	$sqrt{1 - x^2}$	Chebyshev polynomials (second kind)	25.4.40	Chebyshev–Gauss quadrature
[0, ∞)	$e^{-x},$	Laguerre polynomials	25.4.45	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,n-1.

If we pick the nodes to be the zeros of $p_n$ , then there exist weights w_i which make the computed integral exact for all polynomials of degree 2n − 1 or less. Furthermore, all these nodes will lie in the open interval (a, b) .

The polynomial $p_n$ is said to be an orthogonal polynomial of degree n associated to the weight function $omega (x)$ . It is unique up to a constant normalization factor.

Computation of Gaussian quadrature rules

For computing the nodes $x_i$ and weights $w_i$ of Gaussian quadrature rules, the fundamental tool is the three-term recurrence relation satisfied by the set of orthogonal polynomials associated to the corresponding weight function.

If, for instance, $p_n$ is the monic orthogonal polynomial of degree n (the orthogonal polynomial of degree n with the highest degree coefficient equal to one), one can show that such orthogonal polynomials are related through the recurrence relation

p_{n+1}(x)+(B_n-x)p_n (x)+A_n p_{n-1}(x)=0, qquad n=1,2,ldots

From this, nodes and weights can be computed from the eigenvalues and eigenvectors of an associated linear algebra problem. This is usually named as the Golub–Welsch algorithm .

The starting idea comes from the observation that, if $x_i$ is a root of the orthogonal polynomial $p_n$ then, using the previous recurrence formula for $k=0,1,ldots, n-1$ and because $p_n (x_j)=0$ , we have

$Jtilde{P}=x_j tilde{P}$

where $tilde{P}=[p_0 (x_j),p_1 (x_j),...,p_{n-1}(x_j)]^{T}$

and $J$ is the so-called Jacobi matrix:

$mathbf{J}=left(
begin{array}{llllll}
B_0 & 1 & 0 & ldots & ldots & ldots
A_1 & B_1 & 1 & 0 & ldots & ldots
0 & A_2 & B_2 & 1 & 0 & ldots
ldots & ldots & ldots & ldots & ldots & ldots
ldots & ldots & ldots & A_{n-2} & B_{n-2} & 1
ldots & ldots & ldots & ldots & A_{n-1} & B_{n-1}
end{array}
right).$

The nodes of gaussian quadrature can therefore be computed as the eigenvalues of a tridiagonal matrix.

For computing the weights and nodes, it is preferable to consider the symmetric tridiagonal matrix $mathcal{J}$ with elements $mathcal{J}_{i,i}=J_{i,i}$ , $i=1,ldots,n$ and $mathcal{J}_{i-1,i}=mathcal{J}_{i,i-1}=sqrt{J_{i,i-1}J_{i-1,i}},, i=2,ldots,n$ . $mathbf{J}$ and $mathcal{J}$ are equivalent and therefore have the same eigenvalues (the nodes). The weights can be computed from the matrix $J$ . If $phi^{(j)}$ 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 $mu_0$ is the integral of the weight function

$mu_0=int_a^b w(x) dx.$

See, for instance, for further details.

Error estimates

The error of a Gaussian quadrature rule can be stated as follows . For an integrand which has 2n continuous derivatives,

int_a^b omega(x),f(x),dx - sum_{i=1}^n w_i,f(x_i)

= frac{f^{(2n)}(xi)}{(2n)!} , (p_n,p_n)

for some ξ in (a, b), where p_n is the orthogonal polynomial of order n and where

(f,g) = int_a^b omega(x) f(x) g(x) , dx . ,!

In the important special case of ω(x) = 1, we have the error estimate

frac{(b-a)^{2n+1} (n!)^4}{(2n+1)[(2n)!]^3} f^{(2n)} (xi) , qquad a < xi < b . ,!

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.

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 higher-order estimates while re-using the function values of a lower-order estimate. The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error.