Gaussian quadrature

In numerical analysis, an -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for.
The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826. The most common domain of integration for such a rule is taken as, so the rule is stated as
which is exact for polynomials of degree or less. This exact rule is known as the Gauss–Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if is well-approximated by a polynomial of degree or less on.
The Gauss–Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as
where is well-approximated by a low-degree polynomial, then alternative nodes and weights will usually give more accurate quadrature rules. These are known as Gauss–Jacobi quadrature rules, i.e.,
Common weights include and. One may also want to integrate over semi-infinite and infinite intervals.
It can be shown that the quadrature nodes are the roots of a polynomial belonging to a class of orthogonal polynomials. This is a key observation for computing Gauss quadrature nodes and weights.

Gauss–Legendre quadrature

For the simplest integration problem stated above, i.e., is well-approximated by polynomials on, the associated orthogonal polynomials are Legendre polynomials, denoted by. With the -th polynomial normalized to give, the -th Gauss node,, is the -th root of and the weights are given by the formula
Some low-order quadrature rules are tabulated below.

Change of interval

An integral over must be changed into an integral over before applying the Gaussian quadrature rule. This change of interval can be done in the following way:
with
Applying the point Gaussian quadrature rule then results in the following approximation:

Example of two-point Gauss quadrature rule

Use the two-point Gauss quadrature rule to approximate the distance in meters covered by a rocket from to as given by
Change the limits so that one can use the weights and abscissae given in Table 1. Also, find the absolute relative true error. The true value is given as 11061.34 m.
Solution
First, changing the limits of integration from to gives
Next, get the weighting factors and function argument values from Table 1 for the two-point rule,
Now we can use the Gauss quadrature formula
since
Given that the true value is 11061.34 m, the absolute relative true error, is

Other forms

The integration problem can be expressed in a slightly more general way by introducing a positive weight function into the integrand, and allowing an interval other than. That is, the problem is to calculate
for some choices of,, and. For,, and, 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.

Interval	Orthogonal polynomials	A & S	For more information, see...
	Legendre polynomials	25.4.29
	Jacobi polynomials	25.4.33	Gauss–Jacobi quadrature
	Chebyshev polynomials	25.4.38	Chebyshev–Gauss quadrature
	Chebyshev polynomials	25.4.40	Chebyshev–Gauss quadrature
	Laguerre polynomials	25.4.45	Gauss–Laguerre quadrature
	Generalized Laguerre polynomials		Gauss–Laguerre quadrature
	Hermite polynomials	25.4.46	Gauss–Hermite quadrature

Fundamental theorem

Let be a nontrivial polynomial of degree such that
Note that this will be true for all the orthogonal polynomials above, because each is constructed to be orthogonal to the other polynomials for, and is in the span of that set.
If we pick the nodes to be the zeros of, then there exist weights which make the Gaussian quadrature computed integral exact for all polynomials of degree or less. Furthermore, all these nodes will lie in the open interval.
To prove the first part of this claim, let be any polynomial of degree or less. Divide it by the orthogonal polynomial to get
where is the quotient, of degree or less, and is the remainder, also of degree or less. Since is by assumption orthogonal to all monomials of degree less than, it must be orthogonal to the quotient. Therefore
Since the remainder is of degree or less, we can interpolate it exactly using interpolation points with Lagrange polynomials, where
We have
Then its integral will equal
where, the weight associated with the node, is defined to equal the weighted integral of . But all the are roots of, so the division formula above tells us that
for all. Thus we finally have
This proves that for any polynomial of degree or less, its integral is given exactly by the Gaussian quadrature sum.
To prove the second part of the claim, consider the factored form of the polynomial. Any complex conjugate roots will yield a quadratic factor that is either strictly positive or strictly negative over the entire real line. Any factors for roots outside the interval from to will not change sign over that interval. Finally, for factors corresponding to roots inside the interval from to that are of odd multiplicity, multiply by one more factor to make a new polynomial
This polynomial cannot change sign over the interval from to because all its roots there are now of even multiplicity. So the integral
since the weight function is always non-negative. But is orthogonal to all polynomials of degree or less, so the degree of the product
must be at least. Therefore has distinct roots, all real, in the interval from to.

General formula for the weights

The weights can be expressed as
where is the coefficient of in. To prove this, note that using Lagrange interpolation one can express in terms of as
because has degree less than and is thus fixed by the values it attains at different points. Multiplying both sides by and integrating from to yields
The weights are thus given by
This integral expression for can be expressed in terms of the orthogonal polynomials and as follows.
We can write
where is the coefficient of in. Taking the limit of to yields using L'Hôpital's rule
We can thus write the integral expression for the weights as
In the integrand, writing
yields
provided, because
is a polynomial of degree which is then orthogonal to. So, if is a polynomial of at most nth degree we have
We can evaluate the integral on the right hand side for as follows. Because is a polynomial of degree, we have
where is a polynomial of degree. Since is orthogonal to we have
We can then write
The term in the brackets is a polynomial of degree, which is therefore orthogonal to. The integral can thus be written as
According to equation, the weights are obtained by dividing this by and that yields the expression in equation.
can also be expressed in terms of the orthogonal polynomials and now. In the 3-term recurrence relation the term with vanishes, so in Eq. can be replaced by.

Proof that the weights are positive

Consider the following polynomial of degree
where, as above, the are the roots of the polynomial.
Clearly. Since the degree of is less than, the Gaussian quadrature formula involving the weights and nodes obtained from applies. Since for not equal to, we have
Since both and are non-negative functions, it follows that.

Computation of Gaussian quadrature rules

There are many algorithms for computing the nodes and weights of Gaussian quadrature rules. The most popular are the Golub-Welsch algorithm requiring operations, Newton's method for solving using the three-term recurrence for evaluation requiring operations, and asymptotic formulas for large n requiring operations.

Recurrence relation

Orthogonal polynomials with for for a scalar product, degree and leading coefficient one satisfy the recurrence relation
and scalar product defined
for where is the maximal degree which can be taken to be infinity, and where. First of all, the polynomials defined by the recurrence relation starting with have leading coefficient one and correct degree. Given the starting point by, the orthogonality of can be shown by induction. For one has
Now if are orthogonal, then also, because in
all scalar products vanish except for the first one and the one where meets the same orthogonal polynomial. Therefore,
However, if the scalar product satisfies , the recurrence relation reduces to a three-term recurrence relation: For is a polynomial of degree less than or equal to. On the other hand, is orthogonal to every polynomial of degree less than or equal to. Therefore, one has and for. The recurrence relation then simplifies to
or
where
.

The Golub-Welsch algorithm

The three-term recurrence relation can be written in matrix form where, is the th standard basis vector, i.e.,, and is the following tridiagonal matrix, called the Jacobi matrix:
The zeros of the polynomials up to degree, which are used as nodes for the Gaussian quadrature can be found by computing the eigenvalues of this matrix. This procedure is known as Golub–Welsch algorithm.
For computing the weights and nodes, it is preferable to consider the symmetric tridiagonal matrix with elements
That is,
and are similar matrices and therefore have the same eigenvalues. The weights can be computed from the corresponding eigenvectors: If is a normalized eigenvector associated with the eigenvalue, the corresponding weight can be computed from the first component of this eigenvector, namely:
where is the integral of the weight function
See, for instance, for further details.