In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, a quadrature rule is an approximation of the
definite integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, usually stated as a
weighted sum
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
of function values at specified points within the domain of integration. (See
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
for more on
quadrature rules.) An -point Gaussian quadrature rule, named after
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, is a quadrature rule constructed to yield an exact result for
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s 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
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, Dynamics (mechanics), dynamics, differential equations, determinants, and number theory. H ...
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
(
Chebyshev–Gauss) and
. One may also want to integrate over semi-infinite (
Gauss-Laguerre quadrature) and infinite intervals (
Gauss–Hermite quadrature
In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind:
:\int_^ e^ f(x)\,dx.
In this case
:\int_^ e^ f(x)\,dx \approx \sum_^n w_i f(x_i)
where ''n'' is ...
).
It can be shown (see Press, et al., or Stoer and Bulirsch) that the quadrature nodes are the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of a polynomial belonging to a class of
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product.
The most widely used orthogonal polynomial ...
(the class orthogonal with respect to a weighted inner-product). 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