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 ...
, Gauss–Legendre quadrature is a form of
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 .
Th ...
for approximating the
definite integral
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Int ...
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-orie ...
. For integrating over the interval , the rule takes the form:
:
:
where
* ''n'' is the number of sample points used,
* ''w''
''i'' are quadrature weights, and
* ''x''
''i'' are the roots of the ''n''th
Legendre polynomial
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
.
This choice of quadrature weights ''w''
''i'' and quadrature nodes ''x''
''i'' is the unique choice that allows the quadrature rule to integrate degree polynomials exactly.
Many algorithms have been developed for computing Gauss–Legendre quadrature rules. The Golub–Welsch algorithm presented in 1969 reduces the computation of the nodes and weights to an
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
problem which is solved by the
QR algorithm
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a Matrix (mathematics), matrix. The QR algorithm was developed in the late 1950s by John ...
.
This algorithm was popular, but significantly more efficient algorithms exist. Algorithms based on the
Newton–Raphson method
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a ...
are able to compute quadrature rules for significantly larger problem sizes. In 2014, Ignace Bogaert presented explicit asymptotic formulas for the Gauss–Legendre quadrature weights and nodes, which are accurate to within double-precision
machine epsilon
Machine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point number systems. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the ...
for any choice of ''n'' ≥ 21.
This allows for computation of nodes and weights for values of ''n'' exceeding one billion.
History
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
was the first to derive the Gauss–Legendre quadrature rule, doing so by a calculation with
continued fraction
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
s in 1814.
He calculated the nodes and weights to 16 digits up to order ''n''=7 by hand.
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory.
Biography
Jacobi was ...
discovered the connection between the quadrature rule and the orthogonal family of
Legendre polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
. As there is no closed-form formula for the quadrature weights and nodes, for many decades people were only able to hand-compute them for small ''n'', and computing the quadrature had to be done by referencing tables containing the weight and node values. By 1942 these values were only known for up to ''n''=16.
Definition

For integrating ''f'' over