Gausslegendre MethodEdit
The Gauss-Legendre method is a cornerstone of numerical integration, providing a highly efficient way to approximate definite integrals by a weighted sum of function evaluations. It belongs to the family of Gaussian quadrature rules, and it specializes to Gauss-Legendre quadrature by using the roots and weights derived from Legendre polynomials. On the standard interval [-1, 1], the method is exact for every polynomial of degree up to 2n − 1 when n nodes are used, which means that a surprisingly small number of function evaluations can yield very accurate results for smooth functions.
To apply the method to a general interval [a, b], one performs a simple change of variables that maps t in [-1, 1] to x in [a, b]. The integral ∫a^b f(x) dx becomes (b − a)/2 ∑{i=1}^n w_i f(x_i), where x_i = (b − a)/2 · t_i + (a + b)/2, and the t_i are the roots of the nth Legendre polynomial P_n(t) with associated weights w_i = 2 / [(1 − t_i^2) [P'_n(t_i)]^2]. The x_i serve as the evaluation points (nodes), and the w_i as the weights. The formula is the same idea whether one is integrating a simple function or a more complicated one arising in physics, engineering, or applied mathematics. The method’s reliance on orthogonal polynomials and fixed nodes makes it particularly effective for smooth integrands.
History and development
The underlying idea traces back to the work of Carl Friedrich Gauss, who developed quadrature rules that maximize exactness for polynomials by choosing optimal nodes and weights. Gauss’s approach is closely tied to the theory of interpolation and orthogonal polynomials, and in the Legendre case the weight function is constant on the interval, leading to the Legendre polynomials studied by Adrien-Marie Legendre and colleagues. Over the 19th and 20th centuries, the algorithmic aspects—computing the nodes and weights efficiently, handling interval mappings, and implementing stable numerical procedures—were refined as computational methods evolved. Today, Gauss-Legendre quadrature is a standard tool in numerical analysis and is implemented in many scientific libraries, alongside the broader family of Gaussian quadrature methods.
Theory and construction
Nodes and weights: In the standard formulation on [-1, 1], the nodes x_i are the zeros of the nth Legendre polynomial P_n(x). The corresponding weights w_i are given by w_i = 2 / [(1 − x_i^2) [P'n(x_i)]^2]. This choice ensures that the rule ∫{−1}^{1} f(x) dx ≈ ∑_{i=1}^n w_i f(x_i) is exact for all polynomials of degree ≤ 2n − 1.
Interval mapping: For a general interval [a, b], the transformation x = (b − a)/2 · t + (a + b)/2 maps t ∈ [−1, 1] to x ∈ [a, b]. The integral becomes ∫a^b f(x) dx = (b − a)/2 ∑{i=1}^n w_i f( (b − a)/2 · x_i + (a + b)/2 ). Here the same nodes t_i and weights w_i from the standard interval are used after rescaling.
Computation of the nodes and weights: The t_i are the roots of P_n(t); these roots can be obtained by numerical root-finding, or more stably via eigenvalue methods applied to a tridiagonal Jacobi matrix whose entries depend on n. The weights follow from the derivative relation w_i = 2 / [(1 − t_i^2)[P'_n(t_i)]^2]. For small n, tabulated values are common, but for larger n these formulas enable on-the-fly computation.
Convergence and error: Gauss-Legendre quadrature yields spectral-like convergence for analytic or very smooth integrands, meaning the error decreases faster than any power of 1/n as n increases. For functions with limited smoothness, the method remains highly accurate up to the point where the function’s irregularities dominate; in such cases adaptivity or alternative quadrature strategies may be preferable. In practice, the method tends to require far fewer function evaluations than equally spaced rules to achieve the same accuracy, especially when the integrand is well-behaved.
Applications and performance
Efficiency for smooth integrands: Because the rule is exact for polynomials of high degree and uses optimally placed nodes, far fewer evaluations are needed than with many other quadrature schemes to reach a given accuracy. This makes Gauss-Legendre quadrature a frequent choice in physics simulations, computational chemistry, and engineering calculations where smooth integrands arise.
General intervals and weighted integrals: The mapping to arbitrary intervals extends the method to a broad range of problems. It can also be adapted to weight functions other than the uniform weight by considering different families of orthogonal polynomials, though that leads to other Gauss-type rules (for example, the Gauss–Kronrod extension for error estimation). For more on the general concept, see Gaussian quadrature.
Limitations: The method can be less effective for highly oscillatory integrands, integrals with endpoint singularities, or functions with abrupt local features. In such cases, adaptive quadrature, variable transformations, or other specialized quadrature rules may be preferable. The choice often hinges on the trade-off between the cost of evaluating f and the desired accuracy.
Related methods: The Gauss-Legendre approach sits within the broader landscape of numerical integration and orthogonal-polynomial-based methods. See also Legendre polynomials and the general theory of Orthogonal polynomials for the mathematical foundations, or consult Numerical integration for an overview of alternative rules and their use cases.