Search results
Results From The WOW.Com Content Network
The Gaussian quadrature chooses more suitable points instead, so even a linear function approximates the function better (the black dashed line). As the integrand is the third-degree polynomial y(x) = 7x 3 – 8x 2 – 3x + 3, the 2-point Gaussian quadrature rule even returns an exact result.
n is the number of sample points used, w i are quadrature weights, and; x i are the roots of the nth Legendre polynomial. This choice of quadrature weights w i and quadrature nodes x i is the unique choice that allows the quadrature rule to integrate degree 2n − 1 polynomials exactly.
If the interval [a, b] is subdivided, the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at the midpoint for odd numbers of evaluation points), and thus the integrand must be evaluated at every point. Gauss–Kronrod formulas are extensions of the Gauss quadrature formulas generated by ...
If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, such as the Gaussian quadrature formulas. A Gaussian quadrature rule is typically more accurate than a Newton–Cotes rule that uses the same number of function evaluations, if the integrand is smooth (i.e., if it is sufficiently ...
Weights versus x i for four choices of n. In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind: + (). In this case
The Gauss–Legendre method based on s points has order 2s. [2] All Gauss–Legendre methods are A-stable. [3] In fact, one can show that the order of a collocation method corresponds to the order of the quadrature rule that one would get using the collocation points as weights.
"Table of zeros and Gaussian Weights of certain Associated Laguerre Polynomials and the related Hermite Polynomials". Mathematics of Computation. 18 (88): 598– 616. doi: 10.1090/S0025-5718-1964-0166397-1. JSTOR 2002946. MR 0166397. Ehrich, S. (2002). "On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas".
Points inside the integration range are given alternating weights 4/3 and 2/3. ... Gaussian quadrature; Notes References. Atkinson, Kendall E. (1989). ...