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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.
In 1969, Golub and Welsch published their method for computing Gaussian quadrature rules given the three term recurrence relation that the underlying orthogonal polynomials satisfy. [1] They reduce the problem of computing the nodes of a Gaussian quadrature rule to the problem of finding the eigenvalues of a particular symmetric tridiagonal matrix.
1 Example with change of variable. ... "A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems". J. Comput.
Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure (quadrature or squaring), as in the quadrature of the circle. The term is also sometimes used to describe the numerical solution of differential equations .
Gauss–Kronrod formulas are extensions of the Gauss quadrature formulas generated by adding + points to an -point rule in such a way that the resulting rule is exact for polynomials of degree less than or equal to + (Laurie (1997, p. 1133); the corresponding Gauss rule is of order ).
Gauss–Laguerre quadrature — extension of Gaussian quadrature for integrals with weight exp(−x) on [0, ∞] Gauss–Kronrod quadrature formula — nested rule based on Gaussian quadrature; Gauss–Kronrod rules; Tanh-sinh quadrature — variant of Gaussian quadrature which works well with singularities at the end points
Typically, this means that either the function is highly oscillatory or lacks derivatives at certain points. In these cases, Simpson's rule may give very poor results. One common way of handling this problem is by breaking up the interval [,] into > small subintervals. Simpson's rule is then applied to each subinterval, with the results being ...
Special examples are the Gaussian quadrature for polynomials and the Discrete Fourier Transform for plane waves. It should be stressed that the grid points and weights, x i , w i {\displaystyle x_{i},w_{i}} are a function of the basis and the number N {\displaystyle N} .