Gaussian quadrature

It is similar to Gaussian quadrature with the following differences:

Gaussian quadrature often requires noticeably less work for superior accuracy.

Instead, spacing of latitudes is defined by the Gaussian quadrature.

The error of a Gaussian quadrature rule can be stated as follows .

Note that Gaussian quadrature can also be adapted for various weight functions, but the technique is somewhat different.

The integral may be efficiently computed by Gaussian quadrature.

After applying the Gaussian quadrature rule, the following approximation is:

These matrices are usually evaluated numerically using Gaussian quadrature for numerical integration.

Since the degree of f(x) is less than 2n-1, the Gaussian quadrature formula involving the weights and nodes obtained from applies.

Using this and Gaussian quadrature we can construct so that is an approximate (t,t)-design.

Important consequence of the above equation is that Gaussian quadrature of order n is accurate for all polynomials up to degree 2n-1.

If f(x) does not have many derivatives at all points, or if the derivatives become large, then Gaussian quadrature is often insufficient.

These make analytic integrals often easier to evaluate, and Gaussian quadrature tables are often presented in terms of area coordinates.

The classic method of Gaussian quadrature evaluates the integrand at points and is constructed to exactly integrate polynomials up to degree .

It may seem, therefore, that Clenshaw-Curtis is intrinsically worse than Gaussian quadrature, but in reality this does not seem to be the case.

Gaussian quadrature as above will only produce accurate results if the function 'f'('x') is well approximated by a polynomial function within the range -1,1 .

In Gaussian quadrature, different weight functions lead to different orthogonal polynomials, and thus different roots where the integrand is evaluated.

Other quadrature rules, such as Gaussian quadrature or Gauss-Kronrod quadrature, may also be used.

In practice, several authors have observed that Clenshaw-Curtis can have accuracy comparable to that of Gaussian quadrature for the same number of points.

If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, such as the Gaussian quadrature formulas.

Popular methods use one of the Newton-Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature.

If it is possible to evaluate the integrand at unequally-spaced points, then other methods such as Gaussian quadrature and Clenshaw-Curtis quadrature are generally more accurate.

In numerical analysis, Gauss-Jacobi quadrature is a method of numerical quadrature based on Gaussian quadrature.

Gaussian Quadrature by Chris Maes and Anton Antonov, Wolfram Demonstrations Project.

The zeros of the polynomials up to degree which are used as nodes for the Gaussian quadrature can be found by computing the eigenvalues of this tridiagonal matrix.