The Gaussian quadratures provide the flexibility of choosing not only the weighting coefficients (weight factors) but also the locations (abscissas) where the functions are evaluated. As a result, Gaussian quadratures yield twice as many places of accuracy as that of the Newton-Cotes formulas with the same number of function evaluations. When the function is known and smooth, the Gaussian quadratures usually have decisive advantages in efficiency. However, engineering data obtained from measurements are not always smooth or located right on the abscissas which are not uniformly spaced. Therefore, the Gaussian quadratures are not suitable for such cases.
All Gaussian quadratures share the following format:
Gauss-Legendre Formula: The Gauss-Legendre integration formula is the most commonly used form of Gaussian quadratures. Some numerical analysis books refer to the Gauss-Legendre formula as the Gaussian quadratures' definitive form. It is based on the Legendre polynomials of the first kind .
Gauss-Chebyshev Formula: The Gauss-Chebyshev formula based on the Chebyshev polynomials of the first kind has more handy integration points and weights. But, more variations of parameters may be needed.
Gauss-Hermite Formula: The Gauss-Hermite formula uses the Hermite polynomials to deal with the integration interval of .
Gauss-Laguerre Formula: The Gauss-Laguerre Formula uses Laguerre polynomials to treat the integration interval of .