Gaussian Formulas

Gaussian Formulas


formulas relating to various branches of mathematics and bearing the name of K. Gauss.

(1) Gaussian quadrature formulas —formulas of the type

in which the nodes Xk and the coefficients Ak do not depend on the function f(x) and are chosen in such a way that the formula is accurate (that is, Rn = 0) for an arbitrary polynomial of degree 2n — 1. In contrast to the Newton-Cotes quadrature formulas, the nodes in Gaussian quadrature formulas generally are not equally spaced. If p(x) ≥ 0 and Gaussian Formulas, then for any natural n there is only one Gaussian quadrature formula. These formulas are of great practical importance, since in many cases they result in much greater accuracy than do quadrature formulas having the same number of equidistant nodes. In 1816, Gauss himself investigated the case p(x) ≡ 1 .

(2) The Gaussian formula that expresses the total curvature K of a surface in terms of the coefficients of its linear element; in coordinates for which ds2 = λ(du2 + dv2), this Gaussian formula has the form

This formula, published in 1827, shows that the total curvature does not change as the surface is bent. It constitutes the content of one of the main propositions of the intrinsic geometry of a surface devised by Gauss.

(3) The Gaussian formula for Gaussian sums:

This formula was used by Gauss in 1801 in one of the proofs of the reciprocity principle of quadratic residues

where p and q are odd primes and (p/q) is the Legendre symbol. This formula was the first example of the use of the method of trigonometric sums in number theory. This method was further developed in the works of H. Weyl and especially of I. M. Vinogradov and represents one of the most powerful methods of analytic number theory.

(4) The Gaussian formula for the sum of a hypergeometric series. If Re(c — b — a) > 0, then

where Γ(x) is the gamma function. This formula was published in 1812.


References in periodicals archive ?
In [39], effort has been taken in order to determine whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss-Kronrod quadrature formulas to estimate the remainder term of a Gaussian rule.
SPALEVIC, On generalized averaged Gaussian formulas, Math.
39] --, A note on generalized averaged Gaussian formulas, Numer.
It is known that Gaussian formulas ensure a higher order of accuracy and have better stability properties than Newton-Cotes formulas [2].
2, we have compared our results with those obtained by the only other numerical approach we know, that is, DQ methods based on Newton-Cotes formulas (in particular the trapezoidal and Simpson 3/8 formulas) [7]: the errors of the two families of methods are very similar, in spite of the error of approximation due to the interpolation technique used for the Gaussian formulas.
We have studied the convergence properties of the constructed methods and we have proven that their order of convergence is the minimum between the order of convergence of the Gaussian formula and the degree of accuracy of the interpolating polynomial.
The closest parallelism with classical Gaussian formulas and orthogonal polynomials will be specially emphasized.
In both cases results given by Gaussian formulas are strongly improved.