Gaussian quadrature definition

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

When applying this algorithm, the moment set [mo, m, ..., m2N must be known (and realizable). From the definition of the objective function and the properties of Gaussian quadrature,we have /(O) > 0. Thus, as a first step in the bounded-search algorithm, an upper bound cr+ can be determined such that /(cr+) < 0 under the condition that the moment set mj, m, 2iv-i) found from Eq. (3.93) using cr = is realizable. If no such cr+ exists, then can be chosen such that it minimizes /(cr ) and the moment set... 

By extending the concepts of the previous section, it is possible to derive an approximation (which is no longer a real Gaussian quadrature) for the bivariate system. The resulting Appoint quadrature approximation transforms the definition of the general bivariate moment into (Wright et at, 2001b) m = Za=i where, as for the univariate case, Wa... 

The definition of the Gaussian quadrature formula in Eq. E.30 implies that the determination of this formula is determined by the selection of N quadrature points and N quadrature weights that is, we have 2N parameters to be found. With these degrees of freedom (2N parameters), it is possible to fit a polynomial of degree 2N - 1. This means that if Xj and are properly chosen, the Gausssian quadrature formula can exactly integrate a polynomial of degree up to 2N — 1. 

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