Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures

摘要

Asymptotic approximations to the zeros of Hermite and Laguerre polynomialsare given, together with methods for obtaining the coefficients in theexpansions. These approximations can be used as a standalone method ofcomputation of Gaussian quadratures for high enough degrees, with Gaussianweights computed from asymptotic approximations for the orthogonal polynomials.We provide numerical evidence showing that for degrees greater than $100$ theasymptotic methods are enough for a double precision accuracy computation($15$-$16$ digits) of the nodes and weights of the Gauss--Hermite andGauss--Laguerre quadratures.