Numerical construction of Gaussian quadrature formulas for $\int _{0}^{1}(-\textrm {Log}\ x)\cdot x^{\alpha }\cdot f(x)\cdot dx$ and $\int _{0}^{\infty } E_{m}(x)\cdot f(x)\cdot dx$

Abstract:

Most nonclassical Gaussian quadrature rules are difficult to construct because of the loss of significant digits during the generation of the associated orthogonal polynomials. But, in some particular cases, it is possible to develop stable algorithms. This is true for at least two well-known integrals, namely \[ \int _0^1 { - ({\operatorname {Log}}\;x) \cdot {x^\alpha } \cdot f(x) \cdot dx\quad {\text {and}}\quad \int _0^\infty {{E_m}(x) \cdot f(x) \cdot } dx.} \] A new approach is presented, which makes use of known classical Gaussian quadratures and is remarkably well-conditioned since the generation of the orthogonal polynomials requires only the computation of discrete sums of positive quantities. Finally, some numerical results are given.