Addition of Points to Gauss–Laguerre Quadrature Formulas

The Gauss–Laguerre quadrature formula is defined by \[ If = \int_0^\infty {e^{ - x} f(x)dx \simeq \sum_{i = 1}^n {\alpha _i^{(n)} f\left( {\xi _i^{(n)} } \right),} } \] where the numbers ${\alpha _i^{(n)} }$ and ${\xi _i^{(n)} }$ are weights and nodes. A common method of estimating the error of this rule is to evaluate the quadrature rule for two different values of n and to then compare the difference in the answers. Unfortunately, none of the nodes are in common for the two different quadrature rules, and so the function must be evaluated at each separate node. We investigate in this paper the addition of points to the Gauss–Laguerre rule such that the new points are real, lie in the interval of integration, and the associated weights are positive. Such rules enable one to estimate economically the error of quadrature, because the function values at the Gauss–Laguerre abscissae are reused. The weights and nodes for some suitable low-order formulae are given in Table 2.