We are presenting here a class of integrals that has shown its importance in quantum mechanics. It's the class of integrals $\int_{-1}^{1} e^{-\alpha x} f(x) dx$ where $\alpha$ may assume all possible positive real values and the function $f(x)$ is known only approximatively in a tabular form. To evaluate such integrals we use a classical gaussian quadrature formula for which we develop nodes and weights through a new general recursive algorithm using a set of orthogonal polynomials. Such orthogonal polynomials are obtained through a numerical method using a three terms recurrence relation. The numerical results of the algorithm are presented and a special attention is given to obtain a good number of significant digits.
A new recursive algorithm for a Gaussian quadrature formula via orthogonal polynomials
REDIVO ZAGLIA, MICHELA
1991
Abstract
We are presenting here a class of integrals that has shown its importance in quantum mechanics. It's the class of integrals $\int_{-1}^{1} e^{-\alpha x} f(x) dx$ where $\alpha$ may assume all possible positive real values and the function $f(x)$ is known only approximatively in a tabular form. To evaluate such integrals we use a classical gaussian quadrature formula for which we develop nodes and weights through a new general recursive algorithm using a set of orthogonal polynomials. Such orthogonal polynomials are obtained through a numerical method using a three terms recurrence relation. The numerical results of the algorithm are presented and a special attention is given to obtain a good number of significant digits.Pubblicazioni consigliate
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