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Journal Article

Numerical Construction of Gaussian Quadrature Formulas for $$\int^1_0 (-Log x)\cdot x^\alpha \cdot f(x) \cdot dx \quad \text{and}\quad \int^\infty_0 E_m(x) \cdot f(x) \cdot dx$$

Bernard Danloy
Mathematics of Computation
Vol. 27, No. 124 (Oct., 1973), pp. 861-869
DOI: 10.2307/2005521
Stable URL: http://www.jstor.org/stable/2005521
Page Count: 9
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Numerical Construction of Gaussian Quadrature Formulas for $$\int^1_0 (-Log x)\cdot x^\alpha \cdot f(x) \cdot dx \quad \text{and}\quad \int^\infty_0 E_m(x) \cdot f(x) \cdot dx$$
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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 $$\int^1_0 -(\operatorname{Log} x)\cdot x^\alpha \cdot f(x) \cdot dx \quad\text{and}\quad \int^\infty_0 E_m (x)\cdot f(x) \cdot dx$$. 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.

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