- Authors:
- Gene H. Golub,
- John H. Welsch
Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.
Cited By
Stava O, Vanek J, Benes B, Carr N and Měch R (2012). Stress relief, ACM Transactions on Graphics (TOG), 31:4, (1-11), Online publication date: 5-Aug-2012.- Gülpinar M, Gülpinar C and Demiralp M Experimentations on the Gaussian wave type basis set for the matrix representations of certain univariate functions Proceedings of the 11th WSEAS international conference on Mathematical methods and computational techniques in electrical engineering, (406-412)
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