Numerical analysis: an introduction
Numerical analysis: an introduction
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Vectorized adaptive quadrature in MATLAB
Journal of Computational and Applied Mathematics
Is Gauss Quadrature Better than Clenshaw-Curtis?
SIAM Review
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The standard way of dealing with singular integrals is first to write the integral in the form ∫ab: f(x)w(x)dx with a weight function w(x) 0 and a relatively smooth function f(x). Polynomials orthogonal on [a, b] with weight function w(x) are then used to derive accurate formulas for approximating the integral. The approach developed in this paper is to use a change of variable to obtain an integral over a finite interval that has a relatively smooth integrand and no weight function. Popular formulas can be applied to this standard problem to obtain easily alternatives to all the common schemes for weighted quadrature. Moreover, the approach provides a way to apply schemes for estimating the error in the unweighted case to integrals involving weight functions. It can be used with popular codes to approximate integrals with some kinds of strong end point singularities. Implementation of the approach is quite convenient in MATLAB.