Fundamentals of numerical computing
Fundamentals of numerical computing
Anti-Gaussian quadrature formulas
Mathematics of Computation
Fast integration of rapidly oscillatory functions
Journal of Computational and Applied Mathematics
The Mathematica book (4th edition)
The Mathematica book (4th edition)
A Modification of Filon's Method of Numerical Integration
Journal of the ACM (JACM)
Computation of irregularly oscillating integrals
Applied Numerical Mathematics
On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics
Numerical Methods
Journal of Computational and Applied Mathematics
Evaluating infinite range oscillatory integrals using generalised quadrature methods
Applied Numerical Mathematics
Efficient quadrature for highly oscillatory integrals involving critical points
Journal of Computational and Applied Mathematics
Modified anti-Gauss and degree optimal average formulas for Gegenbauer measure
Applied Numerical Mathematics
Efficient quadrature of highly oscillatory integrals with algebraic singularities
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
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The standard classic integration rules give inaccurate results for@!01t^@af(t)sin(@w/t^r)dtand@!01f(t)t^@acos(@w/t^r)dt where @w,r0, @a+r-1 are real numbers and f is any sufficiently smooth function on [0,1]. These integrals have been investigated for the special case @a=0 in Hascelik [A.I. Hascelik, On numerical computation of integrals with integrands of the form f(x)sin(1/x^r) on [0,1] (2007), in press] and for the case (r=1, @a=0) in Gautschi [W. Gautschi, Computing polynomials orthogonal with respect to densely oscillating and exponentially decaying weight functions and related integrals, J. Comput. Appl. Math. 184 (2005) 493-504]. In this work we construct suitable Gauss quadrature rules for approximating these integrals in high accuracy. The required three-term recurrence coefficients are computed by the Chebyshev algorithm using arbitrary precision arithmetic. We also give appropriate Filon-type methods for these integrals, with related error bounds. Some numerical examples are given to test the new methods.