Fun with Filon Quadrature — a Computer Algebra Perspective
نویسنده
چکیده
Filon quadrature, designed specifically for a common class of oscillatory integrals, is easily implemented in a computer algebra system (CAS) using exact or approximate arithmetic. Although quadrature is almost always used in the context of approximation, we see that various properties are exhibited by running an exact algorithm on exact symbolic inputs. This experimental approach to the mathematics allows us to prove the implementation of the algorithm has the expected mathematical correctness properties, and is therefore more likely to be, itself, a correct implementation.
منابع مشابه
A generalization of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals
The Filon–Clenshaw–Curtis method (FCC) for the computation of highly oscillatory integrals has been proposed by Domı́nguez, Graham and Smyshlayev and is known to attain surprisingly high precision. Yet, for large values of frequency ω it is not competitive with other versions of the Filon method, which use high derivatives at critical points and exhibit high asymptotic order. In this paper we pr...
متن کاملQuadrature methods for highly oscillatory singular integrals
We study asymptotic expansions, Filon-type methods and complex-valued Gaussian quadrature for highly oscillatory integrals with power-law and logarithmic singularities. We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the integral. A truncated asymptotic expansion...
متن کاملA Special Case of the Filon Quadrature Formula*
The Filon formula is advantageous over usual numerical integration formulas for smooth fix), especially for large k, since the number of points which need be tabulated depends on the behavior of fix) rather than on fix) sin kx or fix) cos kx. Under certain circumstances the Filon quadrature formula reduces to a simple form, namely (2) S* = ii-l)m/k){fia) fib)} , C* = ii-l)m/k)[fib) fia)} , wher...
متن کاملOn the convergence of Filon quadrature
We analyze the convergence behavior of Filon-type quadrature rules by making explicit the dependence on both k, the parameter that controls the oscillatory behavior of the integrand, and n, the number of function evaluations. We provide explicit conditions on the domain of analyticity of the integrand to ensure convergence for n → ∞.
متن کاملA combined Filon/asymptotic quadrature method for highly oscillatory problems
A cross between the asymptotic expansion of an oscillatory integral and the Filon-type methods is obtained by applying a Filon-type method on the error term in the asymptotic expansion, which is in itself an oscillatory integral. The efficiency of the approach is investigated through analysis and numerical experiments, and a potential for better methods than current ones is revealed. In particu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014