Complex Gaussian quadrature of oscillatory integrals
نویسندگان
چکیده
We construct and analyze Gauss-type quadrature rules with complex-valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with nonstandard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.
منابع مشابه
Complex Gaussian quadrature for oscillatory integral transforms
The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect to an oscillatory weight, yielding methods with complex quadrature nodes and positive weights. These rules are well suited for highly oscillatory integrals because they attain optimal asymptotic order. We show that for the Fourier oscilla...
متن کاملComputing Integrals of Highly Oscillatory Special Functions Using Complex Integration Methods and Gaussian Quadratures
An account on computation of integrals of highly oscillatory functions based on the so-called complex integration methods is presented. Beside the basic idea of this approach some applications in computation of Fourier and Bessel transformations are given. Also, Gaussian quadrature formulas with a modified Hermite weight are considered, including some numerical examples.
متن کاملNumerical methods for highly oscillatory integrals on semi-finite intervals
In highly oscillatory integrals, the integrand fw(x) oscillates rapidly with a frequency ω. For very high values of ω, numerical evaluation of such integrals by Gaussian quadrature rules can be of very low accuracy. In such problems which have many applications in mathematical physics, it is important to devise algorithms with errors which decrease as fast as w−N , for some N > 0. In this paper...
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 Gaussian quadrature rule for oscillatory integrals on a bounded interval
We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function ei!x on the interval [ 1, 1]. We show that such a rule attains high asymptotic order, in the sense that the quadrature error quickly decreases as a function of the frequency !. However, accuracy is maintained for all values of ! and in particular the rule elegantly reduces ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Numerische Mathematik
دوره 112 شماره
صفحات -
تاریخ انتشار 2009