An Asymptotic Expansion for Product Integration Applied to Cauchy Principal Value Integrals
نویسنده
چکیده
Product integration methods for Cauchy principal value integrals based on piecewise Lagrangian interpolation are studied. I t is shown that for this class of quadrature methods the truncation error has an asymptotic expansion in integer powers of the step-size, and that a method with an asymptotic expansion in even powers of the step-size does not exist. The relative merits of a quadrature method which employs values of both the integrand and its first derivative and for which the truncation error has an asymptotic expansion in even powers of the step-size are discussed.
منابع مشابه
On the Convergence of an Interpolatory Product Rule for Evaluating Cauchy Principal Value Integrals*
The authors give convergence theorems for interpolatory product rules for evaluating Cauchy singular integrals and obtain asymptotic estimates of the remainder. Some results, previously established by other authors, are generalized and improved.
متن کاملTWO LOW-ORDER METHODS FOR THE NUMERICAL EVALUATION OF CAUCHY PRINCIPAL VSlLUE INTEGRALS OF OSCILLATORY KIND
In this paper, we develop two piecewise polynomial methods for the numerical evaluation of Cauchy Principal Value integrals of oscillatory kind. The two piecewisepolynomial quadratures are compact, easy to implement, and are numerically stable. Two numerical examples are presented to illustrate the two rules developed, The convergence of the two schemes is proved and some error bounds obtai...
متن کاملPrincipal Value Integrals
A general uniform convergence theorem for numerical integration of Cauchy principal value integrals is proved. Seven special instances of this theorem are given as corollaries.
متن کاملNumerical Evaluation of Cauchy Principal Value Integrals with Singular Integrands
Convergence results are proved for sequences of interpolatory integration rules for Cauchy principal value integrals of the form -l k(x)(f(x)/(x-X))dx, -1<A<1, -i when f(x) is singular at a point { / À and the singularity is ignored or avoided. /:'
متن کاملGauss Type Quadrature Rules for Cauchy Principal Value Integrals
Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integ...
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 2005