Nested quadrature rules for Cauchy principal value integrals

Gauss 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...

A Sinc-hunter Quadrature Rule for Cauchy Principal Value Integrals

A Sine function approach is used to derive a new Hunter type quadrature rule for the evaluation of Cauchy principal value integrals of certain analytic functions. Integration over a general arc in the complex plane is considered. Special treatment is given to integrals over the interval (-1, 1). It is shown that the quadrature error is of order 0(e~ ), where N is the number of nodes used, and w...

Quadrature Rules for Fuzzy Integrals

Generalized Noninterpolatory Rules for Cauchy Principal Value Integrals

Consider the Cauchy principal value integral I(kf;X)=i k{x)F^Ldx, -1<A<1. 7-1 x — A If we approximate f(x) by Yli=oajP¡(x'i w) where {p } is a sequence of orthonormal polynomials with respect to an admissible weight function w and û, = (/. P.), then an approximation to I(kf; X) is given by X!/=o ajl(kp¡ ; ^)If, in turn, we approximate a¡ by ajm = £TM , wimf(x¡m)Pj(xim), then we get a double seq...

Gauss-Kronrod Integration Rules for Cauchy Principal Value Integrals

Kronrod extensions to two classes of Gauss and Lobatto integration rules for the evaluation of Cauchy principal value integrals are derived. Since in one frequently occurring case, the Kronrod extension involves evaluating the derivative of the integrand, a new extension is introduced using n + 2 points which requires only values of the integrand. However, this new rule does not exist for all n...