Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: Basic properties and error estimates

[摘要] Let pi(n-1)[f] be the polynomial of degree n-1 interpolating the function f at the points x(1),x(2),...x(n) with P-n(x(i)) = 0, i.e., at the nodes of the classical Gaussian quadrature formula. For the numerical approximation of the Cauchy principal value integral integral(-1)(1) f(x)(x-lambda)(-1) dx with lambda is an element of(-1,1) and f is an element of C-1[-1,1], we present the quadrature formula Q(n+1)(G3) given Q(n+1)(G3)[f;lambda]:= integral(-1)(1) pi(n-1)[f](x)-pi(n-1)[f](lambda)/x-lambda dx+f(lambda)In1-lambda/1+lambda. We show that this quadrature formula does not have the disadvantages of the other two well-known quadrature formulae based on the same set of nodes. In particular, we prove that the sequence (Q(n+1)(G3)[f;lambda]) converges to the true value of the integral uniformly for all lambda is an element of(-1,1). We give estimates for the error term. Furthermore, we state some relations connecting the present quadrature formula to the previously introduced formulae.

[发布日期] 1995-12-29

[效力级别]  Proceedings Paper

[关键词] Cauchy principal value integrals;Gaussian quadrature formula;uniform convergence