On Product Integration with Gauss-kronrod Nodes
نویسنده
چکیده
Gauss-Kronrod product quadrature formulas for the numerical approximation of R 1 ?1 k(x)f (x)dx are shown to converge for every Riemann integrable f , and to possess optimal stability. Similar results are proved for the product formulas based on the Kronrod nodes only. An application to the uniform convergence of approximate solutions of integral equations is given.
منابع مشابه
Computation of Gauss-kronrod Quadrature Rules with Non-positive Weights
Recently Laurie presented a fast algorithm for the computation of (2n + 1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. We describe modifications of this algorithm that allow the computation of Gauss-Kronrod quadrature rules with complex conjugate nodes and weights or with real nodes and positive and negative weights.
متن کاملCalculation of Gauss-Kronrod quadrature rules
The Jacobi matrix of the (2n+1)-point Gauss-Kronrod quadrature rule for a given measure is calculated efficiently by a five-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for Gaussian quadrature formulas.
متن کاملStieltjes Polynomials and the Error of Gauss-kronrod Quadrature Formulas
The Gauss-Kronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the error of the Gauss-Kronrod scheme. An essential progress was made only recently, based on new bounds and as-ymptotic properties for the S...
متن کاملError estimates for Gauss–Turán quadratures and their Kronrod extensions
We study the kernel Kn,s(z) of the remainder term Rn,s( f ) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L∞-error bounds of Gauss–Turán–Kronrod quadratures. Following Kronrod,...
متن کاملAn Algebraic Study of Gauss-Kronrod Quadrature Formulae for Jacobi Weight Functions*
We study Gauss-Kronrod quadrature formulae for the Jacobi weight function «/"'"'(t) = (l-i)Q(l + t)'3 and its special case a = ß = X^ of the Gegenbauer weight function. We are interested in delineating regions in the (a, /3)-plane, resp. intervals in A, for which the quadrature rule has (a) the interlacing property, i.e., the Gauss nodes and the Kronrod nodes interlace; (b) all nodes contained ...
متن کامل