New block quadrature rules for the approximation of matrix functions

Quadrature rules for rational functions

It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turr an, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. 0. Introduction The idea of constructing quadrature rules that are exact for rational functions with prescribed poles, rather than for polynomials, has received...

Quadrature Rules for Fuzzy Integrals

Generalized averaged Gauss quadrature rules for the approximation of matrix functionals

The need to compute expressions of the form uf(A)v, where A is a large square matrix, u and v are vectors, and f is a function, arises in many applications, including network analysis, quantum chromodynamics, and the solution of linear discrete ill-posed problems. Commonly used approaches first reduce A to a small matrix by a few steps of the Hermitian or non-Hermitian Lanczos processes and the...

Gauss-type Quadrature Rules for Rational Functions

When integrating functions that have poles outside the interval of integration, but are regular otherwise, it is suggested that the quadrature rule in question ought to integrate exactly not only polynomials (if any), but also suitable rational functions. The latter are to be chosen so as to match the most important poles of the integrand. We describe two methods for generating such quadrature ...

Locally Corrected Multidimensional Quadrature Rules for Singular Functions

Accurate numericalintegrationof singularfunctions usually requireseither adaptivity or product integration. Both interfere with fast summation techniques and thus hamper large-scale computations. This paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods. Given the singularity and the N nodes, we rst const...