1,162 research outputs found
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas
In the present paper we introduce a new concept of Hardy type space naturally
defined on the Klein-Dirac quadric. We study different properties of the
functions belonging to these spaces, in particular boundary value problems. We
apply these new spaces to polyharmonic interpolation and to interpolatory
cubature formulas.Comment: 32 page
Design of quadrature rules for MĂźntz and MĂźntz-logarithmic polynomials using monomial transformation
A method for constructing the exact quadratures for MĂźntz and MĂźntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of MĂźntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for MĂźntz polynomial
An explicit kernel-split panel-based Nystr\"om scheme for integral equations on axially symmetric surfaces
A high-order accurate, explicit kernel-split, panel-based, Fourier-Nystr\"om
discretization scheme is developed for integral equations associated with the
Helmholtz equation in axially symmetric domains. Extensive incorporation of
analytic information about singular integral kernels and on-the-fly computation
of nearly singular quadrature rules allow for very high achievable accuracy,
also in the evaluation of fields close to the boundary of the computational
domain.Comment: 30 pages, 5 figures, errata correcte
Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals
High-order derivatives of analytic functions are expressible as Cauchy
integrals over circular contours, which can very effectively be approximated,
e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius
of convergence is equal, numerical stability strongly depends on r. We give a
comprehensive study of this effect; in particular we show that there is a
unique radius that minimizes the loss of accuracy caused by round-off errors.
For large classes of functions, though not for all, this radius actually gives
about full accuracy; a remarkable fact that we explain by the theory of Hardy
spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and
by the saddle-point method of asymptotic analysis. Many examples and
non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature
rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat
Quadrature methods for 2D and 3D problems
AbstractIn this paper we give an overview on well-known stability and convergence results for simple quadrature methods based on low-order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds. First, we explain the methods for the case of second-kind equations. Then we discuss what is known for the analysis of pseudodifferential equations. We explain why these simple methods are not recommended for integral equations over domains with dimension higher than one. Finally, for the solution of a two-dimensional singular integral equation, we prove a new result on a quadrature method based on product rules
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