Explicit Error Bound for Modified Numerical Iterated Integration by Means of Sinc Methods

This paper reinforces numerical iterated integration developed by Muhammad---Mori in the following two points: 1 the approximation formula is modified so that it can achieve a better convergence rate in more general cases, and 2 an explicit error bound is given in a computable form for the modified formula. The formula works quite efficiently, especially if the integrand is of a product type. Numerical examples that confirm it are also presented.

[1]  Automatic integration over a sphere , 1981 .

[2]  Masatake Mori,et al.  Double Exponential Formulas for Numerical Integration , 1973 .

[3]  Avram Sidi,et al.  Extension of a class of periodizing variable transformations for numerical Integration , 2005, Math. Comput..

[4]  Masatake Mori,et al.  Double exponential formulas for numerical indefinite integration , 2003 .

[5]  M. Mori,et al.  The double-exponential transformation in numerical analysis , 2001 .

[6]  Knut Petras,et al.  Principles of verified numerical integration , 2007 .

[7]  Masaaki Sugihara,et al.  Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind , 2010, J. Comput. Appl. Math..

[8]  Nicole Fruehauf.  Numerical Methods Based On Sinc And Analytic Functions , 2016 .

[9]  M. C. Eiermann.  Automatic, guaranteed integration of analytic functions , 1989 .

[10]  Avram Sidi,et al.  A New Variable Transformation for Numerical Integration , 1993 .

[11]  Masao Iri,et al.  On a certain quadrature formula , 1987 .

[12]  Masaaki Sugihara,et al.  Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration , 2013, Numerische Mathematik.

[13]  Gradimir V. Milovanović,et al.  Numerical integration of analytic functions , 2012 .

[14]  Ian Robinson,et al.  d2lri: a nonadaptive algorithm for two-dimensional cubature , 1999 .

[15]  Michael Hill,et al.  Algorithm 816: r2d2lri: an algorithm for automatic two-dimensional cubature , 2002, TOMS.

[16]  Elise de Doncker,et al.  Algorithm 45. Automatic computation of improper integrals over a bounded or unbounded planar region , 2005, Computing.

[17]  C. Schwartz,et al.  Numerical integration of analytic functions , 1969 .

[18]  F. Stenger.  Summary of Sinc numerical methods , 2000 .