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Design of quadrature rules for Müntz and Müntz‐logarithmic polynomials using monomial transformation
Author(s) -
Lombardi Guido
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2684
Subject(s) - mathematics , clenshaw–curtis quadrature , numerical integration , gaussian quadrature , logarithm , gauss–jacobi quadrature , gauss–kronrod quadrature formula , quadrature (astronomy) , tanh sinh quadrature , monomial , singularity , mathematical analysis , gauss–laguerre quadrature , nyström method , boundary value problem , pure mathematics , electrical engineering , engineering
A method for constructing the exact quadratures for Müntz and Müntz‐logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz‐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 Müntz polynomials. Copyright © 2009 John Wiley & Sons, Ltd.