Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Rates of convergence of Gauss, Lobatto, and Radau integration rules for singular integrands


Author: Philip Rabinowitz
Journal: Math. Comp. 47 (1986), 625-638
MSC: Primary 65D30; Secondary 65D32
MathSciNet review: 856707
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Rates of convergence (or divergence) are obtained in the application of Gauss, Lobatto, and Radau integration rules to functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a generalized Jacobi weight function on $ [ - 1,1]$, the error in applying an n-point rule to $ f(x) = \vert x - y{\vert^{ - \delta }}$ is $ O({n^{ - 2 + 2\delta }})$, if $ y = \pm 1$ and $ O({n^{ - 1 + \delta }})$ if $ y \in ( - 1,1)$, provided we avoid the singularity. If we ignore the singularity y, the error is $ O({n^{ - 1 + 2\delta }}{(\log n)^\delta }{(\log \log n)^{\delta (1 + \varepsilon )}})$ for almost all choices of y. These assertions are sharp with respect to order.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D30, 65D32

Retrieve articles in all journals with MSC: 65D30, 65D32


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1986-0856707-6
PII: S 0025-5718(1986)0856707-6
Article copyright: © Copyright 1986 American Mathematical Society