Is Gauss quadrature optimal for analytic functions?
We consider the problem of optimal quadratures for integrands f: [ -1 , 1 ] → R which have an analytic extension f to an open disk D, of radius r about the origin such that |f| â‰¤ 1 on Dr. If r = 1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degree n rather than at optimal points, tends to infinity with n. In particular there is an "infinite" penalty for using Gauss quadrature. On the other hand, if r > l, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings.
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More About This Work
- Academic Units
- Computer Science
- Department of Computer Science, Columbia University
- Columbia University Computer Science Technical Reports, CUCS-081-83
- Published Here
- October 26, 2011