Abstract
The complexities of weighted approximation and weighted integration problems for univariate functions defined over ℝ have recently been found in [7]. Complexity (almost) optimal algorithms have also been provided therein. In this paper, we propose another class of (almost) optimal algorithms that, for a number of instances, are easier to implement. More importantly, these new algorithms have a cost smaller than the original algorithms from [7]. Since both classes of algorithms are (almost) optimal, their costs differ by a multiplicative constant that depends on the specific weight functions and the error demand. In one of our tests we observed this constant to be as large as four, which means a cost reduction by a factor of four.
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References
F. Curbera, Optimal integration of Lipschitz functions with a Gaussian weight, J. Complexity 14 (1998) 122-149.
P.J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic Press, San Diego, 1984).
J. Hays, Optimal algorithms for weighted integration, Master's project, Department of Computer Science, University of Kentucky, Lexington, KY (1998).
P. Mathé , Asymptotically optimal weighted numerical integration, J. Complexity 14 (1998) 34-48.
J.F. Traub, G.W.Wasilkowski and H. Wozniakowski, Information, Uncertainty, Complexity (Addison-Wesley, Reading, MA, 1982).
J.F. Traub, G.W. Wasilkowski and H. Wozniakowski, Information-Based Complexity (Academic Press, New York, 1988).
G.W. Wasilkowski and H. Wozniakowski, Complexity of weighted approximation over R, J. Approx. Theory (2000) to appear.
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Han, L., Wasilkowski, G. A new optimal algorithm for weighted approximation and integration over ℝ. Numerical Algorithms 23, 393–406 (2000). https://doi.org/10.1023/A:1019164403899
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DOI: https://doi.org/10.1023/A:1019164403899