Function classes for double exponential integration formulas

• Published in: Numerische Mathematik Vol. 111; no. 4; pp. 631 - 655
• Main Authors: Tanaka, Ken’ichiro, Sugihara, Masaaki, Murota, Kazuo, Mori, Masatake
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-02-2009; Springer
• Subjects: Exact sciences and technology; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Measure and integration; Numerical Analysis; Numerical analysis. Scientific computation; Numerical and Computational Physics; Numerical approximation; Sciences and techniques of general use; Simulation; Theoretical; 65D30; Numerical approximation; Numerical analysis; Numerical integration
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-008-0195-1

The double exponential (DE) formulas for numerical integration are known to be highly efficient, more efficient than the single exponential (SE) formulas in many cases. Function classes suited to the SE formulas have already been investigated in the literature through rigorous mathematical analysis, whereas this is not the case with the DE formulas. This paper identifies function classes suited to the DE formulas in a way compatible with the existing theoretical results for the SE formulas. The DE formulas are good for more restricted classes of functions, but more efficient for such functions. Two concrete examples demonstrate the subtlety in the behavior of the DE formulas that is revealed by our theoretical analysis.