## Numerical indefinite integration of functions with singularities

HTML articles powered by AMS MathViewer

- by Aeyoung Park Jang and Seymour Haber;
- Math. Comp.
**70**(2001), 205-221 - DOI: https://doi.org/10.1090/S0025-5718-00-01226-6
- Published electronically: March 3, 2000
- PDF | Request permission

## Abstract:

We derive an indefinite quadrature formula, based on a theorem of Ganelius, for $H^p$ functions, for $p>1$, over the interval $(-1,1)$. The main factor in the error of our indefinite quadrature formula is $O(e^{-\pi \sqrt {N/ q}})$, with $2 N$ nodes and $\frac 1 p +\frac 1q=1$. The convergence rate of our formula is better than that of the Stenger-type formulas by a factor of $\sqrt {2}$ in the constant of the exponential. We conjecture that our formula has the best possible value for that constant. The results of numerical examples show that our indefinite quadrature formula is better than Haberβs indefinite quadrature formula for $H^p$-functions.## References

- J. E. Andersson,
*Optimal quadrature of $H^p$ functions*, Math. Z.**172**(1980), 55-62. - J. E. Andersson and B. D. Bojanov,
*A note on the optimal quadrature in $H^p$*, Numer. Math.**44**(1984), 301-308. - B. D. Bojanov,
*optimal rate of integration and $\epsilon$-entropy of a certain class of analytic functions*(in Russian), Mat. Zametki**14**, 1(1973), 3-10. [English transl.: Math. Notes**14**(1973) 551-556.] - β,
*Best quadrature formula for a certain class of analytic functions*, Zastosowania Matematyki Appl. Mat.**XIV**,**3**(1974), 441-447. - P. L. Duren,
*Theory of $H^p$ spaces*, Academic Press, San Diego (1970) - T. Ganelius,
*Rational approximation in the complex plane and on the line*, Ann. Acad. Sci. Fenn. Ser. A. I.**2**(1976), 129-145. - β,
*Some extremal functions and approximation*, Fourier analysis and approximation theory, Proceedings of a Colloquium (Budapest 1976), 371-381, Amsterdam-Oxford-New York, North Holland (1978). - S. Haber,
*The tanh rule for numerical integration*, SIAM J. Numer. Anal.**14**(1977), 668-685. - β,
*Two formulas for numerical indefinite integration*, Math. Comp.**201**(1993), 279-296. - R. B. Kearfott,
*A sinc approximation for the indefinite integral*, Math. Comp.,**41**(1983), 559-572. - H. L. Loeb, and H. Werner,
*Optimal numerical quadrature in $H^p$ spaces*, Math. Z.**138**(1974), 111-117. - D. J. Newman,
*Rational approximation to $|x|$*, Michigan Math. J.**11**(1964), 11-14. - β,
*Quadrature formula for $H^p$ functions*, Math. Z.**166**(1979), 111-115. - C. Schwartz,
*Numerical integration of analytic function*, J. Comput. Phys.**4**(1969), 19-29. - K. Sikorski,
*Optimal Quadrature Algorithms in $H_p$ Spaces*, Numer. Math.**39**(1982), 405-410. - K. Sikorski, F. Stenger, J. Schwing,
*Algorithm 614, A Fortran Subroutine for Numerical Integration in $H_p$*, ACM TOMS, v.10 (1984), 152-160. - K. Sikorski, F. Stenger,
*Optimal Quadratures in $H_p$ Spaces*, ACM TOMS, v.10 (1984), 140-151. - F. Stenger,
*Numerical methods based on Whittaker cardinal, or sinc functions*, SIAM Rev.**23**(1981), 165-224.

## Bibliographic Information

**Aeyoung Park Jang**- Affiliation: Trinity College, 10505 Oakton Ridge Court, Oakton, Virginia 22124
- Email: aeyoung@prodigy.net
**Seymour Haber**- Affiliation: Temple University, Mosaryk 1, Jerusalem, Israel
- Received by editor(s): May 19, 1998
- Received by editor(s) in revised form: January 4, 1999
- Published electronically: March 3, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp.
**70**(2001), 205-221 - MSC (2000): Primary 65D30; Secondary 41A55
- DOI: https://doi.org/10.1090/S0025-5718-00-01226-6
- MathSciNet review: 1709152