Numerical indefinite integration of functions with singularities

Jang, Aeyoung; Haber, Seymour

doi:10.1090/S0025-5718-00-01226-6

Numerical indefinite integration of functions with singularities
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by Aeyoung Park Jang and Seymour Haber PDF

Math. Comp. 70 (2001), 205-221 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 Haber’s 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.]
—, 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.
—, 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.
—, 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.
—, 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.