An error functional expansion for -dimensional quadrature with an integrand function singular at a point

Author:
J. N. Lyness

Journal:
Math. Comp. **30** (1976), 1-23

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1976-0408211-0

MathSciNet review:
0408211

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Abstract: Let *If* be the integral of over an *N*-dimensional hypercube and be the approximation to *If* obtained by subdividing the hypercube into equal subhypercubes and applying the same quadrature rule *Q* to each. In order to extrapolate efficiently for *If* on the basis of several different approximations , it is necessary to know the form of the error functional as an expansion in *m*. When has a singularity, the conventional form (with inverse even powers of *m*) is not usually valid. In this paper we derive the expansion in the case in which has the form

*Q*. For several easily recognizable classes of integrand function and for most familiar quadrature rules some of these coefficients are zero. An analogous expansion for the error functional with integrand function is also derived.

**[1]**Christopher T. H. Baker and Graham S. Hodgson,*Asymptotic expansions for integration formulas in one or more dimensions*, SIAM J. Numer. Anal.**8**(1971), 473–480. MR**0285115**, https://doi.org/10.1137/0708043**[2]**F. L. Bauer, H. Rutishauser, and E. Stiefel,*New aspects in numerical quadrature*, Proc. Sympos. Appl. Math., Vol. XV, Amer. Math. Soc., Providence, R.I., 1963, pp. 199–218. MR**0174177****[3]**J. S. R. Chisholm, A. Genz, and Glenys E. Rowlands,*Accelerated convergence of sequences of quadrature approximations*, J. Computational Phys.**10**(1972), 284–307. MR**0326998****[4]**M. E. A. EL TOM, Oral Contribution, Royal Irish Academy Conference on Numerical Analysis, Dublin, 1972.**[5]**A. C. GENZ, "An adaptive multi-dimensional quadrature procedure,"*Comp. Phys. Comm.*, v. 4, 1972, pp. 11-15.**[6]**A. C. GENZ,*Some Extrapolation Methods for the Numerical Calculation of Multi-Dimensional Integrals*, UKC Preprint No. AM/ACG/2, University of Kent at Canterbury, Kent, England.**[7]**David K. Kahaner,*Numerical quadrature by the 𝜖-algorithm*, Math. Comp.**26**(1972), 689–693. MR**0329210**, https://doi.org/10.1090/S0025-5718-1972-0329210-X**[8]**J. N. LYNESS, "Symmetric integration rules for hypercubes. I, II, III,"*Math. Comp.*, v. 19, 1965, pp. 260-276, 394-407, 625-637. MR**34**#952; #953; #954.**[9]**J. N. Lyness,*Computational techniques based on the Lanczos representation*, Math. Comp.**28**(1974), 81–123. MR**0334458**, https://doi.org/10.1090/S0025-5718-1974-0334458-6**[10]**J. N. Lyness and B. W. Ninham,*Numerical quadrature and asymptotic expansions*, Math. Comp.**21**(1967), 162–178. MR**0225488**, https://doi.org/10.1090/S0025-5718-1967-0225488-X**[11]**J. N. Lyness and B. J. J. McHugh,*On the remainder term in the 𝑁-dimensional Euler Maclaurin expansion.*, Numer. Math.**15**(1970), 333–344. MR**0267734**, https://doi.org/10.1007/BF02165125**[12]**Israel Navot,*An extension of the Euler-Maclaurin summation formula to functions with a branch singularity*, J. Math. and Phys.**40**(1961), 271–276. MR**0140876****[13]**Frank Stenger,*Integration formulae based on the trapezoidal formula*, J. Inst. Math. Appl.**12**(1973), 103–114. MR**0381261****[14]**A. H. Stroud,*Approximate calculation of multiple integrals*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. Prentice-Hall Series in Automatic Computation. MR**0327006****[15]**Hidetosi Takahasi and Masatake Mori,*Quadrature formulas obtained by variable transformation*, Numer. Math.**21**(1973/74), 206–219. MR**0331738**, https://doi.org/10.1007/BF01436624

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0408211-0

Keywords:
Multidimensional quadrature,
singularity,
error functional asymptotic expansion,
Romberg integration,
Euler-Maclaurin expansion

Article copyright:
© Copyright 1976
American Mathematical Society