Numerical quadrature by the algorithm
Author:
David K. Kahaner
Journal:
Math. Comp. 26 (1972), 689693
MSC:
Primary 65D30
MathSciNet review:
0329210
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Abstract: Modifications of Romberg's integration method are applicable to functions with endpoint singularities. Several authors have reached different conclusions on the usefulness of these schemes due to the potential difficulties in calculating certain exponents in the asymptotic error expansion. In this paper, we consider a method based on the algorithm which does not require the user to supply these parameters. Tests show this luxury produces a method substantially better than the unmodified Romberg's method, but not as good as the modified procedure which assumes all the exponents are known. It is possible to design a scheme which incorporates the best of both methods and allows the user to decide how much information he wishes to provide. The algorithm is stable numerically in quadrature applications and converges for functions with endpoint singularities of an algebraic or logarithmic nature.
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 J. N. Lyness & B. W. Ninham, ``Further asymptotic expansions for the error functional,'' Math. Comp., v. 23, 1969, pp. 7183. MR 39 #3682. MR 0242351 (39:3682)
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 C. deBoor, On Writing an Automatic Integration Algorithm, Proc. Math. Software Sympos., Purdue University, West Lafayette, Ind., 1970.
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 [17]
 P. Wynn, Upon a Conjecture Concerning a Method for Solving Linear Equations, and Certain Other Matters, MRC Technical Summary Report No. 626, 1966, p. 10.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819720329210X
PII:
S 00255718(1972)0329210X
Keywords:
algorithm,
acceleration of convergence,
extrapolation,
numerical quadrature
Article copyright:
© Copyright 1972
American Mathematical Society
