Monotonicity in Romberg quadrature

Author:
Torsten Ström

Journal:
Math. Comp. **26** (1972), 461-465

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1972-0307448-5

MathSciNet review:
0307448

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Abstract: Monotonicity of one or more derivatives of the integrand is shown to imply a corresponding property of the approximating Romberg scheme. This is of importance in connection with error estimation by majorants [6]. The monotonicity properties are derived from an elementary study of the kernel functions involved. A possible explanation is given of the monotonicity which frequently occurs in applications where nothing is presupposed about the signs of derivatives of the integrand. Finally, a nonlinear addition to Havie's scheme is suggested.

**[1]**F. L. Bauer, H. Rutishauser & E. Stiefel,*New Aspects in Numerical Quadrature*, Proc. Sympos. Appl. Math., vol. 15, Amer. Math. Soc., Providence, R.I., 1963, pp. 198-218. MR**30**#4384. MR**0174177 (30:4384)****[2]**T. Håvie, ``On a modification of Romberg's algorithm,''*Nordisk Tidskr. Informationsbehandling*, v. 6, 1966, pp. 24-30. MR**33**#3460. MR**0195257 (33:3460)****[3]**T. Håvie, ``Derivation of explicit expressions for the error terms in the ordinary and the modified Romberg algorithms,''*Nordisk Tidskr. Informationsbehandling*, v. 9, 1969, pp. 18-29. MR**40**#5126. MR**0251900 (40:5126)****[4]**J. N. Lyness & L. M. Delves, ``On numerical contour integration round a closed contour,''*Math. Comp.*, v. 21, 1967, pp. 561-577. MR**37**#4962. MR**0229388 (37:4962)****[5]**T. Ström, ``Strict error bounds in Romberg quadrature,''*Nordisk Tidskr. Informationsbehandling*, v. 7, 1967, pp. 314-321. MR**36**#7329. MR**0224285 (36:7329)****[6]**T. Ström,*Absolutely Monotonic Majorants and Their Use for Automatic Strict Error Estimation in Some Areas of Numerical Analysis*, Report NA 71.34, Department of Computer Science, Royal Institute of Technology, -100 44 Stockholm, Sweden.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0307448-5

Keywords:
Romberg,
quadrature

Article copyright:
© Copyright 1972
American Mathematical Society