Abstract
We consider fast methods, based on FFT techniques, for obtaining families of quadrature rules approximating
. The rules are fractional quadrature rules derived from approximations to \(\smallint\!^{nh}_{0}\,\,\varphi({\text{s}}){\text{ds}}\,({\text{n}}\,\,\epsilon\,\,{\text{Z}}_{+}\!)\) generated by implicit linear multistep formulae. Suitable “starting” weights, computed using the Björck-Pereyra algorithm and FFT techniques, produce formulae with good order accuracy for functions φ(s) of the form \(\varphi({\text{s}})\,=\,\varphi_{0}\,+\varphi_{1}{\text{s}}^{1/2}\,+\,\varphi_{2}{\text{s}}\,+\,\varphi_{3}{\text{s}}^{3/2}\,+\,\varphi_{4}{\text{s}}^{2}\,+\,\,\ldots\,{\text{as\,\,\,s}}\,\to\,0\). The discussion is associated with FORTRAN 77 code given in [2].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
BAKER, C. T. H & DERAKHSHAN, M.S. ‘The use of NAG FFT routines in the construction of functions of power series used in fractional quadrature rules.’ Numer Anal Tech Rept. 115, University of Manchester (April 1986).
BAKER, C. T. H & DERAKHSHAN, M.S. ‘A code for fast generation of quadrature rules with special properties’ Appendix Numer Anal Tech Rept, 121 University of Manchester (August 1986).
GOLUB, G. & VAN LOAN, C.F. ‘Matrix Computations’ North Oxford Academic, Oxford 1983.
HAIRER, E. LUBICH, C. & SCHLICHTE, M. ‘Fast numerical solution of weakly singular Volterra integral equations’. Tech Rept Dept Math., University of Geneva, May 1986.
HENRICI, P. ‘Fast Fourier methods in computational complex analysis’. SIAM Review 21 (1979) pp 481–529.
HIGHAM, N. ‘Error analysis of the Björck -Pereyra algorithm for solving Vandermonde systems’ Numer Anal Tech Rept 108, University of. Manchester, Dec.1985.
LUBICH, C. ‘Discretized fractional calculus‘. SIAM J. Math. Anal, (to appear; preprinted 1985).
LUBICH, C. ‘Fractional linear multistep methods for Abel-Volterra integral equations of the second kind’. Math Comp 45 (1985) pp 463–469.
NEVANLINNA, O. ‘Positive quadratures for Volterra equations’ Computing 16 (1976) pp 349–357.
NUMERICAL ALGORITHMS GROUP. The NAG manual (Mark 11) NAG Central Office, Banbury Rd. Oxford.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 D. Reidel Publishing Company
About this chapter
Cite this chapter
Baker, T.H., Derakhshan, M.S. (1987). Fast Generation of Quadrature Rules with Some Special Properties. In: Keast, P., Fairweather, G. (eds) Numerical Integration. NATO ASI Series, vol 203. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3889-2_4
Download citation
DOI: https://doi.org/10.1007/978-94-009-3889-2_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8227-3
Online ISBN: 978-94-009-3889-2
eBook Packages: Springer Book Archive