Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Positivity of the weights of extended Clenshaw-Curtis quadrature rules


Authors: Takemitsu Hasegawa, Hirosi Sugiura and Tatsuo Torii
Journal: Math. Comp. 60 (1993), 719-734
MSC: Primary 65D30; Secondary 65D32
DOI: https://doi.org/10.1090/S0025-5718-1993-1176710-2
MathSciNet review: 1176710
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that some extended Clenshaw-Curtis quadrature rules have all weights positive. We also present extended Filippi rules of open type having all weights positive. Conjectures on the possibility of other positive quadrature rules embedded in Clenshaw-Curtis or Filippi rule are suggested.


References [Enhancements On Off] (What's this?)

  • [1] R. Askey, Positivity of the Cotes numbers for some Jacobi abscissae II, J. Inst. Math. Appl. 24 (1979), 95-98. MR 539375 (80e:65028)
  • [2] M. Branders and R. Piessens, An extension of Clenshaw-Curtis quadrature, J. Comput. Appl. Math. 1 (1975), 55-65. MR 0371022 (51:7245)
  • [3] H. Brass, Eine Fehlerabschätzung für positive Quadraturformeln, Numer. Math. 47 (1985), 395-399. MR 808558 (86k:65019)
  • [4] C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960), 197-205. MR 0117885 (22:8659)
  • [5] G. Criscuolo, G. Mastroianni, and D. Occorsio, Convergence of extended Lagrange interpolation, Math. Comp. 55 (1990), 197-212. MR 1023044 (91c:65008)
  • [6] P. J. Davis and P. Rabinowitz, Methods of numerical integration, 2nd ed., Academic Press, Orlando, 1984. MR 760629 (86d:65004)
  • [7] S. Elhay and J. Kautsky, Algorithm 655-IQPACK: FORTRAN subroutines for the weights of interpolatory quadratures, ACM Trans. Math. Software 13 (1989), 399-415.
  • [8] D. Elliott, Truncation errors in two Chebyshev series approximations, Math. Comp. 19 (1965), 234-248. MR 0181084 (31:5313)
  • [9] H. Engels, Numerical quadrature and cubature, Academic Press, London, 1980. MR 587486 (83g:65002)
  • [10] S. Filippi, Angenäherte Tschebyscheff-Approximation einer Stammfunktion--eine Modifikation des Verfahrens von Clenshaw und Curtis, Numer. Math. 6 (1964), 320-328. MR 0170472 (30:710)
  • [11] W. Gautschi and S. E. Notaris, Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type, J. Comput. Appl. Math. 25 (1989), 199-224. MR 988057 (90d:65045)
  • [12] W. Gautschi and T. Rivlin, A family of Gauss-Kronrod quadrature formulae, Math. Comp. 51 (1988), 749-754. MR 958640 (89m:65029)
  • [13] W. M. Gentleman, Implementing Clenshaw-Curtis quadrature II. Computing the cosine transformation, Comm. ACM 15 (1972), 343-346. MR 0327002 (48:5344)
  • [14] G. Hämmerlin and K. H. Hoffmann, Numerische Mathematik, Springer-Verlag, Berlin, Heidelberg, 1989.
  • [15] T. Hasegawa, T. Torii, and I. Ninomiya, Generalized Chebyshev interpolation and its application to automatic quadrature, Math. Comp. 41 (1983), 537-553. MR 717701 (84m:65037)
  • [16] T. Hasegawa, T. Torii, and H. Sugiura, An algorithm based on the FFT for a generalized Chebyshev interpolation, Math. Comp. 54 (1990), 195-210. MR 990599 (91c:65009)
  • [17] P. Henrici, Applied and computational complex analysis, Vol.2, Wiley, New York, 1977. MR 0453984 (56:12235)
  • [18] J. P. Imhof, On the method for numerical integration of Clenshaw and Curtis, Numer. Math. 5 (1963), 138-141. MR 0157482 (28:715)
  • [19] J. N. Lyness, The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. Part I. Functions whose early derivatives are continuous, Math. Comp. 24 (1970), 101-135. MR 0260230 (41:4858)
  • [20] G. Monegato, Positivity of the weights of extended Gauss-Legendre quadrature rules, Math. Comp. 32 (1978), 243-245. MR 0458809 (56:17009)
  • [21] T. N. L. Patterson, An algorithm for generating interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions, ACM Trans. Math. Software 15 (1989), 123-136. MR 1062489 (91g:65004)
  • [22] F. Peherstorfer, Linear combination of orthogonal polynomials generating positive quadrature formulas, Math. Comp. 55 (1990), 231-241. MR 1023052 (90j:65043)
  • [23] P. Rabinowitz, J. Kautsky, S. Elhay, and J. C. Butcher, On sequences of imbedded integration rules, Numerical Integration: Recent Developments, Software and Applications (P. Keast and G. Fairweather eds.), Reidel, Dordrecht, 1987, pp. 113-139. MR 907115 (88j:65052)
  • [24] T. Torii, Fast Fourier sine and cosine transform based on the midpoint rule, J. Inform. Process. 15 (1974), 670-679, in Japanese. MR 0411136 (53:14875)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D30, 65D32

Retrieve articles in all journals with MSC: 65D30, 65D32


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1176710-2
Keywords: Extended Clenshaw-Curtis rule, numerical integration, positivity of weights
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society