About the Project

computation%20by%20quadrature

AdvancedHelp

(0.002 seconds)

10 matching pages

1: Bibliography G
  • M. J. Gander and A. H. Karp (2001) Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.
  • W. Gautschi (2002a) Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.
  • A. Gil, J. Segura, and N. M. Temme (2002c) Computing complex Airy functions by numerical quadrature. Numer. Algorithms 30 (1), pp. 11–23.
  • A. Gil, J. Segura, and N. M. Temme (2003b) Computing special functions by using quadrature rules. Numer. Algorithms 33 (1-4), pp. 265–275.
  • P. M. W. Gill and S. Chen (2003) Radial quadrature for multi exponential integrands. J. Comput. Chem. 24 (4), pp. 732–740.
  • 2: 19.36 Methods of Computation
    §19.36 Methods of Computation
    The computation is slowest for complete cases. … For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. …
    3: Bibliography B
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • R. Barakat and E. Parshall (1996) Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.
  • F. L. Bauer, H. Rutishauser, and E. Stiefel (1963) New Aspects in Numerical Quadrature. In Proc. Sympos. Appl. Math., Vol. XV, pp. 199–218.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • R. Bulirsch and H. Rutishauser (1968) Interpolation und genäherte Quadratur. In Mathematische Hilfsmittel des Ingenieurs. Teil III, R. Sauer and I. Szabó (Eds.), Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 141, pp. 232–319.
  • 4: Bibliography C
  • R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.
  • B. C. Carlson (1965) On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.
  • B. C. Carlson (1972a) An algorithm for computing logarithms and arctangents. Math. Comp. 26 (118), pp. 543–549.
  • A. D. Chave (1983) Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • 5: Bibliography I
  • Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J 0 ( z ) i J 1 ( z ) and of Bessel functions J m ( z ) of any real order m . Linear Algebra Appl. 194, pp. 35–70.
  • Y. Ikebe, Y. Kikuchi, and I. Fujishiro (1991) Computing zeros and orders of Bessel functions. J. Comput. Appl. Math. 38 (1-3), pp. 169–184.
  • M. Ikonomou, P. Köhler, and A. F. Jacob (1995) Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • A. Iserles, S. P. Nørsett, and S. Olver (2006) Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.
  • 6: Bibliography S
  • R. P. Sagar (1991a) A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals. Comput. Phys. Comm. 66 (2-3), pp. 271–275.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
  • A. H. Stroud and D. Secrest (1966) Gaussian Quadrature Formulas. Prentice-Hall Inc., Englewood Cliffs, N.J..
  • 7: 18.40 Methods of Computation
    §18.40 Methods of Computation
    A numerical approach to the recursion coefficients and quadrature abscissas and weights
    A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let N be a positive integer and define … Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain w ( x ) . … The quadrature points and weights can be put to a more direct and efficient use. …
    8: Bibliography R
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • REDUCE (free interactive system)
  • W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.
  • G. F. Remenets (1973) Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.
  • J. Rys, M. Dupuis, and H. F. King (1983) Computation of electron repulsion integrals using the Rys quadrature method. J. Comput. Chem. 4 (2), pp. 154–175.
  • 9: 18.39 Applications in the Physical Sciences
    Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge. … Full expressions for both A x i , l and B l ( x ) are given in Yamani and Reinhardt (1975) and it is seen that | A x i , l / B l ( x i ) | 2 = w i N / w CP ( x i ) where w i N is the Gaussian-Pollaczek quadrature weight at x = x i , and w CP ( x i ) is the Gaussian-Pollaczek weight function at the same quadrature abscissa. This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional L 2 functions of (18.39.53), provided that such information involves potentials, or projections onto L 2 functions, exactly expressed, or well approximated, in the finite basis of (18.39.44). The equivalent quadrature weight, w i / w CP ( x i ) , also forms the foundation of a novel inversion of the Stieltjes–Perron moment inversion discussed in §18.40(ii). …
    10: Bibliography W
  • J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.
  • J. Wimp (1984) Computation with Recurrence Relations. Pitman, Boston, MA.
  • J. Wimp (1985) Some explicit Padé approximants for the function Φ / Φ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.
  • M. E. Wojcicki (1961) Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.
  • R. Wong (1982) Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.