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Wynn cross rule

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1: Bibliography W
  • J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.
  • T. Watanabe, M. Natori, and T. Oguni (Eds.) (1994) Mathematical Software for the P.C. and Work Stations – A Collection of Fortran 77 Programs. North-Holland Publishing Co., Amsterdam.
  • Wolfram’s Mathworld (website)
  • P. Wynn (1966) Upon systems of recursions which obtain among the quotients of the Padé table. Numer. Math. 8 (3), pp. 264–269.
  • 2: 3.11 Approximation Techniques
    The Padé approximants can be computed by Wynn’s cross rule. Any five approximants arranged in the Padé table as … For the recursive computation of [ n + k / k ] f by Wynn’s epsilon algorithm, see (3.9.11) and the subsequent text. …
    3: Bibliography
  • G. Allasia and R. Besenghi (1987a) Numerical computation of Tricomi’s psi function by the trapezoidal rule. Computing 39 (3), pp. 271–279.
  • G. Allasia and R. Besenghi (1991) Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.
  • G. Allasia and R. Besenghi (1987b) Numerical calculation of incomplete gamma functions by the trapezoidal rule. Numer. Math. 50 (4), pp. 419–428.
  • Arblib (C) Arb: A C Library for Arbitrary Precision Ball Arithmetic.
  • Axiom (free interactive system) Center for Algorithms and Interactive Scientific Software.
  • 4: Bibliography T
  • N. M. Temme (1978) The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.
  • W. J. Thompson (1997) Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners. John Wiley & Sons Inc., New York.
  • 5: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2003b) Computing special functions by using quadrature rules. Numer. Algorithms 33 (1-4), pp. 265–275.
  • G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.
  • E. T. Goodwin (1949a) Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.
  • H. P. W. Gottlieb (1985) On the exceptional zeros of cross-products of derivatives of spherical Bessel functions. Z. Angew. Math. Phys. 36 (3), pp. 491–494.
  • GSL (free C library) GNU Scientific Library The GNU Project.
  • 6: 3.9 Acceleration of Convergence
    The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm: …
    7: 1.6 Vectors and Vector-Valued Functions
    Cross Product (or Vector Product)
    1.6.9 a × b = | i j k a 1 a 2 a 3 b 1 b 2 b 3 | = ( a 2 b 3 - a 3 b 2 ) i + ( a 3 b 1 - a 1 b 3 ) j + ( a 1 b 2 - a 2 b 1 ) k = a b ( sin θ ) n ,
    where n is the unit vector normal to a and b whose direction is determined by the right-hand rule; see Figure 1.6.1.
    See accompanying text
    Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify
    8: Bibliography M
  • Maple (commercial interactive system) Maplesoft.
  • Mathematica (commercial interactive system) Wolfram Research, Inc..
  • Matlab (commercial interactive system) The MathWorks, Inc..
  • Maxima (free interactive system)
  • mpmath (free python library)
  • 9: 2.11 Remainder Terms; Stokes Phenomenon
    As these lines are crossed exponentially-small contributions, such as that in (2.11.7), are “switched on” smoothly, in the manner of the graph in Figure 2.11.1. … Similar improvements are achievable by Aitken’s Δ 2 -process, Wynn’s ϵ -algorithm, and other acceleration transformations. …
    10: Bibliography Z
  • Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.
  • S. Zhang and J. Jin (1996) Computation of Special Functions. John Wiley & Sons Inc., New York.