numerical%20approximation
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11: 3.8 Nonlinear Equations
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►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation:
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►Initial approximations to the zeros can often be found from asymptotic or other approximations to , or by application of the phase principle or Rouché’s theorem; see §1.10(iv).
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►Consider and .
We have and .
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►Corresponding numerical factors in this example for other zeros and other values of are obtained in Gautschi (1984, §4).
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12: Bibliography R
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Rational Chebyshev approximation by Remes’ algorithms.
Numer. Math. 7 (4), pp. 322–330.
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Remark on Algorithm 498: Airy functions using Chebyshev series approximations.
ACM Trans. Math. Software 7 (3), pp. 404–405.
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Composite approximations to the solutions of the Orr-Sommerfeld equation.
Studies in Appl. Math. 51, pp. 341–368.
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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow.
Studies in Appl. Math. 53, pp. 91–110.
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13: 25.11 Hurwitz Zeta Function
14: Bibliography N
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On the numerical evaluation of the generalised Fermi-Dirac integrals.
Comput. Phys. Comm. 76 (1), pp. 48–50.
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Approximations for the Bessel and Struve functions.
Math. Comp. 43 (168), pp. 551–556.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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15: Bibliography K
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Numerical Methods and Software.
Prentice Hall, Englewood Cliffs, N.J..
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The second Painlevé equation in electric probe theory. Some numerical solutions.
Zh. Vychisl. Mat. Mat. Fiz. 38 (6), pp. 992–1000 (Russian).
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The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface.
Technical Physics 49 (1), pp. 1–7.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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On Nielsen’s generalized polylogarithms and their numerical calculation.
Nordisk Tidskr. Informationsbehandling (BIT) 10, pp. 38–73.
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16: Bibliography S
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Computing the gamma function using contour integrals and rational approximations.
SIAM J. Numer. Anal. 45 (2), pp. 558–571.
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Numerical evaluation of the Hankel transform.
Comput. Phys. Comm. 116 (2-3), pp. 278–294.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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On numerical Bessel transformation.
Comput. Phys. Comm. 16 (3), pp. 383–387.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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17: 12.11 Zeros
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►Numerical calculations in this case show that corresponds to the th zero on the string; compare §7.13(ii).
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►For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii).
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12.11.9
►where the numerical coefficients have been rounded off.
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18: Bibliography D
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Methods of Numerical Integration.
2nd edition, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL.
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Texas Instruments, Inc..
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic approximation of Mathieu functions.
Methods Appl. Anal. 1 (2), pp. 143–168.
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Uniform asymptotic approximations for incomplete Riemann zeta functions.
J. Comput. Appl. Math. 190 (1-2), pp. 339–353.
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19: 19.36 Methods of Computation
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►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18).
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►If the iteration of (19.36.6) and (19.36.12) is stopped when ( and being approximated by and , and the infinite series being truncated), then the relative error in and is less than if we neglect terms of order .
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►The cases and require different treatment for numerical purposes, and again precautions are needed to avoid cancellations.
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►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).
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►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.
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20: 18.39 Applications in the Physical Sciences
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►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.
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►Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i.
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►While in the basis of (18.39.44) is simply a variational parameter, care must be taken, or the relationship between the results of the matrix variational approximation and the Pollaczek polynomials is lost, although this has no effect on the use of the variational approximations Reinhardt (2021a, b).
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►This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional functions of (18.39.53), provided that such information involves potentials, or projections onto functions, exactly expressed, or well approximated, in the finite basis of (18.39.44).
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►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics.
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