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1: 19.38 Approximations
Minimax polynomial approximations3.11(i)) for K ( k ) and E ( k ) in terms of m = k 2 with 0 m < 1 can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. …
2: 3.11 Approximation Techniques
§3.11(i) Minimax Polynomial Approximations
If we have a sufficiently close approximationFor examples of minimax polynomial approximations to elementary and special functions see Hart et al. (1968). … Since L 0 = 1 , L n is a monotonically increasing function of n , and (for example) L 1000 = 4.07 , this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile. … The theory of polynomial minimax approximation given in §3.11(i) can be extended to the case when p n ( x ) is replaced by a rational function R k , ( x ) . …
3: 7.24 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for erf x , erfc x and the auxiliary functions f ( x ) and g ( x ) .

  • 4: 6.20 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for E 1 ( x ) + ln x , x e x E 1 ( x ) , and the auxiliary functions f ( x ) and g ( x ) . These are included in Abramowitz and Stegun (1964, Ch. 5).

  • 5: 5.23 Approximations
    Hart et al. (1968) gives minimax polynomial and rational approximations to Γ ( x ) and ln Γ ( x ) in the intervals 0 x 1 , 8 x 1000 , 12 x 1000 ; precision is variable. …
    6: 18.38 Mathematical Applications
    §18.38(i) Classical OP’s: Numerical Analysis
    Approximation Theory
    The scaled Chebyshev polynomial 2 1 - n T n ( x ) , n 1 , enjoys the “minimax” property on the interval [ - 1 , 1 ] , that is, | 2 1 - n T n ( x ) | has the least maximum value among all monic polynomials of degree n . …
    Integrable Systems
    Ultraspherical polynomials are zonal spherical harmonics. …
    7: 25.20 Approximations
    §25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

  • Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of s ζ ( s + 1 ) and ζ ( s + k ) , k = 2 , 3 , 4 , 5 , 8 , for 0 s 1 (23D).

  • Morris (1979) gives rational approximations for Li 2 ( x ) 25.12(i)) for 0.5 x 1 . Precision is varied with a maximum of 24S.

  • Antia (1993) gives minimax rational approximations for Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for the intervals - < x 2 and 2 x < , with s = - 1 2 , 1 2 , 3 2 , 5 2 . For each s there are three sets of approximations, with relative maximum errors 10 - 4 , 10 - 8 , 10 - 12 .

  • 8: Bibliography J
  • L. Jager (1997) Fonctions de Mathieu et polynômes de Klein-Gordon. C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).
  • X.-S. Jin and R. Wong (1998) Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.
  • X.-S. Jin and R. Wong (1999) Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.
  • J. H. Johnson and J. M. Blair (1973) REMES2 — a Fortran program to calculate rational minimax approximations to a given function. Technical Report Technical Report AECL-4210, Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories, Chalk River, Ontario.