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minimax rational approximations

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1: 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 ) .

  • Cody (1969) provides minimax rational approximations for erf x and erfc x . The maximum relative precision is about 20S.

  • Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

  • Cody et al. (1970) gives minimax rational approximations to Dawson’s integral F ( x ) (maximum relative precision 20S–22S).

  • 2: 3.11 Approximation Techniques
    §3.11(iii) Minimax Rational Approximations
    Then the minimax (or best uniform) rational approximationA collection of minimax rational approximations to elementary and special functions can be found in Hart et al. (1968). …
    Example
    3: 5.23 Approximations
    Cody and Hillstrom (1967) gives minimax rational approximations for ln Γ ( x ) for the ranges 0.5 x 1.5 , 1.5 x 4 , 4 x 12 ; precision is variable. 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. Cody et al. (1973) gives minimax rational approximations for ψ ( x ) for the ranges 0.5 x 3 and 3 x < ; precision is variable. …
    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).

  • Cody and Thacher (1968) provides minimax rational approximations for E 1 ( x ) , with accuracies up to 20S.

  • Cody and Thacher (1969) provides minimax rational approximations for Ei ( x ) , with accuracies up to 20S.

  • 5: 25.20 Approximations
  • 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 .

  • 6: 9.19 Approximations
  • Moshier (1989, §6.14) provides minimax rational approximations for calculating Ai ( x ) , Ai ( x ) , Bi ( x ) , Bi ( x ) . They are in terms of the variable ζ , where ζ = 2 3 x 3 / 2 when x is positive, ζ = 2 3 ( - x ) 3 / 2 when x is negative, and ζ = 0 when x = 0 . The approximations apply when 2 ζ < , that is, when 3 2 / 3 x < or - < x - 3 2 / 3 . The precision in the coefficients is 21S.

  • 7: Bibliography J
  • 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.