About the Project
NIST

minimax rational functions

AdvancedHelp

(0.002 seconds)

7 matching pages

1: 3.11 Approximation Techniques
§3.11(iii) Minimax Rational Approximations
Then the minimax (or best uniform) rational approximation … 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 ) . … A collection of minimax rational approximations to elementary and special functions can be found in Hart et al. (1968). …
Example
2: 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 et al. (1970) gives minimax rational approximations to Dawson’s integral F ( x ) (maximum relative precision 20S–22S).

  • 3: 5.23 Approximations
    §5.23(i) Rational 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. … See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of Γ ( z ) . …
    4: 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.
  • 5: 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.

  • 6: 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).

  • 7: 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 .