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1: 19.38 Approximations
Minimax polynomial approximations (§3.11(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. Approximations of the same type for K ( k ) and E ( k ) for 0 < k 1 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. …
2: 3.1 Arithmetics and Error Measures
Let E min E E max with E min < 0 and E max > 0 . …The integers p , E min , and E max are characteristics of the machine. … In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ( N = 32 , p = 24 , E min = - 126 , E max = 127 ), binary64 (previously double precision) ( N = 64 , p = 53 , E min = - 1022 , E max = 1023 ) and binary128 (previously quad precision) ( N = 128 , p = 113 , E min = - 16382 , E max = 16383 ) are as in Figure 3.1.1. … N min x N max , and …Then rounding by chopping or rounding down of x gives x - , with maximum relative error ϵ M . …
3: 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.

  • 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 .

  • 4: 33.25 Approximations
    Maximum relative errors range from 1. …
    5: 7.24 Approximations
  • 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).

  • 6: 14.16 Zeros
    The number of zeros of P ν μ ( x ) in the interval ( - 1 , 1 ) is max ( ν - | μ | , 0 ) if any of the following sets of conditions hold: … The number of zeros of P ν μ ( x ) in the interval ( - 1 , 1 ) is max ( ν - | μ | , 0 ) + 1 if either of the following sets of conditions holds: … Q ν μ ( x ) has max ( ν - | μ | , 0 ) + k zeros in the interval ( - 1 , 1 ) , where k can take one of the values - 1 , 0 , 1 , 2 , subject to max ( ν - | μ | , 0 ) + k being even or odd according as cos ( ν π ) and cos ( μ π ) have opposite signs or the same sign. …
    7: 15.15 Sums
    Here z 0 ( 0 ) is an arbitrary complex constant and the expansion converges when | z - z 0 | > max ( | z 0 | , | z 0 - 1 | ) . …
    8: 29.20 Methods of Computation
    Alternatively, the zeros can be found by locating the maximum of function g in (29.12.11).
    9: 1.10 Functions of a Complex Variable
    §1.10(v) Maximum-Modulus Principle
    Analytic Functions
    If f ( z ) is continuous on D ¯ and analytic in D , then | f ( z ) | attains its maximum on D .
    Harmonic Functions
    Schwarz’s Lemma
    10: Bibliography P
  • K. A. Paciorek (1970) Algorithm 385: Exponential integral Ei ( x ) . Comm. ACM 13 (7), pp. 446–447.
  • W. F. Perger, A. Bhalla, and M. Nardin (1993) A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • G. P. M. Poppe and C. M. J. Wijers (1990) Algorithm 680: Evaluation of the complex error function. ACM Trans. Math. Software 16 (1), pp. 47.
  • M. J. D. Powell (1967) On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.