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1: 2.7 Differential Equations
2.7.23 | ϵ j ( x ) | , 1 2 f - 1 / 2 ( x ) | ϵ j ( x ) | exp ( 1 2 𝒱 a j , x ( F ) ) - 1 , j = 1 , 2 ,
Here F ( x ) is the error-control function
2.7.24 F ( x ) = ( 1 f 1 / 4 d 2 d x 2 ( 1 f 1 / 4 ) - g f 1 / 2 ) d x ,
2.7.25 𝒱 a j , x ( F ) = a j x | ( 1 f 1 / 4 ( t ) d 2 d t 2 ( 1 f 1 / 4 ( t ) ) - g ( t ) f 1 / 2 ( t ) ) d t | .
2: Bibliography K
  • H. Kuki (1972) Algorithm 421. Complex gamma function with error control. Comm. ACM 15 (4), pp. 271–272.
  • 3: 15.19 Methods of Computation
    The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.
    4: 13.29 Methods of Computation
    The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.
    5: 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. … The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for ϕ near π / 2 with the improvements made in the 1970 reference. …