An algorithm is given for computing the incomplete gamma function ratios P ( a , x ) and Q> ( a , x ) for a ⪈ 0, x ⪈ 0, a + x ≠ 0. Temme's uniform asymptotic expansions are used. The algorithm is robust; results accurate to 14 significant digits can be obtained. An' extensive set of coefficients for the Temme expansions is included. An algorithm, employing third-order Schröder iteration supported by Newton-Raphson iteration, is provided for computing x when a , P ( a , x ), and Q ( a , x ) are given. Three iterations at most are required to obtain 10 significant digit accuracy for x .

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New series expansions are developed for computing incomplete elliptic integrals of the first and second kind when the values of the amplitude and modulus are large. The classical series, which are obtained after a binomial expansion of the integrands, are used when the values of the amplitude and modulus are small. The range of use of each series is so selected as to maintain a minimum of rounding error. A special criterion is used to determine when the binomial series should be terminated. The calculation of elliptic integrals by these series expansions is compared with the calculation by the previously established Landen transformation, which has been used by Legendre. The new series yield more accurate results and the average time of computation is 30 per cent shorter. The computing program in the NORC subroutine for the calculation of elliptic integrals is described.

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