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Rational approximations, software and test methods for sine and cosine integrals. (English) Zbl 0863.65004
The present work develops suitable rational approximations to the sine integral Si(x) and cosine integral Ci(x) with maximal accuracy of 20sf., except near the zeros of Ci(x) for x>6. The implementation of these approximations into a robust and reliable code is then considered. Finally, a test procedure, to assess the performance of Si and Ci codes, is developed and applied to various available programs. Use of the tests discovers a major error in the netlib for codes for Si.
MSC:
65D20Computation of special functions, construction of tables
33B20Incomplete beta and gamma functions
Software:
SPECFUN; MACHAR
References:
[1]M. Abramowitz and I.A. Stegun (eds.),Handbook of Mathematical Functions, Dover, New York, 1965.
[2]C.W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Camb. Phil. Soc. 53 (1957) 134–149. · doi:10.1017/S0305004100032072
[3]W.J. Cody, Algorithm 665: MACHAR: A subroutine to dynamically determine machine parameters, ACM Trans. Math. Soft., 14 (1988) 303–311. · Zbl 0665.65049 · doi:10.1145/50063.51907
[4]W.J. Cody, Algorithm 715: SPECFUN – A portable FORTRAN package of special function routines and test drivers, ACM Trans. Math. Softw. 19 (1993) 22–32. · Zbl 0889.65009 · doi:10.1145/151271.151273
[5]W.J. Cody, W. Fraser and J.F. Hart, Rational Chebyshev approximation using linear equations, Numer. Math. 12 (1968) 242–251. · Zbl 0169.19801 · doi:10.1007/BF02162506
[6]W.J. Cody and L. Stoltz, The use of Taylor series to test accuracy of function programs, ACM Trans. Math. Soft. 17 (1991) 55–63. · Zbl 0900.65036 · doi:10.1145/103147.103154
[7]J.J. Dongarra and E. Grosse, Distribution of mathematical software via electronic mail, Comm. ACM 30 (1987) 403–407. · doi:10.1145/22899.22904
[8]W. Gautschi, Algorithm 282 – derivatives ofe x /x, cosx/x, and sinx/x, Comm. ACM 9 (1966) 272. · doi:10.1145/365278.365519
[9]W. Gautschi, Computation of successive derivatives off(z)/z, Math. Comp. 20 (1966) 209–214.
[10]W. Gautschi and B.J. Klein, Recursive computation of certain derivatives – a study of error propogation, Comm. ACM 13 (1970) 7–9. · Zbl 0185.40903 · doi:10.1145/361953.361959
[11]W. Gautschi and B.J. Klein, Remark on Algorithm 282 – Derivatives ofe x /x, cosx/x, and sinx/x, Comm. ACM 13 (1970) 53–54. · doi:10.1145/361953.361988
[12]J.F. Hart, E.W. Cheney, C.L. Lawson, H.J. Maehhly, C.K. Mesztenyi, J.R. Rice, H.G. Thacher and C. Witzgall,Computer Approximations, Wiley, New York, 1968.