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βΊMethods of computation for and symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981).
βΊFor symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989).
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βΊIn the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) (, , , ), binary64 (previously double precision) (, , , ) and binary128 (previously quad precision) (, , , ) are as in Figure 3.1.1.
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βΊFigure 3.1.1: Floating-point arithmetic.
Representation of data in the binary interchange formats for binary32, binary64 and binary128 (previously single, double and quad precision).
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βΊA transposition is a permutation that consists of a single cycle of length two.
…A permutation that consists of a single cycle of length can be written as the composition of two-cycles (read from right to left):
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βΊWith the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer.
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In association with the DLMF we will provide an index of all software for the
computation of special functions covered by the DLMF. It is our intention that
this will become an exhaustive list of sources of software for special functions.
In each case we will maintain a single link where readers can obtain more information
about the listed software. We welcome requests from software authors
(or distributors) for new items to list.
Note that here we will only include software with capabilities that go beyond the
computation of elementary functions in standard precisions since such software is
nearly universal in scientific computing environments.
D. E. Amos (1986)Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order.
ACM Trans. Math. Software12 (3), pp. 265–273.
β
Notes:
Single and double precision, maximum accuracy 18S.