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21: Errata

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In ¶IEEE Standard (in §3.1(i)), the description was modified to reflect the most recent IEEE 754-2019 Floating-Point Arithmetic Standard IEEE (2019). In the new standard, single, double and quad floating-point precisions are replaced with new standard names of binary32, binary64 and binary128. Figure 3.1.1 has been expanded to include the binary128 floating-point memory positions and the caption has been updated using the terminology of the 2019 standard. A sentence at the end of Subsection 3.1(ii) has been added referring readers to the IEEE Standards for Interval Arithmetic IEEE (2015, 2018).

Suggested by Nicola Torracca.

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22: Notices

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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.

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23: DLMF Project News

error generating summary

24: 3.11 Approximation Techniques

… ►Here the single prime on the summation symbol means that the first term is to be halved. … ►More precisely, it is known that for the interval $[a,b]$, the ratio of the maximum value of the remainder … ►For many applications a spline function is a more adaptable approximating tool than the Lagrange interpolation polynomial involving a comparable number of parameters; see §3.3(i), where a single polynomial is used for interpolating $f(x)$ on the complete interval $[a,b]$. …

25: 18.39 Applications in the Physical Sciences

… ►which in one dimensional systems are typically non-degenerate, namely there is only a single eigenfunction corresponding to each ${ϵ}_n$, $n\ge 0$. … ►The non-relativistic Schrödinger equation describing a single, bound (negative energy) electron, in an $L^2$ eigenstate of energy $E$ is: … ►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with $Z$ protons, involve Laguerre polynomials of fractional index. … ►A major difficulty in such calculations, loss of precision, is addressed in Gautschi (2009) where use of variable precision arithmetic is discussed and employed. … ►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …