computer arithmetic

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11: 27.17 Other Applications

… ►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …

12: 4.48 Software

… ►All scientific programming languages, libraries, and systems support computation of at least some of the elementary functions in standard floating-point arithmetic (§3.1(i)). … ►A more complete list of available software for computing these functions is found in the Software Index; again, software that uses only standard floating-point arithmetic is excluded. …

13: Bibliography Y

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The calculation of the Riemann zeta function in the complex domain. USSR Comput. Math. and Math. Phys. 25 (2), pp. 111–119.

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External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i).

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Computation of the derivatives of the Riemann zeta-function in the complex domain. USSR Comput. Math. and Math. Phys. 28 (4), pp. 115–124.

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Notes:

Includes Fortran programs.

External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i), §25.21(iii).

Encodings:

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Software for interval arithmetic: A reasonably portable package. ACM Trans. Math. Software 5 (1), pp. 50–63.

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External Links:

ISSN 0098-3500, Document

Referenced by:

§4.48(ii).

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Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

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Notes:

Japanese with English summary. Double-precision Fortran.

External Links:

ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

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14: 3.6 Linear Difference Equations

… ►Difference equations are simple and attractive for computation. … ►Unless exact arithmetic is being used, however, each step of the calculation introduces rounding errors. … ►

Example 1. Bessel Functions

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Example 2. Weber Function

… ►The results of the computations are displayed in Table 3.6.1. …

15: 15.19 Methods of Computation

§15.19 Methods of Computation

… ►For fast computation of $F(a,b;c;z)$ with $a,b$ and $c$ complex, and with application to Pöschl–Teller–Ginocchio potential wave functions, see Michel and Stoitsov (2008). … ►The representation (15.6.1) can be used to compute the hypergeometric function in the sector . … ►The relations in §15.5(ii) can be used to compute $F(a,b;c;z)$, provided that care is taken to apply these relations in a stable manner; see §3.6(ii). … ►The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.

16: 33.23 Methods of Computation

§33.23 Methods of Computation

… ►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. …Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. …

17: 13.29 Methods of Computation

§13.29 Methods of Computation

… ►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. In Allasia and Besenghi (1991) and Allasia and Besenghi (1987a) the high accuracy of the trapezoidal rule for the computation of Kummer functions is described. … ►The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. … ►The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.

18: Bibliography B

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Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.

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External Links:

ISSN 1064-8275, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.74(vi), §3.8(iii).

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A program for computing the Fermi-Dirac functions. Comput. Phys. Comm. 21 (3), pp. 315–322.

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Notes:

Double-precision Fortran, maximum accuracy 10D.

External Links:

Document, Code

Referenced by:

2nd item.

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Computation of the polygamma functions. Comm. Statist. B—Simulation Comput. 13 (3), pp. 409–415.

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External Links:

ISSN 0361-0918, Document, MathReview (S. Conde)

Referenced by:

§5.24(iii).

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A Fortran multiple-precision arithmetic package. ACM Trans. Math. Software 4 (1), pp. 57–70.

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External Links:

ISSN 0098-3500, Document

Referenced by:

1st item.

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Algorithm 524: MP, A Fortran multiple-precision arithmetic package [A1]. ACM Trans. Math. Software 4 (1), pp. 71–81.

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External Links:

ISSN 0098-3500, Document, Remark. GAMS (module 524; package TOMS)

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3rd item, 1st item, 1st item, 1st item, 3rd item, 1st item, 1st item.

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19: Bibliography C

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On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.

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External Links:

MathReview (R. Nicolovius), ZentralBlatt

Referenced by:

§19.36(ii), §19.36(ii).

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An algorithm for computing logarithms and arctangents. Math. Comp. 26 (118), pp. 543–549.

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External Links:

FullText, MathReview (F. Gotze), ZentralBlatt

Referenced by:

§4.45(i).

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Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.1(iv).

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The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

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External Links:

ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.8(i).

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Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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External Links:

ISSN 0002-9920, MathReview (Willard Parker)

Referenced by:

§19.8(i).

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20: Bibliography K

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Algorithm 910: a portable C++ multiple-precision system for special-function calculations. ACM Trans. Math. Software 37 (4), pp. Art. 45, 27.

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Notes:

This system provides a uniform interface in C++, named e_float, to perform arithmetic operations and to calculate mathematical functions that are implemented in several different MP packages. It is a free, open-source, multiple precision package that allows the computation of many special functions. It supports calculations with 30….300 decimal digits and is interoperable with Microsoft’s CLR, Python and Mathematica.

External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

Erratum (V1.0.9) for References, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

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