level-index arithmetic
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1: 19.8 Quadratic Transformations
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§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)
… ►As , and converge to a common limit called the AGM (Arithmetic-Geometric Mean) of and . …showing that the convergence of to 0 and of and to is quadratic in each case. … ►
19.8.5
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►Again, and converge quadratically to and 0, respectively, and converges to 0 faster than quadratically.
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2: 3.1 Arithmetics and Error Measures
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§3.1(ii) Interval Arithmetic
… ►§3.1(iii) Rational Arithmetics
… ►§3.1(iv) Level-Index Arithmetic
… ►In level-index arithmetic is represented by (or for negative numbers). … ►For further references on level-index arithmetic (and also other arithmetics) see Anuta et al. (1996). …3: Bibliography C
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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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The arithmetic-geometric mean of Gauss.
Enseign. Math. (2) 30 (3-4), pp. 275–330.
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Gauss and the arithmetic-geometric mean.
Notices Amer. Math. Soc. 32 (2), pp. 147–151.
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4: 4.44 Other Applications
§4.44 Other Applications
… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). …5: 27.17 Other Applications
§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. …6: 4.48 Software
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►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)).
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►Here we provide links to the research literature describing the implementation of algorithms in software for the evaluation of functions described in this chapter when the arithmetic is nonstandard.
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►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.
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§4.48(ii) Interval Arithmetic
…7: 15.17 Mathematical Applications
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►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)).
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8: 22.20 Methods of Computation
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§22.20(ii) Arithmetic-Geometric Mean
… ►Then as sequences , converge to a common limit , the arithmetic-geometric mean of . … ►The rate of convergence is similar to that for the arithmetic-geometric mean. … ►using the arithmetic-geometric mean. … ►Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute . …9: Annie A. M. Cuyt
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►As a consequence her expertise spans a wide range of activities from pure abstract mathematics to computer arithmetic and different engineering applications.
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