machine number
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1: 3.1 Arithmetics and Error Measures
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►The set of machine numbers
is the union of and the set
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►The lower and upper bounds for the absolute values of the nonzero machine numbers are given by
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Rounding
… ►Interval arithmetic is intended for bounding the total effect of rounding errors of calculations with machine numbers. …2: 27.15 Chinese Remainder Theorem
§27.15 Chinese Remainder Theorem
… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product has 20 digits, twice the number of digits in the data. … ►Even though the lengthy calculation is repeated four times, once for each modulus, most of it only uses five-digit integers and is accomplished quickly without overwhelming the machine’s memory. Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …3: 27.16 Cryptography
§27.16 Cryptography
… ►The primes are kept secret but their product , an 800-digit number, is made public. …With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer. For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large numbers. … ►If and are known, and can be determined (mod ) by straightforward calculations that require only a few minutes of machine time. …4: Bibliography C
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Some congruences for the Bernoulli numbers.
Amer. J. Math. 75 (1), pp. 163–172.
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-Bernoulli and Eulerian numbers.
Trans. Amer. Math. Soc. 76 (2), pp. 332–350.
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A note on Euler numbers and polynomials.
Nagoya Math. J. 7, pp. 35–43.
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Expansions of -Bernoulli numbers.
Duke Math. J. 25 (2), pp. 355–364.
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An algorithm for the machine calculation of complex Fourier series.
Math. Comp. 19 (90), pp. 297–301.
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