Numerical solution of some classical differential-difference equations
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- by George Marsaglia, Arif Zaman and John C. W. Marsaglia PDF
- Math. Comp. 53 (1989), 191-201 Request permission
Abstract:
For differential-difference equations, we provide a method that gives numerical solutions accurate to hundreds or even thousands of digits. We illustrate with numerical solutions to three classical problems. With a few exceptions, previous claims of extended accuracy for these problems are found to be wrong.References
- R. Bellman and B. Kotkin, On the numerical solution of a differential-difference equation arising in analytic number theory, Math. Comp. 16 (1962), 473–475. MR 148248, DOI 10.1090/S0025-5718-1962-0148248-2
- N. G. De Bruijn, On the number of uncancelled elements in the sieve of Eratosthenes, Nederl. Akad. Wetensch., Proc. 53 (1950), 803–812 = Indagationes Math. 12, 247–256 (1950). MR 35785 A. Buchstab, "Asymptotic estimates of a general number-theoretic function," Mat. Sb. (N.S.), v. 44, 1937, pp. 1239-1246. (In Russian)
- B. Edwin Blaisdell and Herbert Solomon, On random sequential packing in the plane and a conjecture of Palasti, J. Appl. Probability 7 (1970), 667–698. MR 282389, DOI 10.1017/s0021900200110630
- H. Davenport and P. Erdös, The distribution of quadratic and higher residues, Publ. Math. Debrecen 2 (1952), 252–265. MR 55368 K. Dickman, "On the frequency of numbers containing prime factors of a certain relative magnitude," Ark. Mat. Astronom. Fys., v. 22A, 10, 1930, pp. 1-14.
- A. Dvoretzky and H. Robbins, On the “parking” problem, Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964), 209–225 (English, with Russian summary). MR 173275
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York-London, 1965. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin; Translated from the Russian by Scripta Technica, Inc; Translation edited by Alan Jeffrey. MR 0197789
- Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms. MR 633878
- M. Lal and P. Gillard, Evaluation of a constant associated with a parking problem, Math. Comp. 28 (1974), 561–564. MR 341814, DOI 10.1090/S0025-5718-1974-0341814-9 M. Lal & P. Gillard, Numerical Solution of Two Differential-Difference Equations of Analytic Theory of Numbers, Lecture Notes in Math., vol. 109, Springer-Verlag, Berlin and New York, 1969, pp. 179-187.
- J. van de Lune and E. Wattel, On the numerical solution of a differential-difference equation arising in analytic number theory, Math. Comp. 23 (1969), 417–421. MR 247789, DOI 10.1090/S0025-5718-1969-0247789-3
- David Mannion, Random space-filling in one dimension, Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964), 143–154 (English, with Russian summary). MR 177435
- V. Ramaswami, On the number of positive integers less than $x$ and free of prime divisors greater than $x^c$, Bull. Amer. Math. Soc. 55 (1949), 1122–1127. MR 31958, DOI 10.1090/S0002-9904-1949-09337-0
- Alfréd Rényi, On a one-dimensional problem concerning random space filling, Magyar Tud. Akad. Mat. Kutató Int. Közl. 3 (1958), no. 1-2, 109–127 (Hungarian, with English and Russian summaries). MR 104284
- Karl K. Norton, Numbers with small prime factors, and the least $k$th power non-residue, Memoirs of the American Mathematical Society, No. 106, American Mathematical Society, Providence, R.I., 1971. MR 0286739
- Sigmund Selberg, The number of cancelled elements in the sieve of Eratosthenes, Norsk Mat. Tidsskr. 26 (1944), 79–84 (Norwegian). MR 18691
- Dura W. Sweeney, On the computation of Euler’s constant, Math. Comp. 17 (1963), 170–178. MR 160308, DOI 10.1090/S0025-5718-1963-0160308-X
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp. 53 (1989), 191-201
- MSC: Primary 65L05; Secondary 65Q05
- DOI: https://doi.org/10.1090/S0025-5718-1989-0969490-3
- MathSciNet review: 969490