Numerical solution of some classical differential-difference equations

Authors:
George Marsaglia, Arif Zaman and John C. W. Marsaglia

Journal:
Math. Comp. **53** (1989), 191-201

MSC:
Primary 65L05; Secondary 65Q05

MathSciNet review:
969490

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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**0148248**, 10.1090/S0025-5718-1962-0148248-2**[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**0035785****[3]**A. Buchstab, "Asymptotic estimates of a general number-theoretic function,"*Mat. Sb. (N.S.)*, v. 44, 1937, pp. 1239-1246. (In Russian)**[4]**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**0282389****[5]**H. Davenport and P. Erdös,*The distribution of quadratic and higher residues*, Publ. Math. Debrecen**2**(1952), 252–265. MR**0055368****[6]**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.**[7]**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**0173275****[8]**I. S. Gradshteyn and I. M. Ryzhik,*Table of integrals, series, and products*, Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey, Academic Press, New York-London, 1965. MR**0197789****[9]**Donald E. Knuth,*The art of computer programming. Vol. 2*, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR**633878****[10]**M. Lal and P. Gillard,*Evaluation of a constant associated with a parking problem*, Math. Comp.**28**(1974), 561–564. MR**0341814**, 10.1090/S0025-5718-1974-0341814-9**[11]**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.**[12]**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**0247789**, 10.1090/S0025-5718-1969-0247789-3**[13]**David Mannion,*Random space-filling in one dimension*, Magyar Tud. Akad. Mat. Kutató Int. Közl**9**(1964), 143–154 (English, with Russian summary). MR**0177435****[14]**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**0031958**, 10.1090/S0002-9904-1949-09337-0**[15]**Alfréd Rényi,*On a one-dimensional problem concerning random space filling*, Magyar Tud. Akad. Mat. Kutató Int. Közl.**3**(1958), no. no 1/2, 109–127 (Hungarian, with Russian and English summaries). MR**0104284****[16]**Karl K. Norton,*Numbers with small prime factors, and the least 𝑘th power non-residue*, Memoirs of the American Mathematical Society, No. 106, American Mathematical Society, Providence, R.I., 1971. MR**0286739****[17]**Sigmund Selberg,*The number of cancelled elements in the sieve of Eratosthenes*, Norsk Mat. Tidsskr.**26**(1944), 79–84 (Norwegian). MR**0018691****[18]**Dura W. Sweeney,*On the computation of Euler’s constant*, Math. Comp.**17**(1963), 170–178. MR**0160308**, 10.1090/S0025-5718-1963-0160308-X

Retrieve articles in *Mathematics of Computation*
with MSC:
65L05,
65Q05

Retrieve articles in all journals with MSC: 65L05, 65Q05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0969490-3

Keywords:
Difference equations,
differential-difference equations,
numerical solutions

Article copyright:
© Copyright 1989
American Mathematical Society