Numerical solution of some classical differential-difference equations

Marsaglia, George; Zaman, Arif; Marsaglia, John C. W.

doi:10.1090/S0025-5718-1989-0969490-3

Numerical solution of some classical differential-difference equations
HTML articles powered by AMS MathViewer

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

Mat. Sb. (N.S.)

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

Ark. Mat. Astronom. Fys.

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

Numerical Solution of Two Differential-Difference Equations of Analytic Theory of Numbers

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