Arif Zaman

Toward a universal random number generator

This article describes an approach towards a random number generator that passes all of the stringent tests for randomness we have put to it, and that is able to produce exactly the same sequence of uniform random variables in a wide variety of computers, including TRS80, Apple, Macintosh, Commodore, Kaypro, IBM PC, AT, PC and AT clones, Sun, Vax, IBM , 3090, Amdahl, CDC Cyber and even 205 and ETA supercomputers.

Monkey tests for random number generators

This article describes some very simple, as well as some quite sophisticated, tests that shed light on the suitability of certain random number generators.

A random number generator for PC's

Abstract It is now possible to do serious scientific work on personal computers (PCs). Many simulation studies, whether exploratory or for production runs, call for random numbers. We offer here a new kind of random number generator with implementation tailored specifically for PCs using Intel 8088/8086 or 80286/80386 processors. A floating-point coprocessor is not required or even useful for the generator, although, of course, a coprocessor may help other parts of a simulation. The generator has an extremely long period — some 2 1407 — requires only 43 stored values and uses only one arithmetic operation: subtraction. It is one of a new class of generators that we have recently developed. They are called add-with-carry and subtract-with-borrow generators. Related to lagged-Fibonacci generators, the new class has an interesting underlying theory, astonishingly long periods and provable uniformity for full sequences. This article describes a machine language subroutine that provides 32-bit random integers as well as uniform (single precision) reals with standard 24-bit fractions.

Some portable very-long-period random number generators

It is found that a proposed random number generator ran2, recently presented in the Numerical Recipes column [W. H. Press and S. A. Teukolsky, Comput. Phys. 6, 521–524 (1992)], is a good one, but a number of generators are presented that are at least as good and are simpler, much faster, and with periods ‘‘billions and billions’’ of times longer. They are presented not necessarily to supplant ran2, but to put it in perspective. Any serious user of Monte Carlo methods should have a variety of random number generators from which to choose. In addition to two specific programs, one in Fortran and one in C, a framework is offered within which the readers can easily fashion their own generators with periods ranging from 1027–10101.

Rapid evaluation of the inverse of the normal distribution function

Here is a method for very fast evaluation of the inverse of the normal distribution--in two versions. The first, given u, rapidly produces the solution x to 2 , to within the accuracy available in single precision arithmetic. The second is faster. Using one less term in an expansion, it provides accuracy to within 0.000002--suitable for generating a normal random variable by direct inversion of its distribution function.

Numerical solution of some classical differential-difference equations

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.

The distance between random points in rectangles

Since 1943, numerous papers have discussed the problem of the distribution of the distance between random points in rectangles, considering special cases such as two points in the same square, points in adjacent squares, two rectangles sharing a side and others. The problems arise in a variety of settings: operations research, population studies, urban planning, physical chemistry, chemical physics and materials science. Reported results are all of special cases with formulas specific to each case. It is possible to put such problems in a general setting with a single formula that handles all the particular cases. The method is well suited to computing and use of graphics. Now that computers and graphic output are commonplace it seems worthwhile to describe the general method and provide program outlines for computing and plotting the resulting distributions. We do that in this article.