.HE "/ /Description of Program Spectral/\*(DA/" .na INTRODUCTION .sp The purpose of program spectral is to check the quality of multiplier and modulus of linear congruential random number generators. One-parameter and two-parameter generators can be handled. .sp Program spectral implements Knuth's Algorithm S (The Spectral Test) [1, pp. 89-113]. The underlying theory is described on pages 89-98 of [1]. The algorithm implemented is described on pages 98-105. The generalization described in exercise 24, page 113 of [1] has also been implemented. .sp Program spectral employs software written by Steve Moshier [2] to do 100-digit extended precision arithmetic. .sp This document describes how to use program spectral. .sp 2 HOW TO USE PROGRAM SPECTRAL .sp Program spectral resides on disk as spectral.exe. Standard units are used for input and output: .sp .ai stderr - Prompts for inputs and and error messages stdin - Input data stdout - Results .sp .fi Start program spectral by issuing the command .sp spectral .sp at the command line. .sp My favorite input technique is to prepare an input file with the required data, one item per line. Then the redirect the input unit with: .sp sepctral < [file name] .sp If you decide to enter the data directly, you will be prompted for input data as follows: .sp .ai How Many Multipliers? [1 or 2]: Enter Multiplier: Enter Modulus: Enter Maximum Dimension: .sp .fi The text shown is produced when you respond '1' to the first prompt. You must enter either 1 or 2. If you don't, program spectral issues a warning message and asks you to reenter the input: .sp .ai How Many Multipliers? [1 or 2]: 3 Incorrect Response: 3 How Many Multipliers? [1 or 2]: .sp .fi .ne 2 The following prompts are produced when you respond '2' to the first prompt: .sp .ai How Many Multipliers? [1 or 2]: 2 Enter First Multiplier: Enter Second Multiplier: Enter Modulus: Enter Maximum Dimension: .sp .fi For example, to replicate the numbers on line 27, page 102 of [1] your responses must be: .sp .ai How Many Multipliers? [1 or 2]: 1 Enter Multiplier: 314159269 Enter Modulus: 2147483647 Enter Maximum Dimension: 6 .sp .fi The only two-parameter generator illustrated in [1] is given on line~28. Here is what your responses must be: .sp .ai How Many Multipliers? [1 or 2]: 2 Enter First Multiplier: 271828183 Enter Second Multiplier: -314159269 Enter Modulus: 2147483647 Enter Maximum Dimension: 6 .sp .fi A requirement of the Spectral Test is that the multiplier(s) and the modulus be relatively prime. When this is not the case, you will be advised with a printout like the following: .sp .ai Multiplier and modulus not relatively prime. GCD of Multiplier and Modulus = xxxx Error(s) detected in input data .sp .fi The minimum dimension that is tested is 2. If you enter something less than that your input is silently ignored. .sp 2 OUTPUT FROM PROGRAM SPECTRAL .sp Standard Printouts .br ------------------ .sp The first few lines of output echo your inputs. Thereafter, the results of the Spectral Test are printed - two lines per dimension requested. For example, when the test cited above for line 27 from [1] is run, the following output is produced: .sp .ai 1 Number of Multipliers 314159269 Multiplier 2147483647 Modulus 6 Maximum Dimension of Test 2-Dimensional Accuracy = 1432232969 No. Bits: 15.2 2-Dimensional Merit = 2.10 (Good > 0.1, Great > 1.0) 3-Dimensional Accuracy = 899290 No. Bits: 9.9 3-Dimensional Merit = 1.66 (Good > 0.1, Great > 1.0) 4-Dimensional Accuracy = 36985 No. Bits: 7.6 4-Dimensional Merit = 3.14 (Good > 0.1, Great > 1.0) 5-Dimensional Accuracy = 3427 No. Bits: 5.9 5-Dimensional Merit = 1.69 (Good > 0.1, Great > 1.0) 6-Dimensional Accuracy = 1144 No. Bits: 5.1 6-Dimensional Merit = 3.60 (Good > 0.1, Great > 1.0) .sp .fi The numbers that you see in this example exactly replicate those given in [1]. The output you see here has been cropped to fit on the page. .sp Error Printouts .br --------------- .sp As stated previously, you must enter 1 or 2 to the first prompt. Failure to do so can result is severe mental anguish, to wit: .sp .ai How Many Multipliers? [1 or 2]: 3 <-- Uncalled for Incorrect Response: 3 How Many Multipliers? [1 or 2]: .sp .fi It is a requirement that the multiplier(s) under test and the modulus be relatively prime. When this is not the case, you are shown the error of your ways as follows: .sp .ai How Many Multipliers? [1 or 2]: 1 Enter Multiplier: 314159269 Enter Modulus: 2147483639 Multiplier and modulus not relatively prime. GCD of Multiplier and Modulus = 17 Error(s) detected in input data .sp .fi REFERENCES .sp 1.~Donald E. Knuth, The Art of Computer Programming, Second Edition, Vol. 2, Seminumerical Algorithms, Addison-Wesley, Reading, Mass. (1981). .sp 2.~Steven L. Moshier, Methods and Programs for Mathematical Functions, Prentice-Hall (1989)