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  • What does linear congruential generator do?

    A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. The method represents one of the oldest and best-known pseudorandom number generator algorithms.

  • What is the formula for linear congruential generator?

    An LCG is defined by the equation Ln ? (a ? Ln-1 + c) mod m, where the values of m (the modulus, a positive integer), a (the multiplier, a positive integer less than m) and c (the increment, either 0 or a positive integer less than m) are chosen by the designer of the LCG.

  • How do you calculate LCG period?

    Full-Period Theorem (Hull and Dobell, 1966) In general, cycle length determined by parameters m, a, and c: The LCG Zi = (aZi-1 + c) (mod m) has full period (m) if and only all three of the following hold: 1. c and m are relatively prime (i.e., the only positive integer that divides both c and m is 1).
  • The simple form of the LCG algorithm is as follows: X n+1 = (a X n +b) mod m , n? 0 ; A constant in the above algorithm is called a multiplier, the constant b is called an increment, and the constant m is called modulus.
[PDF] Chapter 6 - Random-Number Generation 6.1

Chapter 6

Random-Number Generation

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.2

Contents

¥AEProperties of Random Numbers

¥AEPseudo-Random Numbers ¥AEGenerating Random Numbers ¥AELinear Congruential Method ¥AECombined Linear Congruential Method ¥AETests for Random Numbers ¥AEReal Random Numbers

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.3

Overview

¥AEDiscuss characteristics and the generation of random numbers.

¥AESubsequently, introduce

tests for randomness:

¥AEFrequency test ¥AEAutocorrelation test

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.4

Overview

¥AEHistorically

¥AEThrow dices ¥AEDeal out cards ¥AEDraw numbered balls ¥AEUse digits of AE ¥AEMechanical devices (spinning disc, etc.) ¥AEElectric circuits

¥AEElectronic Random Number Indicator (ERNIE)

¥AECounting gamma rays

¥AEIn combination with a computer

¥AEHook up an electronic device to the computer ¥AERead-in a table of random numbers

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.5

Pseudo-Random Numbers

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.6

Pseudo-Random Numbers

¥AEApproach: Arithmetically generation (calculation) of random numbers ¥AEÒPseudoÓ, because generating numbers using a known method removes the potential for true randomness.

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number Ñ there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method. John von Neumann, 1951

6.7 Pseudo-Random Numbers ¥AEGoal: To produce a sequence of numbers in [0,1] that simulates, or imitates, the ideal properties of random numbers (RN).

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

É probably É can not be justified, but should merely be judged by their results. Some statistical study of the digits generated by a given recipe should be made, but exhaustive tests are impractical. If the digits work well on one problem, they seem usually to be successful with others of the same type. John von Neumann, 1951

6.8

Pseudo-Random Numbers

¥AEImportant properties of good random number routines: ¥AEFast ¥AEPortable to different computers ¥AEHave sufficiently long cycle ¥AEReplicable ¥AEVerification and debugging ¥AEUse identical stream of random numbers for different systems ¥AEClosely approximate the ideal statistical properties of

¥AEuniformity and ¥AEindependence

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.9

Pseudo-Random Numbers: Properties

¥AETwo important statistical properties:

¥AEUniformity ¥AEIndependence

¥AERandom number R

must be independently drawn from a uniform distribution with PDF:

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

otherwise ,0

10 ,1

xdxRE

PDF for random numbers

0 1 f(x) x

6.10

Pseudo-Random Numbers

¥AEProblems when generating pseudo-random numbers

¥AEThe generated numbers might not be uniformly distributed ¥AEThe generated numbers might be discrete-valued instead of

continuous-valued ¥AEThe mean of the generated numbers might be too high or too low ¥AEThe variance of the generated numbers might be too high or too low

¥AEThere might be dependence:

¥AEAutocorrelation between numbers ¥AENumbers successively higher or lower than adjacent numbers ¥AESeveral numbers above the mean followed by several

numbers below the mean

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.11

Generating Random Numbers

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.12

Generating Random Numbers

¥AEMidsquare method

¥AELinear Congruential Method (LCM) ¥AECombined Linear Congruential Generators (CLCG) ¥AERandom-Number Streams

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.13

Midsquare method

Generating Random Numbers

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.14

Midsquare method

¥AEFirst arithmetic generator: Midsquare method

¥AEvon Neumann and Metropolis in 1940s

¥AEThe Midsquare method:

¥AEStart with a four-digit positive integer Z

¥AECompute: to obtain an integer with up to eight digits ¥AETake the middle four digits for the next four-digit number

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

ZZZAE=

i Z

0 7182 - 51581124 1 5811 0.5811 33767721 2 7677 0.7677 58936329 3 9363 0.9363 87665769 É

6.15

Midsquare method

¥AEProblem: Generated numbers tend to 0

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

i Z AEZ

0 7182 - 51581124 1 5811 0,5811 33767721 2 7677 0,7677 58936329 3 9363 0,9363 87665769 4 6657 0,6657 44315649 5 3156 0,3156 09960336 6 9603 0,9603 92217609 7 2176 0,2176 04734976 8 7349 0,7349 54007801 9 78 0,0078 00006084 10 60 0,006 00003600 11 36 0,0036 00001296 12 12 0,0012 00000144 13 1 0,0001 00000001 14 0 0 00000000 15 0 0 00000000

6.16

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

É random numbers should not be generated with a method chosen at random. Some theory should be used.

Donald E. Knuth, The Art of Computer Programming, Vol. 2 6.17

Linear Congruential Method

Generating Random Numbers

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

6.18

Linear Congruential Method

¥AETo produce a sequence of integers X

, X , É between 0 and m-1 by following a recursive relationship:

¥AEAssumption: m > 0 and a < m, c < m, X

< m ¥AEThe selection of the values for a, c, m, and X drastically affects the statistical properties and the cycle length

¥AEThe random integers X

are being generated in [0, m-1]

Prof. Dr. Mesut GŸne

Ch. 6 Random-Number Generation

,...2,1,0 , mod )( imcaXX

The multiplier The increment The modulus

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