Utility

A linear congruential generator (LCG) is an algorithm that yields a sequence of randomized numbers calculated with a linear equation. The method represents one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is easy to understand, and they are easily implemented and fast, especially on computer hardware which can provide modulo arithmetic by storage-bit truncation.

The generator is defined by the recurrence relation:

$X_{n+1}\equiv \left(aX_{n}+c\right)~~{\pmod {m}}$

where $X$ is the sequence of pseudorandom values, and

$m,\,0<m$ – the "modulus"

$a,\,0<a<m$ – the "multiplier"

$c,\,0\leq c<m$ – the "increment"

$X_{0},\,0\leq X_{0}<m$ – the "seed" or "start value"

are integer constants that specify the generator. If c = 0, the generator is often called a multiplicative congruential generator (MCG), or Lehmer RNG. If c ≠ 0, the method is called a mixed congruential generator.

Period length

The period of a general LCG is at most m, and for some choices of factor a much less than that. Provided that the offset c is nonzero, the LCG will have a full period for all seed values if and only if:

1. $\,c$ and $\,m$ are relatively prime,
2. $\,a-1$ is divisible by all prime factors of $\,m$,
3. $\,a-1$ is a multiple of 4 if $\,m$ is a multiple of 4.

These three requirements are referred to as the Hull-Dobell Theorem. While LCGs are capable of producing pseudorandom numbers which can pass formal tests for randomness, this is extremely sensitive to the choice of the parameters c, m, and a.

Historically, poor choices had led to ineffective implementations of LCGs. A particularly illustrative example of this is RANDU, which was widely used in the early 1970s and led to many results which are currently being questioned because of the use of this poor LCG.

Parameters in common use

The most efficient LCGs have an m equal to a power of 2, most often m = 2³² or m = 2⁶⁴, because this allows the modulus operation to be computed by merely truncating all but the rightmost 32 or 64 bits. The following table lists the parameters of LCGs in common use, including built-in rand() functions in runtime libraries of various compilers.

Source	m	(multiplier) aTemplate:Ns	(increment) c	output bits of seed in rand() / Random(L)
Numerical Recipes	232	1664525	1013904223
Borland C/C++	232	22695477	1	bits 30..16 in rand(), 30..0 in lrand()
glibc (used by GCC)	231	1103515245	12345	bits 30..0
ANSI C: Watcom, Digital Mars, CodeWarrior, IBM VisualAge C/C++	231	1103515245	12345	bits 30..16
Borland Delphi, Virtual Pascal	232	134775813	1	bits 63..32 of (seed * L)
Microsoft Visual/Quick C/C++	232	214013 (343FD16)	2531011 (269EC316)	bits 30..16
Microsoft Visual Basic (6 and earlier)	224	1140671485 (43FD43FD16)	12820163 (C39EC316)
RtlUniform from Native API	231 − 1	2147483629 (7FFFFFED16)	2147483587 (7FFFFFC316)
Apple CarbonLib, C++11's minstd_rand0	231 − 1	16807	0	see MINSTD
C++11's minstd_rand	231 − 1	48271	0	see MINSTD
MMIX by Donald Knuth	264	6364136223846793005	1442695040888963407
Newlib	264	6364136223846793005	1	bits 63...32
VAX's MTH$RANDOM, old versions of glibc	232	69069	1
Java's java.utilTemplate:Not a typoRandom, glibc [ld]rand48[_r]()	248	25214903917	11	bits 47...16
Formerly common:
Template:Midsize	231	65539	0

As shown above, LCGs do not always use all of the bits in the values they produce. For example, the Java implementation operates with 48-bit values at each iteration but returns only their 32 most significant bits. This is because the higher-order bits have longer periods than the lower-order bits (see below). LCGs that use this truncation technique produce statistically better values than those that do not.

Advantages and disadvantages of LCGs

LCGs are fast and require minimal memory (typically 32 or 64 bits) to retain state. This makes them valuable for simulating multiple independent streams.

Hyperplanes of a linear congruential generator in three dimensions

LCGs should not be used for applications where high-quality randomness is critical. For example, it is not suitable for a Monte Carlo simulation because of the serial correlation (among other things). They should also not be used for cryptographic applications; see cryptographically secure pseudo-random number generator for more suitable generators. If a linear congruential generator is seeded with a character and then iterated once, the result is a simple classical cipher called an affine cipher; this cipher is easily broken by standard frequency analysis.

LCGs tend to exhibit some severe defects. For instance, if an LCG is used to choose points in an n-dimensional space, the points will lie on, at most, m^1/n hyperplanes (Marsaglia's Theorem, developed by George Marsaglia). This is due to serial correlation between successive values of the sequence X_n. The spectral test, which is a simple test of an LCG's quality, is based on this fact.

A further problem of LCGs is that the lower-order bits of the generated sequence have a far shorter period than the sequence as a whole if m is set to a power of 2. In general, the nth least significant digit in the base b representation of the output sequence, where b^k = m for some integer k, repeats with at most period bⁿ.

Nevertheless, LCGs may be a good option. For instance, in an embedded system, the amount of memory available is often severely limited. Similarly, in an environment such as a video game console taking a small number of high-order bits of an LCG may well suffice. The low-order bits of LCGs when m is a power of 2 should never be relied on for any degree of randomness whatsoever. Indeed, simply substituting 2ⁿ for the modulus term reveals that the low order bits go through very short cycles. In particular, any full-cycle LCG when m is a power of 2 will produce alternately odd and even results.

Comparison with other PRNGs

If higher-quality random numbers are needed, and sufficient memory is available (~ 2 kilobytes), then the Mersenne twister algorithm provides a vastly longer period (2¹⁹⁹³⁷ − 1) and variate uniformity. The Mersenne twister generates higher-quality deviates than almost any LCG.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. A common Mersenne twister implementation, interestingly enough, uses an LCG to generate seed data.

A Linear Feedback Shift Register PRNG can be implemented with essentially the same amount of memory and produces a stream of pseudorandom numbers with better randomness qualitiesPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. when considering streams of bits, albeit with a bit more computation.

The linear feedback shift register has a strong relationship to linear congruential generators. Given a few values in the sequence, some techniques can predict the following values in the sequence for not only linear congruent generators but any other polynomial congruent generator.

See also

• Full cycle
• Inversive congruential generator
• Multiply-with-carry
• Lehmer RNG (sometimes called the Park-Miller RNG)
• Combined Linear Congruential Generator

Notes

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References

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  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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• D. E. Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 3.2.1: The Linear Congruential Method, pp. 10–26.
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• Gentle, James E., (2003). Random Number Generation and Monte Carlo Methods, 2nd edition, Springer, ISBN 0-387-00178-6.
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang (in this paper, efficient algorithms are given for inferring sequences produced by certain pseudo-random number generators).

External links

• The simulation Linear Congruential Generator visualizes the correlations between the pseudo-random numbers when manipulating the parameters.
• Security of Random Number Generation: An Annotated Bibliography
• Linear Congruential Generators post to sci.math
• The "Death of Art" computer art project at Goldstein Technologies LLC, uses an LCG to generate 33,554,432 images
• P. L'Ecuyer and R. Simard, "TestU01: A C Library for Empirical Testing of Random Number Generators", May 2006, revised November 2006, ACM Transactions on Mathematical Software, 33, 4, Article 22, August 2007.