Vitale's random Brunn–Minkowski inequality

In computer science, multiply-with-carry (MWC) is a method invented by George Marsaglia for generating sequences of random integers based on an initial set from two to many thousands of randomly chosen seed values. The main advantages of the MWC method are that it invokes simple computer integer arithmetic and leads to very fast generation of sequences of random numbers with immense periods, ranging from around 2⁶⁰ to 2^2000000.

As with most pseudorandom number generators, the resulting sequences are functions of the randomly chosen seed values.

General theory

A MWC sequence is based on arithmetic modulo a base b, usually b = 2³², because arithmetic modulo of that b is automatic in most computers. However, sometimes a base such as b = 2³² − 1 is used, because arithmetic for modulus 2³² − 1 requires only a simple adjustment from that for 2³², and theory for MWC sequences based on modulus 2³² has some nagging difficulties avoided by using b = 2³² − 1.

In its most common form, a lag-r MWC generator requires a base b, a multiplier a, and a set of r+1 random seed values, consisting of r residues of b,

x₀, x₁, x₂ ,..., x_r−1,

and an initial carry cr−1 < a.

The lag-r MWC sequence is then a sequence of pairs x_n, c_n determined by

$x_n=(a{x}_{n-r}+c_{n-1})modb,c_n=\lfloor \frac{a{x}_{n-r}+c_{n-1}}{b}\rfloor ,n\ge r,$

and the MWC generator output is the sequence of x's,

x_r , x_r+1 , x_r+2, ...

The period of a lag-r MWC generator is the order of b in the multiplicative group of numbers modulo ab^r − 1. It is customary to choose a's so that p = ab^r − 1 is a prime for which the order of b can be determined. Because 2 is a quadratic residue of numbers of the form 8k±1, b = 2³² cannot be a primitive root of p = ab^r − 1. Therefore there are no MWC generators for base 2³² that have the maximum possible period, one of the difficulties that use of b = 2³² − 1 overcomes.

A theoretical problem with MWC generators, pointed out by Couture and l'Ecuyer (1997) is that the most significant bits are slightly biased; complementary-multiply-with-carry generators do not share this problem: "We shall see that, for the complementary MWC, each bit of the output value is fair, that is, the two binary digits will appear equally often in a full period, a property not shared by MWC generators." They do not appear to elaborate further as to the extent of the bias. Complementary-multiply-with-carry generators also require slightly more computation time per iteration, so there is a tradeoff to evaluate depending on implementation requirements.

Comparisons with linear congruential generators

Linear congruential generators are implemented as

$x_{n+1}=(a{x}_n+c)mod{2}^{32},$

because most arithmetic processors are able to put the multiplier a and the current x in 32-bit registers, form the 64-bit product in adjoining registers, and take the lower 32 bits as the product, that is, form

$a\times xmod{2}^{32}$.

Adding the 32-bit c to that lower half then provides (ax+c) mod 2³². If a mod 8 is 1 or 5 and c is odd, the resulting base 2³² congruential sequence will have period 2³².

A lag-1 multiply-with-carry generator allows us to make the period nearly 2⁶³ by using those same computer operations, except that this time the top half of the 64-bit product is used rather than ignored after the 64 bits are formed. It's used as a new carry value c rather than the fixed carry value of the standard congruential sequence: Get ax+c in 64-bits, then form a new c as the top half of those 64 bits, and the new x as the bottom half.

With multiplier a specified, each pair of input values x, c is converted to a new pair,

$x\leftarrow (ax+c)mod{2}^{32},c\leftarrow \lfloor \frac{ax+c}{2^{32}}\rfloor .$

If x and c are not both zero, then the period of the resulting multiply-with-carry sequence will be the order of b = 2³² in the multiplicative group of residues modulo ab − 1, that is, the smallest n such that bⁿ = 1 mod (ab − 1). If we choose an a of 28 to 31 bits such that ab−1 is a "safe prime", that is both ab − 1 and ab/2 − 1 are prime, then the period will be ab/2 − 1, approaching 2⁶³, which in practice may be an acceptably large subset of the number of possible 32-bit pairs (x, c).

Following are some maximal values of a for computer applications which satisfy the above safe prime condition:

Bits in a	b	Maximum a Such That ab-1 is a Safe Prime	Period
15	216	32,718	1,072,103,423
16	216	65,184	2,135,949,311
31	232	2,147,483,085	4,611,684,809,394,094,079
32	232	4,294,967,118	9,223,371,654,602,686,463
64	264	18,446,744,073,709,550,874	170,141,183,460,469,224,887,945,252,369,640,456,191
128	2128	2128-10,408	2127(2128-10,408)-1
256	2256	2256-9166	2255(2256-9166)-1
512	2512	2512-150,736	2511(2512-150,736)-1

However, as being a safe prime does not affect the randomness of the sequence, one may instead simply choose a such that the order of b is ab/2 − 1. The following are again maximum values of a of various sizes.

Bits in a	b	Maximum a Such that b has order ab/2−1	Period
15	216	32,739	1,072,791,551
16	216	65,514	2,146,762,751
31	232	2,147,483,580	4,611,685,872,398,499,839
32	232	4,294,967,220	9,223,371,873,646,018,559
63	264	9,223,372,036,854,775,668	85,070,591,730,234,614,574,571,566,698,273,439,743
64	264	18,446,744,073,709,551,500	170,141,183,460,469,230,661,776,147,440,730,111,999

Here is a comparison of congruential and MWC sequences for the simple case of arithmetic modulo 10; here the "registers" are a single digit, adjoining registers are two digits:

Starting with $x_0=1$, the congruential sequence

$x_n=(7x_{n-1}+3)mod10,$

has this sequence of adjoining registers:

$10,03,24,31,10,03,24,31,10,\dots ,$

and the output sequence of x's, (the rightmost register), has period 4:

$0,3,4,1,0,3,4,1,0,3,4,1,\dots$

Starting with $x_0=1,c_0=3$, the MWC sequence

$x_n=(7x_{n-1}+c_{n-1})mod10,c_n=\lfloor \frac{7x_{n-1}+c_{n-1}}{10}\rfloor ,$

has this sequence of adjoining registers

10,01,07,49,67,55,40,04,28,58,61,13,22,16,43,25,37,52,19,64,34,31 10,01,07,...

with output sequence of x's having period 22:

0,1,7,9,7,5,0,4,8,8,1,3,2,6,3,5,7,2,9,4,4,1 0,1,7,9,7,5,0,...

Notice that if those repeated segments of x values are put in reverse order starting from a $x_{22n+20}$,

$449275\cdots 97101449275\cdots 9710144\cdots$

we get the expansion j/(ab−1) with a=7, b=10, j=31:

$\frac{31}{69}=.44927536231884057971014492753623\dots$

This is true in general: The sequence of x's produced by a lag-r MWC generator:

$x_n=(a{x}_{n-r}+c_{n-1})modb,c_n=\lfloor \frac{a{x}_{n-r}+c_{n-1}}{b}\rfloor ,$

when put in reverse order, will be the base-b expansion of a rational j/(ab^r − 1) for some 0 < j < ab^r.

Also notice that if, starting with $x_0=34$, we generate the ordinary congruential sequence

$x_n=7x_{n-1}mod69$,

we get the period 22 sequence

31,10,1,7,49,67,55,40,4,28,58,61,13,22,16,43,25,37,52,19,64,34, 31,10,1,7,...

and that sequence, reduced mod 10, is

1,0,1,7,9,7,5,0,4,8,8,1,3,2,6,3,5,7,2,9,4,4, 1,0,1,7,9,7,5,0,...

the same sequence of x's resulting from the MWC sequence.

This is true in general, (but apparently only for lag-1 MWC sequences):

Given initial values $x_0,c_0$, the sequence $x_1,x_2,\dots$ resulting from the lag-1 MWC sequence

$x_n=(a{x}_{n-1}+c_{n-1})modb,c_n=\lfloor \frac{a{x}_{n-1}+c_{n-1}}{b}\rfloor$

is exactly the congruential sequence y_n = ay_n − 1 mod(ab − 1), reduced modulo b.

Choice of initial value y₀ merely rotates the cycle of x's.

Complementary-multiply-with-carry generators

Establishing the period of a lag-r MWC generator usually entails choosing multiplier a so that p=ab^r − 1 is prime. If p is a safe prime, then the order of b will be p − 1 or (p − 1)/2. Otherwise, it is likely that p − 1 will have to be factored in order to find the order of b mod p, and p = ab^r − 1 may be difficult to factor.

But a prime of the form p = ab^r + 1 will make p−1 easy to factor, so a version of multiply-with-carry that involves the order of b for a prime p = ab^r + 1 would reduce considerably the computational number theory required to establish the period of a MWC sequence.

Fortunately, a slight modification of the MWC procedure leads to primes of the form ab^r + 1. The new procedure is called complementary-multiply-with-carry (CMWC),

and the setup is the same as that for lag-r MWC: multiplier a, base b, r + 1 seeds

x₀, x₁, x₂, ..., x_r−1, and c_r − 1.

There is a slight change in the generation of a new pair (x, c): $x_n=(b-1)-(a{x}_{n-r}+c_{n-1})modb,c_n=\lfloor \frac{a{x}_{n-r}+c_{n-1}}{b}\rfloor .$

That is, take the complement, (b−1)−x, when forming the new x.

The resulting sequence of x's produced by the CMWC RNG will have period the order of b in the multiplicative group of residues modulo ab^r+1, and the output x's, in reverse order, will form the base b expansion of j/(ab^r+1) for some 0<j<ab^r.

Use of lag-r CMWC makes it much easier to find periods for r's as large as 512, 1024, 2048, etc. (Making r a power of 2 makes it slightly easier (and faster) to access elements in the array containing the r most recent x's.)

Some examples: With b=2³², the period of the lag-1024 CMWC

$x_n=(b-1)-(a{x}_{n-1024}+c_{n-1})modb,c_n=\lfloor \frac{a{x}_{n-1024}+c_{n-1}}{b}\rfloor .$

will be a$\cdot$2³²⁷⁶², about 10⁹⁸⁶⁷ for these three as: 109111 or 108798 or 108517.

With b = 2³² and a = 3636507990, p = ab¹³⁵⁹ − 1 is a safe prime, so the MWC sequence based on that a has period 3636507990$\cdot$2⁴³⁴⁸⁷ $\approx$10¹³¹⁰¹.

With b = 2³², a CMWC RNG with near record period may be based on the prime p = 15455296b⁴²⁶⁵⁸ + 1. The order of b for that prime is 241489*2^1365056, about 10⁴¹⁰⁹²⁸.

Implementation

The following is an implementation of the CMWC algorithm in the C programming language. Also, included in the program is a sample initialization function. In this implementation the lag r=4096. The period of the resulting generator is about $2^{131104}$.

#include <stdint.h>

#define PHI 0x9e3779b9

static uint32_t Q[4096], c = 362436;

void init_rand(uint32_t x)
{
	int i;

	Q[0] = x;
	Q[1] = x + PHI;
	Q[2] = x + PHI + PHI;

	for (i = 3; i < 4096; i++)
		Q[i] = Q[i - 3] ^ Q[i - 2] ^ PHI ^ i;
}

uint32_t rand_cmwc(void) {
        static uint32_t i = 4095;
        uint64_t t;
 
        i = (i + 1) & 4095;
        t = (18705ULL * Q[i]) + c;
        c = t >> 32;
        Q[i] = 0xfffffffe - t;
 
        return Q[i];
}

See also

• Pseudorandom number generator

References

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  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
• 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
• 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
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí.