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50-549: The Mersenne Twister was designed specifically to rectify most of the flaws found in older PRNGs. The most commonly used version of the Mersenne Twister algorithm is based on the Mersenne prime 2 19937 − 1 {\displaystyle 2^{19937}-1} . The standard implementation of that, MT19937, uses a 32-bit word length. There is another implementation (with five variants) that uses

100-405: A x 0 = 1 {\displaystyle {\boldsymbol {x}}A={\begin{cases}{\boldsymbol {x}}\gg 1&x_{0}=0\\({\boldsymbol {x}}\gg 1)\oplus {\boldsymbol {a}}&x_{0}=1\end{cases}}} where x 0 {\displaystyle x_{0}} is the lowest order bit of x {\displaystyle x} . As like TGFSR(R), the Mersenne Twister

150-403: A 0 ) ) {\displaystyle A={\begin{pmatrix}0&I_{w-1}\\a_{w-1}&(a_{w-2},\ldots ,a_{0})\end{pmatrix}}} with I w − 1 {\displaystyle I_{w-1}} as the ( w − 1 ) ( w − 1 ) {\displaystyle (w-1)(w-1)} identity matrix. The rational normal form has

200-459: A processor , memory , and other major system components that operate on data in 32- bit units. Compared to smaller bit widths, 32-bit computers can perform large calculations more efficiently and process more data per clock cycle. Typical 32-bit personal computers also have a 32-bit address bus , permitting up to 4 GB of RAM to be accessed, far more than previous generations of system architecture allowed. 32-bit designs have been used since

250-488: A 64-bit word length, MT19937-64; it generates a different sequence. A pseudorandom sequence x i {\displaystyle x_{i}} of w -bit integers of period P is said to be k-distributed to v -bit accuracy if the following holds. For a w -bit word length, the Mersenne Twister generates integers in the range [ 0 , 2 w − 1 ] {\displaystyle [0,2^{w}-1]} . The Mersenne Twister algorithm

300-605: A Mersenne Twister implementation is an array of n values of w bits each. To initialize the array, a w -bit seed value is used to supply x 0 {\displaystyle x_{0}} through x n − 1 {\displaystyle x_{n-1}} by setting x 0 {\displaystyle x_{0}} to the seed value and thereafter setting for i {\displaystyle i} from 1 {\displaystyle 1} to n − 1 {\displaystyle n-1} . In order to achieve

350-518: A mirror surface. HDR imagery allows for the reflection of highlights that can still be seen as bright white areas, instead of dull grey shapes. A 32-bit file format is a binary file format for which each elementary information is defined on 32 bits (or 4 bytes ). An example of such a format is the Enhanced Metafile Format . Concatenation In formal language theory and computer programming , string concatenation

400-623: A period of 2 127 − 1 {\displaystyle 2^{127}-1} , far shorter than the original, so it is only recommended by the authors in cases where memory is at a premium. Advantages: Disadvantages: The Mersenne Twister is used as default PRNG by the following software: It is also available in Apache Commons , in the standard C++ library (since C++11 ), and in Mathematica . Add-on implementations are provided in many program libraries, including

450-487: A simple recurrence relation, and then output numbers of the form x i T {\displaystyle x_{i}^{T}} , where T is an invertible F 2 {\displaystyle {\textbf {F}}_{2}} -matrix called a tempering matrix . The general algorithm is characterized by the following quantities: with the restriction that 2 n w − r − 1 {\displaystyle 2^{nw-r}-1}

500-410: A specific sequence to produce a grammatically correct sentence that is announced throughout the facility. One of the principles of relational database design is that the fields of data tables should reflect a single characteristic of the table's subject, which means that they should not contain concatenated strings. When concatenation is desired in a report, it should be provided at the time of running

550-399: A tempering transform to be easily computable, and so do not actually construct T itself. This tempering is defined in the case of Mersenne Twister as where x {\displaystyle x} is the next value from the series, y {\displaystyle y} is a temporary intermediate value, and z {\displaystyle z} is the value returned from

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600-413: A total of 96 bits per pixel. 32-bit-per-channel images are used to represent values brighter than what sRGB color space allows (brighter than white); these values can then be used to more accurately retain bright highlights when either lowering the exposure of the image or when it is seen through a dark filter or dull reflection. For example, a reflection in an oil slick is only a fraction of that seen in

650-572: Is patented . MTGP is a variant of Mersenne Twister optimised for graphics processing units published by Mutsuo Saito and Makoto Matsumoto. The basic linear recurrence operations are extended from MT and parameters are chosen to allow many threads to compute the recursion in parallel, while sharing their state space to reduce memory load. The paper claims improved equidistribution over MT and performance on an old (2008-era) GPU ( Nvidia GTX260 with 192 cores) of 4.7 ms for 5×10 random 32-bit integers. The SFMT ( SIMD -oriented Fast Mersenne Twister)

700-402: Is a 32-bit machine, with 32-bit registers and instructions that manipulate 32-bit quantities, but the external address bus is 36 bits wide, giving a larger address space than 4 GB, and the external data bus is 64 bits wide, primarily in order to permit a more efficient prefetch of instructions and data. Prominent 32-bit instruction set architectures used in general-purpose computing include

750-455: Is a Mersenne prime. This choice simplifies the primitivity test and k -distribution test that are needed in the parameter search. The series x {\displaystyle x} is defined as a series of w -bit quantities with the recurrence relation: where ∣ {\displaystyle \mid } denotes concatenation of bit vectors (with upper bits on the left), ⊕ {\displaystyle \oplus }

800-555: Is a variant of Mersenne Twister, introduced in 2006, designed to be fast when it runs on 128-bit SIMD. Intel SSE2 and PowerPC AltiVec are supported by SFMT. It is also used for games with the Cell BE in the PlayStation 3 . TinyMT is a variant of Mersenne Twister, proposed by Saito and Matsumoto in 2011. TinyMT uses just 127 bits of state space, a significant decrease compared to the original's 2.5 KiB of state. However, it has

850-432: Is based on a matrix linear recurrence over a finite binary field F 2 {\displaystyle {\textbf {F}}_{2}} . The algorithm is a twisted generalised feedback shift register (twisted GFSR, or TGFSR) of rational normal form (TGFSR(R)), with state bit reflection and tempering. The basic idea is to define a series x i {\displaystyle x_{i}} through

900-485: Is cascaded with a tempering transform to compensate for the reduced dimensionality of equidistribution (because of the choice of A being in the rational normal form). Note that this is equivalent to using the matrix A where A = T − 1 ∗ A T {\displaystyle A=T^{-1}*AT} for T an invertible matrix, and therefore the analysis of characteristic polynomial mentioned below still holds. As with A , we choose

950-946: Is required to reach the upper bound of equidistribution for the upper bits. The coefficients for MT19937 are: ( w , n , m , r ) = ( 32 , 624 , 397 , 31 ) a = 9908B0DF 16 ( u , d ) = ( 11 , FFFFFFFF 16 ) ( s , b ) = ( 7 , 9D2C5680 16 ) ( t , c ) = ( 15 , EFC60000 16 ) l = 18 {\displaystyle {\begin{aligned}(w,n,m,r)&=(32,624,397,31)\\a&={\textrm {9908B0DF}}_{16}\\(u,d)&=(11,{\textrm {FFFFFFFF}}_{16})\\(s,b)&=(7,{\textrm {9D2C5680}}_{16})\\(t,c)&=(15,{\textrm {EFC60000}}_{16})\\l&=18\\\end{aligned}}} Note that 32-bit implementations of

1000-565: Is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalizations of concatenation theory , also called string theory, string concatenation is a primitive notion . In many programming languages , string concatenation is a binary infix operator , and in some it is written without an operator. This is implemented in different ways: In programming, string concatenation generally occurs at run time, as string values are typically not known until run time. However, in

1050-404: Is usually expressed as simple juxtaposition (as with multiplication ). The strings over an alphabet, with the concatenation operation, form an associative algebraic structure with identity element the null string —a free monoid . Sets of strings with concatenation and alternation form a semiring , with concatenation (*) distributing over alternation (+); 0 is the empty set and 1

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1100-461: The 2 n w − r − 1 {\displaystyle 2^{nw-r}-1} theoretical upper limit of the period in a T GFSR , ϕ B ( t ) {\displaystyle \phi _{B}(t)} must be a primitive polynomial , ϕ B ( t ) {\displaystyle \phi _{B}(t)} being the characteristic polynomial of The twist transformation improves

1150-606: The 8088/8086 or 80286 , 16-bit microprocessors with a segmented address space where programs had to switch between segments to reach more than 64 kilobytes of code or data. As this is quite time-consuming in comparison to other machine operations, the performance may suffer. Furthermore, programming with segments tend to become complicated; special far and near keywords or memory models had to be used (with care), not only in assembly language but also in high level languages such as Pascal , compiled BASIC , Fortran , C , etc. The 80386 and its successors fully support

1200-827: The Boost C++ Libraries , the CUDA Library , and the NAG Numerical Library . The Mersenne Twister is one of two PRNGs in SPSS : the other generator is kept only for compatibility with older programs, and the Mersenne Twister is stated to be "more reliable". The Mersenne Twister is similarly one of the PRNGs in SAS : the other generators are older and deprecated. The Mersenne Twister is the default PRNG in Stata ,

1250-824: The IBM System/360 , IBM System/370 (which had 24-bit addressing), System/370-XA , ESA/370 , and ESA/390 (which had 31-bit addressing), the DEC VAX , the NS320xx , the Motorola 68000 family (the first two models of which had 24-bit addressing), the Intel IA-32 32-bit version of the x86 architecture, and the 32-bit versions of the ARM , SPARC , MIPS , PowerPC and PA-RISC architectures. 32-bit instruction set architectures used for embedded computing include

1300-551: The IBM System/360 Model 30 had an 8-bit ALU, 8-bit internal data paths, and an 8-bit path to memory, and the original Motorola 68000 had a 16-bit data ALU and a 16-bit external data bus, but had 32-bit registers and a 32-bit oriented instruction set. The 68000 design was sometimes referred to as 16/32-bit . However, the opposite is often true for newer 32-bit designs. For example, the Pentium Pro processor

1350-469: The concatenation S 1 S 2 consists of all strings of the form vw where v is a string from S 1 and w is a string from S 2 , or formally S 1 S 2 = { vw  : v ∈ S 1 , w ∈ S 2 } . Many authors also use concatenation of a string set and a single string, and vice versa, which are defined similarly by S 1 w = { vw  : v ∈ S 1 } and vS 2 = { vw  : w ∈ S 2 } . In these definitions,

1400-448: The integer representation used. With the two most common representations, the range is 0 through 4,294,967,295 (2 − 1) for representation as an ( unsigned ) binary number , and −2,147,483,648 (−2 ) through 2,147,483,647 (2 − 1) for representation as two's complement . One important consequence is that a processor with 32-bit memory addresses can directly access at most 4  GiB of byte-addressable memory (though in practice

1450-586: The k -distribution properties. The ACORN family (published 1989) is another k -distributed PRNG, which shows similar computational speed to MT, and better statistical properties as it satisfies all the current (2019) TestU01 criteria; when used with appropriate choices of parameters, ACORN can have arbitrarily long period and precision. The PCG family is a more modern long-period generator, with better cache locality, and less detectable bias using modern analysis methods. 32-bit In computer architecture , 32-bit computing refers to computer systems with

1500-533: The 16-bit segments of the 80286 but also segments for 32-bit address offsets (using the new 32-bit width of the main registers). If the base address of all 32-bit segments is set to 0, and segment registers are not used explicitly, the segmentation can be forgotten and the processor appears as having a simple linear 32-bit address space. Operating systems like Windows or OS/2 provide the possibility to run 16-bit (segmented) programs as well as 32-bit programs. The former possibility exists for backward compatibility and

1550-423: The 68000 family and ColdFire , x86, ARM, MIPS, PowerPC, and Infineon TriCore architectures. On the x86 architecture , a 32-bit application normally means software that typically (not necessarily) uses the 32-bit linear address space (or flat memory model ) possible with the 80386 and later chips. In this context, the term came about because DOS , Microsoft Windows and OS/2 were originally written for

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1600-535: The LHS, x k {\displaystyle x_{k}} , is the next generated value in the series in terms of values generated in the past, which are on the RHS. The twist transformation A is defined in rational normal form as: A = ( 0 I w − 1 a w − 1 ( a w − 2 , … ,

1650-1051: The Mersenne Twister generally have d  = FFFFFFFF 16 . As a result, the d is occasionally omitted from the algorithm description, since the bitwise and with d in that case has no effect. The coefficients for MT19937-64 are: ( w , n , m , r ) = ( 64 , 312 , 156 , 31 ) a = B5026F5AA96619E9 16 ( u , d ) = ( 29 , 5555555555555555 16 ) ( s , b ) = ( 17 , 71D67FFFEDA60000 16 ) ( t , c ) = ( 37 , FFF7EEE000000000 16 ) l = 43 {\displaystyle {\begin{aligned}(w,n,m,r)=(64,312,156,31)\\a={\textrm {B5026F5AA96619E9}}_{16}\\(u,d)=(29,{\textrm {5555555555555555}}_{16})\\(s,b)=(17,{\textrm {71D67FFFEDA60000}}_{16})\\(t,c)=(37,{\textrm {FFF7EEE000000000}}_{16})\\l=43\\\end{aligned}}} The state needed for

1700-552: The algorithm, with ≪ {\displaystyle \ll } and ≫ {\displaystyle \gg } as the bitwise left and right shifts , and & {\displaystyle \&} as the bitwise AND . The first and last transforms are added in order to improve lower-bit equidistribution. From the property of TGFSR, s + t ≥ ⌊ w 2 ⌋ − 1 {\displaystyle s+t\geq \left\lfloor {\frac {w}{2}}\right\rfloor -1}

1750-426: The benefit that multiplication by A can be efficiently expressed as: (remember that here matrix multiplication is being done in F 2 {\displaystyle {\textbf {F}}_{2}} , and therefore bitwise XOR takes the place of addition) x A = { x ≫ 1 x 0 = 0 ( x ≫ 1 ) ⊕

1800-460: The bitwise exclusive or (XOR), x k u {\displaystyle x_{k}^{u}} means the upper w − r bits of x k {\displaystyle x_{k}} , and x k + 1 l {\displaystyle x_{k+1}^{l}} means the lower r bits of x k + 1 {\displaystyle x_{k+1}} . The subscripts may all be offset by -n where now

1850-467: The case of string literals, the values are known at compile time, and thus string concatenation can be done at compile time, either via string literal concatenation or via constant folding , a potential run-time optimization. In formal language theory and pattern matching (including regular expressions ), the concatenation operation on strings is generalised to an operation on sets of strings as follows: For two sets of strings S 1 and S 2 ,

1900-405: The classical GFSR with the following key properties: CryptMT is a stream cipher and cryptographically secure pseudorandom number generator which uses Mersenne Twister internally. It was developed by Matsumoto and Nishimura alongside Mariko Hagita and Mutsuo Saito. It has been submitted to the eSTREAM project of the eCRYPT network. Unlike Mersenne Twister or its other derivatives, CryptMT

1950-418: The concatenation of the seven fields should happen upon running the report. The reason for such principles is that without them, the entry and updating of large volumes of data becomes error-prone and labor-intensive. Separately entering the city, state, ZIP code, and nation allows data-entry validation (such as detecting an invalid state abbreviation). Then those separate items can be used for sorting or indexing

2000-490: The earliest days of electronic computing, in experimental systems and then in large mainframe and minicomputer systems. The first hybrid 16/32-bit microprocessor , the Motorola 68000 , was introduced in the late 1970s and used in systems such as the original Apple Macintosh . Fully 32-bit microprocessors such as the HP FOCUS , Motorola 68020 and Intel 80386 were launched in the early to mid 1980s and became dominant by

2050-722: The early 1990s. This generation of personal computers coincided with and enabled the first mass-adoption of the World Wide Web . While 32-bit architectures are still widely-used in specific applications, the PC and server market has moved on to 64 bits with x86-64 and other 64-bit architectures since the mid-2000s with installed memory often exceeding the 32-bit 4G RAM address limits on entry level computers. The latest generation of smartphones have also switched to 64 bits. A 32-bit register can store 2 different values. The range of integer values that can be stored in 32 bits depends on

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2100-574: The first decades of 32-bit architectures (the 1960s to the 1980s). Older 32-bit processor families (or simpler, cheaper variants thereof) could therefore have many compromises and limitations in order to cut costs. This could be a 16-bit ALU , for instance, or external (or internal) buses narrower than 32 bits, limiting memory size or demanding more cycles for instruction fetch, execution or write back. Despite this, such processors could be labeled 32-bit , since they still had 32-bit registers and instructions able to manipulate 32-bit quantities. For example,

2150-412: The latter is usually meant to be used for new software development . In digital images/pictures, 32-bit usually refers to RGBA color space ; that is, 24-bit truecolor images with an additional 8-bit alpha channel . Other image formats also specify 32 bits per pixel, such as RGBE . In digital images, 32-bit sometimes refers to high-dynamic-range imaging (HDR) formats that use 32 bits per channel,

2200-468: The limit may be lower). The world's first stored-program electronic computer , the Manchester Baby , used a 32-bit architecture in 1948, although it was only a proof of concept and had little practical capacity. It held only 32 32-bit words of RAM on a Williams tube , and had no addition operation, only subtraction. Memory, as well as other digital circuits and wiring, was expensive during

2250-528: The other one is KISS , for compatibility with older versions of Stata. An alternative generator, WELL ("Well Equidistributed Long-period Linear"), offers quicker recovery, and equal randomness, and nearly equal speed. Marsaglia's xorshift generators and variants are the fastest in the class of LFSRs. 64-bit MELGs ("64-bit Maximally Equidistributed F 2 {\displaystyle {\textbf {F}}_{2}} -Linear Generators with Mersenne Prime Period") are completely optimized in terms of

2300-556: The other provides a grammatically correct sentence to the listener. This technique is also used in number change announcements, voice mail systems, or most telephony applications that provide dynamic feedback to the caller (e.g. moviefone , tellme , and others). Programming for any kind of computerised public address system can also employ concatenation for dynamic public announcements (for example, flights in an airport). The system would archive recorded speech of numbers, routes or airlines, destinations, times, etc. and play them back in

2350-415: The records, such as all with "Boulder" as the city name. In recreational mathematics , many problems concern the properties of numbers under concatenation of their numerals in some base . Examples include home primes (primes obtained by repeatedly factoring the increasing concatenation of prime factors of a given number), Smarandache–Wellin numbers (the concatenations of the first prime numbers ), and

2400-403: The report. For example, to display the physical address of a certain customer, the data might include building number, street name, building sub-unit number, city name, state/province name, postal code, and country name, e.g., "123 Fake St Apt 4, Boulder, CO 80302, USA", which combines seven fields. However, the customers data table should not use one field to store that concatenated string; rather,

2450-482: The set consisting of just the null string. In programming for telephony, concatenation is used to provide dynamic audio feedback to a user. For example, in a "time of day" speaking clock , concatenation is used to give the correct time by playing the appropriate recordings concatenated together. For example: "at the tone, the time will be", "eight", "thirty", "five", "and", "twenty", "five", "seconds". The recordings themselves exist separately, but playing them one after

2500-460: The string vw is the ordinary concatenation of strings v and w as defined in the introductory section. For example, if F = { a, b, c, d, e, f, g, h } , and R = { 1, 2, 3, 4, 5, 6, 7, 8 } , then FR denotes the set of all chess board coordinates in algebraic notation , while e R denotes the set of all coordinates of the kings' file . In this context, sets of strings are often referred to as formal languages. The concatenation operator

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