Random Numbers

Random number generation in Julia uses theXoshiro256++ algorithm by default, with per-Task state. Other RNG types can be plugged in by inheriting theAbstractRNG type; they can then be used to obtain multiple streams of random numbers.

The PRNGs (pseudorandom number generators) exported by theRandom package are:

• TaskLocalRNG: a token that represents use of the currently active Task-local stream, deterministically seeded from the parent task, or byRandomDevice (with system randomness) at program start
• Xoshiro: generates a high-quality stream of random numbers with a small state vector and high performance using the Xoshiro256++ algorithm
• RandomDevice: for OS-provided entropy. This may be used for cryptographically secure random numbers (CS(P)RNG).
• MersenneTwister: an alternate high-quality PRNG which was the default in older versions of Julia, and is also quite fast, but requires much more space to store the state vector and generate a random sequence.

Most functions related to random generation accept an optionalAbstractRNG object as first argument. Some also accept dimension specificationsdims... (which can also be given as a tuple) to generate arrays of random values. In a multi-threaded program, you should generally use different RNG objects from different threads or tasks in order to be thread-safe. However, the default RNG is thread-safe as of Julia 1.3 (using a per-thread RNG up to version 1.6, and per-task thereafter).

The provided RNGs can generate uniform random numbers of the following types:Float16,Float32,Float64,BigFloat,Bool,Int8,UInt8,Int16,UInt16,Int32,UInt32,Int64,UInt64,Int128,UInt128,BigInt (or complex numbers of those types). Random floating point numbers are generated uniformly in$[0, 1)$. AsBigInt represents unbounded integers, the interval must be specified (e.g.rand(big.(1:6))).

Additionally, normal and exponential distributions are implemented for someAbstractFloat andComplex types, seerandn andrandexp for details.

To generate random numbers from other distributions, see theDistributions.jl package.

Random numbers module

Random

Random generation functions

rand([rng=default_rng()], [S], [dims...])

julia> rand(Int, 2)2-element Array{Int64,1}: 1339893410598768192 1575814717733606317julia> using Randomjulia> rand(Xoshiro(0), Dict(1=>2, 3=>4))3 => 4julia> rand((2, 3))3julia> rand(Float64, (2, 3))2×3 Array{Float64,2}: 0.999717  0.0143835  0.540787 0.696556  0.783855   0.938235

rand!([rng=default_rng()], A, [S=eltype(A)])

julia> rand!(Xoshiro(123), zeros(5))5-element Vector{Float64}: 0.521213795535383 0.5868067574533484 0.8908786980927811 0.19090669902576285 0.5256623915420473

bitrand([rng=default_rng()], [dims...])

julia> bitrand(Xoshiro(123), 10)10-element BitVector: 0 1 0 1 0 1 0 0 1 1

randn([rng=default_rng()], [T=Float64], [dims...])

julia> randn()-0.942481877315864

julia> randn(2,3)2×3 Matrix{Float64}:  1.18786   -0.678616   1.49463 -0.342792  -0.134299  -1.45005

julia> using Randomjulia> rng = Xoshiro(123);julia> randn(rng, Float32)-0.6457307f0julia> randn(rng, ComplexF32, (2, 3))2×3 Matrix{ComplexF32}:  -1.03467-1.14806im  0.693657+0.056538im   0.291442+0.419454im -0.153912+0.34807im    1.0954-0.948661im  -0.543347-0.0538589im

randn!([rng=default_rng()], A::AbstractArray) -> A

julia> randn!(Xoshiro(123), zeros(5))5-element Vector{Float64}: -0.6457306721039767 -1.4632513788889214 -1.6236037455860806 -0.21766510678354617  0.4922456865251828

randexp([rng=default_rng()], [T=Float64], [dims...])

julia> rng = Xoshiro(123);julia> randexp(rng, Float32)1.1757717f0julia> randexp(rng, 3, 3)3×3 Matrix{Float64}: 1.37766  0.456653  0.236418 3.40007  0.229917  0.0684921 0.48096  0.577481  0.71835

randexp!([rng=default_rng()], A::AbstractArray) -> A

julia> randexp!(Xoshiro(123), zeros(5))5-element Vector{Float64}: 1.1757716836348473 1.758884569451514 1.0083623637301151 0.3510644315565272 0.6348266443720407

randstring([rng=default_rng()], [chars], [len=8])

julia> Random.seed!(3); randstring()"Lxz5hUwn"julia> randstring(Xoshiro(3), 'a':'z', 6)"iyzcsm"julia> randstring("ACGT")"TGCTCCTC"

Subsequences, permutations and shuffling

randsubseq([rng=default_rng(),] A, p) -> Vector

julia> randsubseq(Xoshiro(123), 1:8, 0.3)2-element Vector{Int64}: 4 7

randsubseq!([rng=default_rng(),] S, A, p)

julia> S = Int64[];julia> randsubseq!(Xoshiro(123), S, 1:8, 0.3)2-element Vector{Int64}: 4 7julia> S2-element Vector{Int64}: 4 7

randperm([rng=default_rng(),] n::Integer)

julia> randperm(Xoshiro(123), 4)4-element Vector{Int64}: 1 4 2 3

randperm!([rng=default_rng(),] A::Array{<:Integer})

julia> randperm!(Xoshiro(123), Vector{Int}(undef, 4))4-element Vector{Int64}: 1 4 2 3

randcycle([rng=default_rng(),] n::Integer)

julia> randcycle(Xoshiro(123), 6)6-element Vector{Int64}: 5 4 2 6 3 1

randcycle!([rng=default_rng(),] A::Array{<:Integer})

julia> randcycle!(Xoshiro(123), Vector{Int}(undef, 6))6-element Vector{Int64}: 5 4 2 6 3 1

shuffle([rng=default_rng(),] v::AbstractArray)

julia> shuffle(Xoshiro(123), Vector(1:10))10-element Vector{Int64}:  5  4  2  3  6 10  8  1  9  7

shuffle!([rng=default_rng(),] v::AbstractArray)

julia> shuffle!(Xoshiro(123), Vector(1:10))10-element Vector{Int64}:  5  4  2  3  6 10  8  1  9  7

Generators (creation and seeding)

Random.default_rng() -> rng

seed!([rng=default_rng()], seed) -> rngseed!([rng=default_rng()]) -> rng

julia> Random.seed!(1234);julia> x1 = rand(2)2-element Vector{Float64}: 0.32597672886359486 0.5490511363155669julia> Random.seed!(1234);julia> x2 = rand(2)2-element Vector{Float64}: 0.32597672886359486 0.5490511363155669julia> x1 == x2truejulia> rng = Xoshiro(1234); rand(rng, 2) == x1truejulia> Xoshiro(1) == Random.seed!(rng, 1)truejulia> rand(Random.seed!(rng), Bool) # not reproducibletruejulia> rand(Random.seed!(rng), Bool) # not reproducible eitherfalsejulia> rand(Xoshiro(), Bool) # not reproducible eithertrue

AbstractRNG

TaskLocalRNG

Xoshiro(seed::Union{Integer, AbstractString})Xoshiro()

julia> using Randomjulia> rng = Xoshiro(1234);julia> x1 = rand(rng, 2)2-element Vector{Float64}: 0.32597672886359486 0.5490511363155669julia> rng = Xoshiro(1234);julia> x2 = rand(rng, 2)2-element Vector{Float64}: 0.32597672886359486 0.5490511363155669julia> x1 == x2true

MersenneTwister(seed)MersenneTwister()

julia> rng = MersenneTwister(123);julia> x1 = rand(rng, 2)2-element Vector{Float64}: 0.37453777969575874 0.8735343642013971julia> x2 = rand(MersenneTwister(123), 2)2-element Vector{Float64}: 0.37453777969575874 0.8735343642013971julia> x1 == x2true

RandomDevice()

Hooking into theRandom API

There are two mostly orthogonal ways to extendRandom functionalities:

1. generating random values of custom types
2. creating new generators

The API for 1) is quite functional, but is relatively recent so it may still have to evolve in subsequent releases of theRandom module. For example, it's typically sufficient to implement onerand method in order to have all other usual methods work automatically.

The API for 2) is still rudimentary, and may require more work than strictly necessary from the implementor, in order to support usual types of generated values.

Generating random values of custom types

Generating random values for some distributions may involve various trade-offs.Pre-computed values, such as analias table for discrete distributions, or“squeezing” functions for univariate distributions, can speed up sampling considerably. How much information should be pre-computed can depend on the number of values we plan to draw from a distribution. Also, some random number generators can have certain properties that various algorithms may want to exploit.

TheRandom module defines a customizable framework for obtaining random values that can address these issues. Each invocation ofrand generates asampler which can be customized with the above trade-offs in mind, by adding methods toSampler, which in turn can dispatch on the random number generator, the object that characterizes the distribution, and a suggestion for the number of repetitions. Currently, for the latter,Val{1} (for a single sample) andVal{Inf} (for an arbitrary number) are used, withRandom.Repetition an alias for both.

The object returned bySampler is then used to generate the random values. When implementing the random generation interface for a valueX that can be sampled from, the implementor should define the method

rand(rng, sampler)

for the particularsampler returned bySampler(rng, X, repetition).

Samplers can be arbitrary values that implementrand(rng, sampler), but for most applications the following predefined samplers may be sufficient:

1. SamplerType{T}() can be used for implementing samplers that draw from typeT (e.g.rand(Int)). This is the default returned bySampler fortypes.

2. SamplerTrivial(self) is a simple wrapper forself, which can be accessed with[]. This is the recommended sampler when no pre-computed information is needed (e.g.rand(1:3)), and is the default returned bySampler forvalues.

3. SamplerSimple(self, data) also contains the additionaldata field, which can be used to store arbitrary pre-computed values, which should be computed in acustom method ofSampler.

We provide examples for each of these. We assume here that the choice of algorithm is independent of the RNG, so we useAbstractRNG in our signatures.

Sampler(rng, x, repetition = Val(Inf))

SamplerType{T}()

SamplerTrivial(x)

SamplerSimple(x, data)

Decoupling pre-computation from actually generating the values is part of the API, and is also available to the user. As an example, assume thatrand(rng, 1:20) has to be called repeatedly in a loop: the way to take advantage of this decoupling is as follows:

rng = Xoshiro()sp = Random.Sampler(rng, 1:20) # or Random.Sampler(Xoshiro, 1:20)for x in X    n = rand(rng, sp) # similar to n = rand(rng, 1:20)    # use nend

This is the mechanism that is also used in the standard library, e.g. by the default implementation of random array generation (like inrand(1:20, 10)).

Generating values from a type

Given a typeT, it's currently assumed that ifrand(T) is defined, an object of typeT will be produced.SamplerType is thedefault sampler for types. In order to define random generation of values of typeT, therand(rng::AbstractRNG, ::Random.SamplerType{T}) method should be defined, and should return values whatrand(rng, T) is expected to return.

Let's take the following example: we implement aDie type, with a variable numbern of sides, numbered from1 ton. We wantrand(Die) to produce aDie with a random number of up to 20 sides (and at least 4):

struct Die    nsides::Int # number of sidesendRandom.rand(rng::AbstractRNG, ::Random.SamplerType{Die}) = Die(rand(rng, 4:20))# output

Scalar and array methods forDie now work as expected:

julia> rand(Die)Die(5)julia> rand(Xoshiro(0), Die)Die(10)julia> rand(Die, 3)3-element Vector{Die}: Die(9) Die(15) Die(14)julia> a = Vector{Die}(undef, 3); rand!(a)3-element Vector{Die}: Die(19) Die(7) Die(17)

A simple sampler without pre-computed data

Here we define a sampler for a collection. If no pre-computed data is required, it can be implemented with aSamplerTrivial sampler, which is in fact thedefault fallback for values.

In order to define random generation out of objects of typeS, the following method should be defined:rand(rng::AbstractRNG, sp::Random.SamplerTrivial{S}). Here,sp simply wraps an object of typeS, which can be accessed viasp[]. Continuing theDie example, we want now to definerand(d::Die) to produce anInt corresponding to one ofd's sides:

julia> Random.rand(rng::AbstractRNG, d::Random.SamplerTrivial{Die}) = rand(rng, 1:d[].nsides);julia> rand(Die(4))1julia> rand(Die(4), 3)3-element Vector{Any}: 2 3 3

Given a collection typeS, it's currently assumed that ifrand(::S) is defined, an object of typeeltype(S) will be produced. In the last example, aVector{Any} is produced; the reason is thateltype(Die) == Any. The remedy is to defineBase.eltype(::Type{Die}) = Int.

Generating values for anAbstractFloat type

AbstractFloat types are special-cased, because by default random values are not produced in the whole type domain, but rather in[0,1). The following method should be implemented forT <: AbstractFloat:Random.rand(::AbstractRNG, ::Random.SamplerTrivial{Random.CloseOpen01{T}})

An optimized sampler with pre-computed data

Consider a discrete distribution, where numbers1:n are drawn with given probabilities that sum to one. When many values are needed from this distribution, the fastest method is using analias table. We don't provide the algorithm for building such a table here, but suppose it is available inmake_alias_table(probabilities) instead, anddraw_number(rng, alias_table) can be used to draw a random number from it.

Suppose that the distribution is described by

struct DiscreteDistribution{V <: AbstractVector}    probabilities::Vend

and that wealways want to build an alias table, regardless of the number of values needed (we learn how to customize this below). The methods

Random.eltype(::Type{<:DiscreteDistribution}) = Intfunction Random.Sampler(::Type{<:AbstractRNG}, distribution::DiscreteDistribution, ::Repetition)    SamplerSimple(distribution, make_alias_table(distribution.probabilities))end

should be defined to return a sampler with pre-computed data, then

function rand(rng::AbstractRNG, sp::SamplerSimple{<:DiscreteDistribution})    draw_number(rng, sp.data)end

will be used to draw the values.

Custom sampler types

TheSamplerSimple type is sufficient for most use cases with precomputed data. However, in order to demonstrate how to use custom sampler types, here we implement something similar toSamplerSimple.

Going back to ourDie example:rand(::Die) uses random generation from a range, so there is an opportunity for this optimization. We call our custom samplerSamplerDie.

import Random: Sampler, randstruct SamplerDie <: Sampler{Int} # generates values of type Int    die::Die    sp::Sampler{Int} # this is an abstract type, so this could be improvedendSampler(RNG::Type{<:AbstractRNG}, die::Die, r::Random.Repetition) =    SamplerDie(die, Sampler(RNG, 1:die.nsides, r))# the `r` parameter will be explained later onrand(rng::AbstractRNG, sp::SamplerDie) = rand(rng, sp.sp)

It's now possible to get a sampler withsp = Sampler(rng, die), and usesp instead ofdie in anyrand call involvingrng. In the simplistic example above,die doesn't need to be stored inSamplerDie but this is often the case in practice.

Of course, this pattern is so frequent that the helper type used above, namelyRandom.SamplerSimple, is available, saving us the definition ofSamplerDie: we could have implemented our decoupling with:

Sampler(RNG::Type{<:AbstractRNG}, die::Die, r::Random.Repetition) =    SamplerSimple(die, Sampler(RNG, 1:die.nsides, r))rand(rng::AbstractRNG, sp::SamplerSimple{Die}) = rand(rng, sp.data)

Here,sp.data refers to the second parameter in the call to theSamplerSimple constructor (in this case equal toSampler(rng, 1:die.nsides, r)), while theDie object can be accessed viasp[].

LikeSamplerDie, any custom sampler must be a subtype ofSampler{T} whereT is the type of the generated values. Note thatSamplerSimple(x, data) isa Sampler{eltype(x)}, so this constrains what the first argument toSamplerSimple can be (it's recommended to useSamplerSimple like in theDie example, wherex is simply forwarded while defining aSampler method). Similarly,SamplerTrivial(x) isa Sampler{eltype(x)}.

Another helper type is currently available for other cases,Random.SamplerTag, but is considered as internal API, and can break at any time without proper deprecations.

Using distinct algorithms for scalar or array generation

In some cases, whether one wants to generate only a handful of values or a large number of values will have an impact on the choice of algorithm. This is handled with the third parameter of theSampler constructor. Let's assume we defined two helper types forDie, saySamplerDie1 which should be used to generate only few random values, andSamplerDieMany for many values. We can use those types as follows:

Sampler(RNG::Type{<:AbstractRNG}, die::Die, ::Val{1}) = SamplerDie1(...)Sampler(RNG::Type{<:AbstractRNG}, die::Die, ::Val{Inf}) = SamplerDieMany(...)

Of course,rand must also be defined on those types (i.e.rand(::AbstractRNG, ::SamplerDie1) andrand(::AbstractRNG, ::SamplerDieMany)). Note that, as usual,SamplerTrivial andSamplerSimple can be used if custom types are not necessary.

Note:Sampler(rng, x) is simply a shorthand forSampler(rng, x, Val(Inf)), andRandom.Repetition is an alias forUnion{Val{1}, Val{Inf}}.

Creating new generators

The API is not clearly defined yet, but as a rule of thumb:

1. anyrand method producing "basic" types (isbitstype integer and floating types inBase) should be defined for this specific RNG, if they are needed;
2. other documentedrand methods accepting anAbstractRNG should work out of the box, (provided the methods from 1) what are relied on are implemented), but can of course be specialized for this RNG if there is room for optimization;
3. copy for pseudo-RNGs should return an independent copy that generates the exact same random sequence as the original from that point when called in the same way. When this is not feasible (e.g. hardware-based RNGs),copy must not be implemented.

Concerning 1), arand method may happen to work automatically, but it's not officially supported and may break without warnings in a subsequent release.

To define a newrand method for an hypotheticalMyRNG generator, and a value specifications (e.g.s == Int, ors == 1:10) of typeS==typeof(s) orS==Type{s} ifs is a type, the same two methods as we saw before must be defined:

1. Sampler(::Type{MyRNG}, ::S, ::Repetition), which returns an object of type saySamplerS
2. rand(rng::MyRNG, sp::SamplerS)

It can happen thatSampler(rng::AbstractRNG, ::S, ::Repetition) is already defined in theRandom module. It would then be possible to skip step 1) in practice (if one wants to specialize generation for this particular RNG type), but the correspondingSamplerS type is considered as internal detail, and may be changed without warning.

Specializing array generation

In some cases, for a given RNG type, generating an array of random values can be more efficient with a specialized method than by merely using the decoupling technique explained before. This is for example the case forMersenneTwister, which natively writes random values in an array.

To implement this specialization forMyRNG and for a specifications, producing elements of typeS, the following method can be defined:rand!(rng::MyRNG, a::AbstractArray{S}, ::SamplerS), whereSamplerS is the type of the sampler returned bySampler(MyRNG, s, Val(Inf)). Instead ofAbstractArray, it's possible to implement the functionality only for a subtype, e.g.Array{S}. The non-mutating array method ofrand will automatically call this specialization internally.

Reproducibility

By using an RNG parameter initialized with a given seed, you can reproduce the same pseudorandom number sequence when running your program multiple times. However, a minor release of Julia (e.g. 1.3 to 1.4)may change the sequence of pseudorandom numbers generated from a specific seed, in particular ifMersenneTwister is used. (Even if the sequence produced by a low-level function likerand does not change, the output of higher-level functions likerandsubseq may change due to algorithm updates.) Rationale: guaranteeing that pseudorandom streams never change prohibits many algorithmic improvements.

If you need to guarantee exact reproducibility of random data, it is advisable to simplysave the data (e.g. as a supplementary attachment in a scientific publication). (You can also, of course, specify a particular Julia version and package manifest, especially if you require bit reproducibility.)

Software tests that rely onspecific "random" data should also generally either save the data, embed it into the test code, or use third-party packages likeStableRNGs.jl. On the other hand, tests that should pass formost random data (e.g. testingA \ (A*x) ≈ x for a random matrixA = randn(n,n)) can use an RNG with a fixed seed to ensure that simply running the test many times does not encounter a failure due to very improbable data (e.g. an extremely ill-conditioned matrix).

The statisticaldistribution from which random samples are drawnis guaranteed to be the same across any minor Julia releases.