Mastering Efficient Array Operations with StaticArrays.jl in Julia – Beragampengetahuan

By: Steven Whitaker

Re-posted from:

The Julia programming languageis known for being a high-level languagethat can still compete with Cin terms of performance.As such,Julia already has performant data structures built-in,such as arrays.But what if arrays could be even faster?That’s where the StaticArrays.jl package comes in.

StaticArrays.jl provides drop-in replacements for Array,the standard Julia array type.These StaticArrays work just like Arrays,but they provide one additional piece of informationin the type:the size of the array.Consequently,you can’t insert or remove elements of a StaticArray;they are statically sized arrays(hence the name).However,this restriction allows more informationto be given to Julia’s compiler,which in turn results in more efficient machine code(for example, via loop unrolling and SIMD operations).The resulting speed-up can often be 10x or more!

In this post,we will learn how to use StaticArrays.jland compare the performance of StaticArraysto that of regular Arraysfor several different operations.

Note that the code examples in this postassume StaticArrays.jl has been installed and loaded:

pkg> add StaticArraysjulia> using StaticArrays

(Check out our post on the Julia REPLfor more details about the package promptand navigating the REPL.)

When working with StaticArrays.jl,typically one will use the SVector typeor the SMatrix type.(There is also the SArray type for N-dimensional arrays,but we will focus on 1D and 2D arrays in this post.)SVectors and SMatrixes have both static sizeand static data,meaning the data contained in such objectscannot be modified.For statically sized arrayswhose contents can be modified,StaticArrays.jl provides MVector and MMatrix (and MArray).We will stick with SVectors and SMatrixes in this postunless we specifically need mutability.

Constructors

There are three ways to construct StaticArrays.

1. Convenience constructor SA:

   julia> SA[1, 2, 3]3-element SVector{3, Int64} with indices SOneTo(3): 1 2 3julia> SA[1 2; 3 4]22 SMatrix{2, 2, Int64, 4} with indices SOneTo(2)SOneTo(2): 1  2 3  4

2. Normal constructor functions:

   julia> SVector(1, 2)2-element SVector{2, Int64} with indices SOneTo(2): 1 2julia> SMatrix{2,3}(1, 2, 3, 4, 5, 6)23 SMatrix{2, 3, Int64, 6} with indices SOneTo(2)SOneTo(3): 1  3  5 2  4  6

3. Macros:

   julia> @SVector [1, 2, 3]3-element SVector{3, Int64} with indices SOneTo(3): 1 2 3julia> @SMatrix [1 2; 3 4]22 SMatrix{2, 2, Int64, 4} with indices SOneTo(2)SOneTo(2): 1  2 3  4

   Note that using macrosalso enables a convenient wayto create StaticArrays from common array-creation functions(eliminating the need to create an Array firstjust to convert it immediately to a StaticArray):

   @SVector [10 * i for i = 1:10]@SVector zeros(5)@SVector rand(7)@SMatrix [(i, j) for i = 1:2, j = 1:3]@SMatrix zeros(2, 2)@SMatrix randn(6, 6)

Conversion to/from Array

It may occasionally be necessaryto convert to or from Arrays.To convert from an Array to a StaticArray,use the appropriate constructor function.However, because Arrays do not have size information in the type,we ourselves must provide the size to the constructor:

SVector{3}([1, 2, 3])SMatrix{4,4}(zeros(4, 4))

To convert back to an Array, use the collect function:

julia> collect(SVector(1, 2))2-element Vector{Int64}: 1 2

Once a StaticArray is created,it can be operated on in the same wayas an Array.To illustrate,we will run a simple benchmark,both to compare the run-time speedsof the two types of arraysand to show that the same code can workwith either type of array.

Here’s the benchmark code,inspired by StaticArrays.jl’s benchmark:

using BenchmarkTools, StaticArrays, LinearAlgebra, Printfadd!(C, A, B) = C .= A .+ Bfunction run_benchmarks(N)    A = rand(N, N); A = A' * A    B = rand(N, N)    C = Matrix{eltype(A)}(undef, N, N)    D = rand(N)    SA = SMatrix{N,N}(A)    SB = SMatrix{N,N}(B)    MA = MMatrix{N,N}(A)    MB = MMatrix{N,N}(B)    MC = MMatrix{N,N}(C)    SD = SVector{N}(D)    speedup = [        @belapsed($A + $B) / @belapsed($SA + $SB),        @belapsed(add!($C, $A, $B)) / @belapsed(add!($MC, $MA, $MB)),        @belapsed($A * $B) / @belapsed($SA * $SB),        @belapsed(mul!($C, $A, $B)) / @belapsed(mul!($MC, $MA, $MB)),        @belapsed(norm($D)) / @belapsed(norm($SD)),        @belapsed(det($A)) / @belapsed(det($SA)),        @belapsed(inv($A)) / @belapsed(inv($SA)),        @belapsed($A \ $D) / @belapsed($SA \ $SD),        @belapsed(eigen($A)) / @belapsed(eigen($SA)),        @belapsed(map(abs, $A)) / @belapsed(map(abs, $SA)),        @belapsed(sum($D)) / @belapsed(sum($SD)),        @belapsed(sort($D)) / @belapsed(sort($SD)),    ]    return speedupendfunction main()    benchmarks = [        "Addition",        "Addition (in-place)",        "Multiplication",        "Multiplication (in-place)",        "L2 Norm",        "Determinant",        "Inverse",        "Linear Solve (A \\ b)",        "Symmetric Eigendecomposition",        "`map`",        "Sum of Elements",        "Sorting",    ]    N = [3, 5, 10, 30]    speedups = map(run_benchmarks, N)    fmt_header = Printf.Format("%-$(maximum(length.(benchmarks)))s" * " | %7s"^length(N))    header = Printf.format(fmt_header, "Benchmark", string.("N = ", N)...)    println(header)    println("="^length(header))    fmt = Printf.Format("%-$(maximum(length.(benchmarks)))s" * " | %7.1f"^length(N))    for i = 1:length(benchmarks)        println(Printf.format(fmt, benchmarks[i], getindex.(speedups, i)...))    endendmain()

Notice that all the functions calledwhen creating the array speedupin run_benchmarksare the same whether using Arrays or StaticArrays,illustrating that StaticArraysare drop-in replacements for standard Arrays.

Running the above codeprints the following results on my laptop(the numbers indicate the speedupof StaticArrays over normal Arrays;e.g., a value of 17.7 meansusing StaticArrays was 17.7 times fasterthan using Arrays):

Benchmark                    |   N = 3 |   N = 5 |  N = 10 |  N = 30====================================================================Addition                     |    17.7 |    14.5 |     7.9 |     2.0Addition (in-place)          |     1.6 |     1.3 |     1.4 |     0.7Multiplication               |     8.2 |     7.0 |     4.2 |     2.6Multiplication (in-place)    |     1.9 |     5.9 |     3.0 |     1.0L2 Norm                      |     4.2 |     4.0 |     5.4 |     9.7Determinant                  |    66.6 |     2.5 |     1.3 |     0.9Inverse                      |    54.8 |     5.9 |     1.8 |     0.9Linear Solve (A \ b)         |    65.5 |     3.7 |     1.8 |     0.9Symmetric Eigendecomposition |     3.7 |     1.0 |     1.0 |     1.0`map`                        |    10.6 |     8.2 |     4.9 |     2.1Sum of Elements              |     1.5 |     1.1 |     1.7 |     2.1Sorting                      |     7.1 |     2.9 |     1.5 |     1.1

There are two main conclusions from this table.First,using StaticArrays instead of Arrayscan result in some nice speed-ups!Second,the gains from using StaticArrays tend to diminishas the sizes of the arrays increase.So,you can’t expect StaticArrays.jlto always magically make your code faster,but if your arrays are small enough(the recommendation being fewer than about 100 elements)then you can expect to see some good speed-ups.

Of course,the above code timed just individual operations;how much faster a particular application would beis a different matter.

For example,consider a physical simulationwhere many 3D vectorsare manipulated over several time steps.Since 3D vectors are static in size(i.e., are 1D arrays with exactly three elements),such a situation is a prime exampleof where StaticArrays.jl is useful.To illustrate,here is an example(taken from the field of magnetic resonance imaging)of a physical simulationusing Arrays vs using StaticArrays:

using BenchmarkTools, StaticArrays, LinearAlgebrafunction sim_arrays(N)    M = Matrix{Float64}(undef, 3, N)    M[1,:] .= 0.0    M[2,:] .= 0.0    M[3,:] .= 1.0    M2 = similar(M)    (sin, cos) = sincosd(30)    R = [1 0 0; 0 cos sin; 0 -sin cos]    E1 = exp(-0.01)    E2 = exp(-0.1)    (sin, cos) = sincosd(1)    F = [E2 * cos E2 * sin 0; -E2 * sin E2 * cos 0; 0 0 E1]    FR = F * R    C = [0, 0, 1 - E1]        for t = 1:50        mul!(M2, FR, M)        M2 .+= C        mul!(M, FR, M2)        M .+= C    end    total = sum(M; dims = 2)    return complex(total[1], total[2])endfunction sim_staticarrays(N)    M = fill(SVector(0.0, 0.0, 1.0), N)    (sin, cos) = sincosd(30)    R = @SMatrix [1 0 0; 0 cos sin; 0 -sin cos]    E1 = exp(-0.01)    E2 = exp(-0.1)    (sin, cos) = sincosd(1)    F = @SMatrix [E2 * cos E2 * sin 0; -E2 * sin E2 * cos 0; 0 0 E1]    FR = F * R    C = @SVector [0, 0, 1 - E1]        for t = 1:100                for i = 1:length(M)            M[i] = FR * M[i] + C        end    end    total = sum(M)    return complex(total[1], total[2])endfunction main(N)    r1 = @btime sim_arrays($N)    r2 = @btime sim_staticarrays($N)    @assert r1  r2 end

The speed-ups on my laptopfor different values of Nwere as follows:

• N = 10: 14.6x faster
• N = 100: 7.1x faster
• N = 1000: 5.2x faster

(Here, N is the number of 3D vectors in the simulation,not the size of the StaticArrays.)

Note also that I wrote sim_arraysto be as performant as possibleby doing in-place operations(like mul!),which has the unfortunate side effectof making the code a bit harder to read.Therefore,sim_staticarrays is both faster and easier to read!

As another exampleof how StaticArrays.jlcan speed up a more involved application,see the DifferentialEquations.jl docs.

In this post,we discussed StaticArrays.jl.We saw that StaticArrays are drop-in replacementsfor regular Julia Arrays.We also saw that using StaticArrayscan result in some nice speed-upsover using Arrays,at least when the sizes of the arraysare not too big.

Are array operations a bottleneck in your code?Try out StaticArrays.jland then comment below how it helps!

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