MutableArithmetics.jl

MutableArithmetics (MA for short) is a Julia package which allows:

• for mutable types to implement mutable arithmetics;
• for algorithms that could exploit mutable arithmetics to exploit them while still being completely generic.

While in some cases, similar features have been included in packages idiosyncratically, the goal of this package is to provide a generic interface to allow anyone to make use of mutability when desired.

The package allows a given type to declare itself mutable through the MA.mutability trait. Then the user can use the MA.operate!! function to write generic code that works for arbitrary type while exploiting mutability of the type if possible. More precisely:

• The MA.operate!!(op::Function, x, args...) redirects to op(x, args...) if x is not mutable or if the result of the operation cannot be stored in x. Otherwise, it redirects to MA.operate!(op, x, args...).
• MA.operate!(op::Function, x, args...) stores the result of the operation in x. It is a MethodError if x is not mutable or if the result of the operation cannot be stored in x.

So from a generic code, MA.operate!! can be used when the value of x is not used anywhere else to recycle it if possible. This allows the code to both work for mutable and for non-mutable type.

When the type is known to be mutable, MA.operate! can be used to make sure the operation is done in-place. If it is not possible, the MethodError allows to easily fix the issue while MA.operate!! would have silently fallen back to the non-mutating function.

In conclusion, the distinction between MA.operate!! and MA.operate! allows to cover all use case while having an universal convention accross all operations.

The following types implement the MutableArithmetics API:

• The API is implemented for Base.BigInt in src/bigint.jl.
• The API is implemented for Base.BigFloat in src/bigfloat.jl.
• The API is implemented for Base.Array in src/linear_algebra.jl.
• The Polynomial type of Polynomials.jl.
• The interface for multivariate polynomials MultivariatePolynomials as well as its two implementations DynamicPolynomials and TypedPolynomials.
• The scalar and quadratic functions used to define an Optimization Program in MathOptInterface.
• The scalar and quadratic expressions used to model optimization in JuMP.

The algorithms from the following libraries use the MutableArithmetics API to exploit the mutability of the type when possible:

• The multivariate polynomials implemented in MultivariatePolynomials, DynamicPolynomials and TypedPolynomials work with any type and exploit the mutability of the type through the MA API.

In addition, the implementation of the following functionalities available from Base are reimplemented on top of the MA API:

• Matrix-matrix, matrix-vector and array-scalar multiplication including SparseArrays.AbstractSparseArray, LinearAlgebra.Adjoint, LinearAlgebra.Transpose, LinearAlgebra.Symmetric.
• Base.sum, LinearAlgebra.dot and LinearAlgebra.diagm.

These methods are reimplemented in this package for several reasons:

• The implementation in Base does not exploit the mutability of the type (except for sum(::Vector{BigInt}) which has a specialized method) and are hence much slower.
• Some implementations in Base assume the following for the types S, T used satisfy:
  • typeof(zero(T)) == T, typeof(one(T)) == T, typeof(S + T) == promote_type(S, T) or typeof(S * T) == promote_type(S, T) which is not true for instance if T is a polynomial variable or the decision variable of an optimization model.
  • The multiplication between elements of type S and T is commutative which is not true for matrices or non-commutative polynomial variables.

The trait defined in this package cannot make the methods for the functions defined in Base to be dispatched to the implementations of this package. For these to be used for a given type, it needs to inherit from MA.AbstractMutable. Not that subtypes of MA.AbstractMutable are not necessarily mutable, for instance, polynomial variables and the decision variable of an optimization model are subtypes of MA.AbstractMutable but are not mutable. The only purpose of this abstract type is to have Base methods to be dispatched to the implementations of this package. See src/dispatch.jl for more details.

Documentation

• STABLE — most recently tagged version of the documentation.
• LATEST — in-development version of the documentation.

Quick Example & Benchmark

using BenchmarkTools
using MutableArithmetics
const MA = MutableArithmetics

n = 200
A = rand(-10:10, n, n)
b = rand(-10:10, n)
c = rand(-10:10, n)

# MA.mul works for arbitrary types
MA.mul(A, b)

A2 = big.(A)
b2 = big.(b)
c2 = big.(c)

The default implementation LinearAlgebra.generic_matvecmul! does not exploit the mutability of BigInt is quite slow and allocates a lot:

using LinearAlgebra
trial = @benchmark LinearAlgebra.mul!($c2, $A2, $b2)
display(trial)

# output

BenchmarkTools.Trial: 407 samples with 1 evaluation.
 Range (min … max):   5.268 ms … 161.929 ms  ┊ GC (min … max):  0.00% … 73.90%
 Time  (median):      5.900 ms               ┊ GC (median):     0.00%
 Time  (mean ± σ):   12.286 ms ±  21.539 ms  ┊ GC (mean ± σ):  29.47% ± 14.50%

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  ██▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▅█▆▇▅▅ ▆
  5.27 ms       Histogram: log(frequency) by time      80.6 ms <

 Memory estimate: 3.66 MiB, allocs estimate: 197732.

In MA.operate!(::typeof(MA.add_mul), ::Vector, ::Matrix, ::Vector), we exploit the mutability of BigInt through the MutableArithmetics API. This provides a significant speedup and a drastic reduction of memory usage:

trial2 = @benchmark MA.add_mul!!($c2, $A2, $b2)
display(trial2)

# output

BenchmarkTools.Trial: 4878 samples with 1 evaluation.
 Range (min … max):  908.860 μs …   1.758 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):       1.001 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):     1.021 ms ± 102.381 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

  █▅
  ██▂▂▂▇▅▇▇▅▅▅▇▅▆▄▄▅▄▄▃▄▄▃▃▂▃▃▂▃▂▂▂▂▂▂▂▂▂▂▂▁▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
  909 μs           Histogram: frequency by time         1.36 ms <

 Memory estimate: 48 bytes, allocs estimate: 3.

There is still 48 bytes that are allocated, where does this come from ? MA.operate!(::typeof(MA.add_mul), ::BigInt, ::BigInt, ::BigInt) allocates a temporary BigInt to hold the result of the multiplication. This buffer is allocated only once for the whole matrix-vector multiplication through the system of buffers of MutableArithmetics. If may Matrix-Vector products need to be computed, the buffer can even be allocated outside of the matrix-vector product as follows:

buffer = MA.buffer_for(MA.add_mul, typeof(c2), typeof(A2), typeof(b2))
trial3 = @benchmark MA.buffered_operate!!($buffer, MA.add_mul, $c2, $A2, $b2)
display(trial3)

# output

BenchmarkTools.Trial: 4910 samples with 1 evaluation.
 Range (min … max):  908.414 μs …   1.774 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     990.964 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):     1.014 ms ± 103.364 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

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  ██▃▂▂▄▄▅▆▃▄▄▅▄▄▃▃▄▃▃▃▃▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
  908 μs           Histogram: frequency by time         1.35 ms <

 Memory estimate: 0 bytes, allocs estimate: 0.

Note that there are now 0 allocations.

This package started out as a GSoC '19 project.