ConjugateGradients.jl is a flexible, non-allocating Julia implementation of the conjugate gradient and biconjugate gradient stabilized methods.


  • Julia 1.2 and up


julia> Pkg.add("")

Why use ConjugateGradients.jl?

There are a few great iterative solver packages available for Julia: IterativeSolvers.jl, KrylovMethods.jl, and Krylov.jl. These are all very well rounded and complete packages.

This package, ConjugateGradients.jl, is built around reducing allocations as much as possible for a particular type of problem. As far as I know, if your program will be using an iterative solver within another iterative process, this module will result in less allocations compared to the previously mentioned packages. Hint: take a look at ILUZero.jl if this type of solver would be beneficial to your project. Combined, these packages can help reduce allocations in those hot paths.

Also, in other iterative solvers, calls to BLAS functions are preferred for obvious reasons. This package uses Julia's multiple dispatch functionality to decide whether to use BLAS or native Julia code to make calculations based on the type associated with the arrays. This gives greater flexibility with types not represented by floating point numbers.

How to use

julia> using ConjugateGradients

For the conjugate gradient method to solve for x in Ax=b:

x, exit_code, num_iters = cg(A, b; kwargs...)
exit_code, num_iters = cg!(A, b, x; kwargs...)

For the biconjugate gradient stabilized method:

x, exit_code, num_iters = bicgstab(A, b; kwargs...)
exit_code, num_iters = bicgstab!(A, b, x; kwargs...)

Where A must be able to be applied as a function such that A(b, x) and the kwargs are:

  • tol = 1e-6: The tolerance of the minimum residual before convergence is accepted.
  • maxIter = 100: The maximum number of iterations to perform.
  • tolRho = 1e-40: [bicgstab only] The tolerance of dot(current residual, initial residual).
  • precon = nothing: The preconditioner. The preconditioner must act as an in-place function of the form f(out, in).
  • data = nothing: The preallocation of the arrays used for solving the system.


The data keyword points to an object containing the preallocated vectors necessary for the functions. If nothing is provided, these vectors will be allocated at each call. The data objects can be created like so:

CGD = CGData(n, T)
BCGD = BiCGStabData(n, T)

Here, n is the dimension of the problem and T is the type of the elements in the problem (e.g. Float64).

Deciphering the exit code

The exit_code can be read with the following function:

exit_string = reader(exit_code)

A tip for A and the preconditioner

The operator A and the preconditioner must be expressed as functions. If A is a matrix, one can do:

x, exit_code, num_iters = cg((x,y) -> mul!(x,A,y), b; kwargs...)

Another useful representation of A is a custom struct. For example, let's consider (B*C + D)x = b. Instead of wasting time to build B*C + D, we can create a non-allocating version of it.

struct MyA

function (t::MyA)(out::Vector{Float64}, x::Vector{Float64})
    mul!(t.cacheVec, t.C, x)
    mul!(out, t.B, t.cacheVec)
    mul!(t.cacheVec, t.D, x)
    out .+= t.cacheVec

A = MyA(B, C, D, zeros(n))