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https://github.com/mcovalt/ConjugateGradients.jl provides CG and BiCGStab. In its README.md, it says
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.
I haven't tested it, but this sounds at least interesting enough to provide wrappers in LinearSolve.jl.
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