Is there any conjugate gradient (CG) implementation in Julia allowing one to specify the inner product (with a fixed symmetric and positive definite matrix `M`

, say)?

In my use case, I know my system matrix `A`

is symmetric and positive definite with respect to `M`

, i.e., `x' * M * A * x > 0`

for all `x != 0`

and `y' * M * (A * x) == (A * y)' * M * x`

for all `x, y`

. Ideally, I would like to avoid scaling the system matrix and vectors myself for using a CG method. Just curious to know whether there’s such a flexible package out there.

I don’t know the answer to the question about specific implementations of CG, but since your property of being symmetric and positive definite with respect to M is equivalent to MA being symmetric positive definite, it seems reasonable to just solve MA x = Mb using the CG routine from `IterativeSolvers.jl`

and form MA as a product of `LinearMaps`

:

```
using IterativeSolvers
using LinearMaps
C = LinearMap(M) * LinearMap(A)
x=cg(C, M*b)
```

That’s pretty easy. In fact, since CG includes more inner products than matrix-vector multiplies, I think this actually involves fewer multiplies by M than if you tried to implement CG for a different inner product. If multiplying by M is nontrivial, this is probably faster than what you were hoping to do.

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Conjugate Gradients · IterativeSolvers.jl has the optional argument `Pl`

to add a left precondition. (That should correspond to a change to the inner product as @mstewart explained)

As the documentation say it uses a different algorithm which is specialised for this setting.

It’s a tempting idea, and there may be some trick to make use of a preconditioner, but doing it in a direct way for this problem doesn’t seem like an option, given that PCG assumes that both the preconditioner and A are symmetric positive definite. That’s not the case here.

KrylovKit supports any vector type that have custom dot and norm methods, without assuming they are the standard ones. See eg Introduction · KrylovKit.jl

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Thank you all! I can only select one answer and have a hard time selecting one

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