# How to solve this Ax=b faster?
**URL:** https://discourse.julialang.org/t/how-to-solve-this-ax-b-faster/81693
**Category:** Numerics
**Tags:** linearalgebra, linearsolve
**Created:** [May 26, 2022, 5:34am UTC](https://discourse.julialang.org/t/how-to-solve-this-ax-b-faster/81693 "2022-05-26T05:34:42Z")
**Posts on this page:** 5
**Page:** 2
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [May 28, 2022, 10:05pm UTC](https://discourse.julialang.org/t/how-to-solve-this-ax-b-faster/81693/21 "2022-05-28T22:05:02Z")
> [@rayegun](#):
>
> It looks like it’s calling CHOLMOD, by dispatching on `ishermitian` . Should we duplicate this codepath @ChrisRackauckas? I’m not even sure if we wrap CHOLMOD very well in LinearSolve.jl yet.
Oh, that completely changes the game. Yes, Cholesky factorizations are going to a way faster than any other factorization if the matrix is SPD! I didn’t catch that detail (now that I re-read, there was a quick mention of SPD).
Yes, we should wrap CHOLMOD for this case. Also, for SPD you shouldn’t use GMRES, instead you should use CG (`KrylovJL_CG`).
---
### Author:  [@Andrea\_Vigliotti](https://discourse.julialang.org/u/Andrea_Vigliotti)
#### Post date: [May 28, 2022, 10:47pm UTC](https://discourse.julialang.org/t/how-to-solve-this-ax-b-faster/81693/22 "2022-05-28T22:47:11Z")
[here](https://drive.google.com/drive/folders/16M8BBqg0odWY2CkKAw7e8yNbosihTNe1?usp=sharing) I uploaded a `JLD2` file and a small script that runs the tests.
following the output of the script
```julia
loading variables ...
A, r,
size(A) = (162372, 162372)
ishermitian(A) = true
norm(r) = 9027.184755276294
doing u = A\r ...
3.878 s (52 allocations: 1.39 GiB)
norm(r - A * u) = 5.766109468498555e-12
doing u = lu(A)\r ...
13.797 s (72 allocations: 2.39 GiB)
norm(r - A * u) = 4.227052608619862e-12
doing LinearSolve.solve(prob, UMFPACKFactorization()) ...
12.981 s (94 allocations: 2.71 GiB)
norm(r - A * sol.u) = 4.227052608619862e-12
doing LinearSolve.solve(prob, KrylovJL_CG()) ...
70.360 s (50 allocations: 179.79 MiB)
norm(r - A * sol.u) = 0.00013389304531825536
```
you can tell `KrylovJL_CG()` is a conjugate gradient method by looking at how little memory it allocates, I would also have expected a CG method would have been faster on a SPD problem (probably the the matrix is not big enough to outperform a very well optimized direct method)
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### Author:  [@amontoison](https://discourse.julialang.org/u/amontoison)
#### Post date: [June 7, 2022, 2:43am UTC](https://discourse.julialang.org/t/how-to-solve-this-ax-b-faster/81693/23 "2022-06-07T02:43:47Z")
@Andrea_Vigliotti, the convergence rate of CG hightly depends on the distribution of the eigenvalues of `A`.
If you have a good preconditioner, the Krylov method could only requires a few iterations.
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### Author:  [@PetrKryslUCSD](https://discourse.julialang.org/u/PetrKryslUCSD)
#### Post date: [June 7, 2022, 2:50am UTC](https://discourse.julialang.org/t/how-to-solve-this-ax-b-faster/81693/24 "2022-06-07T02:50:54Z")
The `\` is probably running `cholesky`, which is about 3x - 4x faster than `lu`.
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### Author:  [@Andrea\_Vigliotti](https://discourse.julialang.org/u/Andrea_Vigliotti)
#### Post date: [June 7, 2022, 8:40am UTC](https://discourse.julialang.org/t/how-to-solve-this-ax-b-faster/81693/25 "2022-06-07T08:40:32Z")
you are right, `u_ch=cholesky(A)\r` and `u=A\r` on this matrix take the same time, do the same number of allocation and find the same solution, `norm(u_ch-u)` is `0.0`
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