I thought that inverting a symmetric matrix would be faster, since there are optimised algorithms and I also saw that there is a method explicity for Symmetric, but maybe I am overlooking some trivial things, so please help :wink: This is the method I found with methods(inv) [9] inv(A::LinearAlgeb…

Not an expert here, but when you use BenchmarkTools, don’t forget to interpolate all global variables with a dollar sign to avoid skewed results. I’m assuming it doesn’t change much in this case?

As far as I can tell from the source code, both code paths lead to an lu call and the Symmetric method mostly makes sure that the output also becomes Symmetric. I suspect that you are drawing too strong complexity conclusions from your samples.

If you follow the chain of calls (tip: use @edit inv(Symmetric(randn(2,2)))) you’ll see that in the end the same method (lu) is called. You can access LAPACK’s specialized symmetric routines with something like function inv_sym(A) A, ipiv, info = LAPACK.sytrf!('U', A) LAPACK.sytri!('U', A, …

[image] tamasgal: also I see O(N^2), so I guess it’s using the Coppersmith–Winograd algorithm and not something specialised like LDU: This has nothing to do with the underlying algorithm (which is not coppersmith-winograd or any other fancy algorithm, and is just plain old N^3), and is probab…

No worries, I checked that before of course. No problems here :wink:

[image] tamasgal: M = Symmetric(A'A) This isn’t just symmetric, it’s symmetric positive-definite, in which case you could call the cholesky factorization, which really is faster than lu. (Of course, the usual caveats about matrix inversion apply — most of the time, you don’t need to compute …

Thanks, I am a bit confused since the inv_sym() is twice as slow compared to inv(), at least on my M1. I don’t have access to MKL on this machine since it’s not Intel. Anyways, I think it would make sense to use the power of (multiple) dispatch to call the optimised algorithm, otherwise I don’t rea…

Anyways, I think it would make sense to use the power of (multiple) dispatch to call the optimised algorithm, otherwise I don’t really see the point of a special method for int() . The problem is that which algorithm is faster very much depends on your matrix size, BLAS implementation and hardwar…

julia> N = 4000; A = rand(N, N); M = Symmetric(A'A); julia> @benchmark inv($M) BenchmarkTools.Trial: 10 samples with 1 evaluation. Range (min … max): 537.033 ms … 539.106 ms ┊ GC (min … max): 0.14% … 0.08% Time (median): 538.214 ms ┊ GC (median): 0.14% Time (mean ± σ): …

WRT the quadratic time you’re seeing, that’s just because LU factorizations have high quadratic overhead and look like they have quadratic looking runtime for a shockingly long time. Looking at the plot below, you can see that for sizes between 8x8 and 512x512, the time for inv is almost completely…

Inverting a symmetric matrix is not faster than inverting a random one

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Inverting a symmetric matrix is not faster than inverting a random one

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