ralphas / genericschur.jl Goto Github PK

This package type pirates the eigen! function to provide support for bigfloat numbers:

This is no longer effective on 1.7 when one calls eigen.:

1.7:

julia> R = rand(BigFloat, 20, 20);

julia> using GenericSchur, LinearAlgebra

julia> eigen(R)
ERROR: MethodError: no method matching eigen!(::Matrix{BigFloat}; permute=true, scale=true, sortby=LinearAlgebra.eigsortby, jvl=false, jvr=true, jce=false, jcv=false)

1.6:

julia> R = rand(BigFloat, 20, 20);

julia> using GenericSchur, LinearAlgebra

julia> eigen(R)
Eigen{Complex{BigFloat}, Complex{BigFloat}, Matrix{Complex{BigFloat}}, Vector{Complex{BigFloat}}}
...

I am pondering to propose a roadmap for the extension of the MatrixEquations package to address the solution of Lyapunov, Sylvester and even Riccati equations for data types not covered by LAPACK (e.g., BigFloat, DoubleFloats). In the matrix equation solvers a central role is played by the Schur decomposition of the involved matrices or matrix pairs, both real and complex. In MatrixEquations , schur(A) and schur(A,B) are used for all element types covered by BlasReal and BlasComplex types. I wonder if I could entirely rely on the GenericSchur package to cover the required functionality for extended precision data types.

Hello there 👋

I am wondering if it would make sense to allow ForwardDiff.Dual numbers through the schur factorization in this package? Currently, many methods are restricted to

StridedMatrix{Complex{T}} where T<:AbstractFloat

which prevents the use of ForwardDiff.Dual since they are not <:AbstractFloat.

ilo, ihi, scale = LAPACK.gebal!('B', A)    # modifies A

It looks like this is:

A, B = balance!(A::AbstractMatrix{T})
ilo, ihi, scale  = B.ilo, B.ihi, ?

What matches the scale?

https://github.com/SciML/ExponentialUtilities.jl/blob/25a7ad9985db8fb11ea3aa5a40a1276e0a62c494/src/exp_noalloc.jl#L77-L87

Is this B.D?

MWE:

using GenericSchur
A = [big"2937189730080557577"   big"9536995145808582886"  big"11892438427558067162";big"6599805415728025309"  big"21429433573366650048"  big"26722066585196691901";big"5292633011830041853"  big"17185071439388109015"  big"21429433573366650048"]
cond(A) # does not work
cond(BigFloat.(A)) # does not work

Same thing works with GenericLinearAlgebra.jl

using GenericLinearAlgebra
A = [big"2937189730080557577"   big"9536995145808582886"  big"11892438427558067162";big"6599805415728025309"  big"21429433573366650048"  big"26722066585196691901";big"5292633011830041853"  big"17185071439388109015"  big"21429433573366650048"]
cond(A) # works 4.791463623157561751482371727718662967237981800410446716813214586738373018559972e+46
cond(BigFloat.(A)) # works 4.791463623157561751482371727718662967237981800410446716813214586738373018559972e+46

Tested on 1.8.5 and 1.9.

I noticed that you implemented Hessenberg solvers in https://github.com/RalphAS/GenericSchur.jl/blob/a361308be3b52faee82a9db30ea87d40d82fb0c4/src/hessenberg.jl

In JuliaLang/julia#31853 we are adding such solvers to LinearAlgebra — the algorithm we are using seems to be significantly faster (5-10x) than what you are doing now because it combines the RQ factorization and the triangular backsolve, and is able to do them both without modifying H.

In the near future, if you already have an upper-Hessenberg matrix H, you will be able to do Hessenberg(H) \ b to use the LinearAlgebra solver. If you need any additional functionality, please comment.

Hi,

I tried running GenericSchur on GPU (via Google Colab) and unfortunately get stuck very early on. He is resorting to LAPACK Schur, which is not implemented for GPUs.

Having a GPU implementation of generalised Schur for non-symmetric matrices would be a great addition. So far, neither cublas, arrayfire, nor any other blas/lapack alternative has it. Julia seems to be promising in getting there.

I am not proficient enough to debug the code myself but happy to help testing. I ran the following notebook to test and added the following code snippets to have GPU support in Julia:
using Pkg Pkg.add(["BenchmarkTools","CUDA","GenericSchur","GenericLinearAlgebra"]) using BenchmarkTools,CUDA,GenericLinearAlgebra import GenericSchur size = 50 arand = rand(size,size) brand = rand(size,size) GenericSchur.schur(arand,brand) agpu = CuArray(arand) bgpu = CuArray(brand) GenericSchur.schur(agpu,bgpu)

I hope this helps in getting closer to the issue.

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This came up in a nanosoldier run. I noticed that in this line, a vector is allocated whose eltype is abstract, since there is no eltype to Givens:

In every subsequent setindex!, this calls a convert routine (which does nothing I guess), but in case it's possible, it may be beneficial to precompute the correct eltype and set it upfront?