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This is a library of numerically stable linear algebra routines for performing inversions as they appear in the calculation of Green's functions in determinant Quantum Monte Carlo.

For more details, check out the documentation and the accompanyig paper, in which we describe and benchmark a few specific algorithms. The plots in this paper have been generated with the notebooks in the notebook directory of this repository.

Feel free to give feedback, open issues, or contribute useful algorithms yourself! 🙂

The package is registered in Julia's General registry. You can install it using

] add StableDQMC

The package has only one dependency, Requires.jl.

Left: Slice matrix product. Right: Equal-times Green's function.

Note that "SVD (D&C)" is the algorithm used by Julia's svd function.

In DQMC, we commonly perform inversions like G = [1 + B]^-1 to obtain the equal-times Green's function and G = [A + B]^-1 for the time-displaced pendant. The following methods are exported to facilitate these tasks.

• inv_one_plus, inv_one_plus!
• inv_sum, inv_sum!

When function names are suffixed with _loh, i.e. inv_one_plus_loh, a more sophisticated method is used for numerical stabilization (see the paper linked above for more details).

julia> using LinearAlgebra

julia> B = rand(ComplexF64, 100,100);

julia> Bfact = udt(B);

julia> G = inv_one_plus_loh(Bfact);

Since the matrix B is well-conditioned in this case, we have

julia> G ≈ inv(I + B)
true

Based on the QR decomposition, we introduce a UDT factorization, where U is unitary, D is real-valued and diagonal, and T is upper-triangular. To decompose a given matrix M the udt function is exported.

julia> M = rand(10,10);

julia> udt(M)
UDT{Float64,Float64,Array{Float64,2}}([-0.246588 0.12668 … 0.582208 0.206435; -0.373953 -0.300804 … 0.152994 0.0523203; … ; -0.214686 -0.403362 … -0.124248 -0.390502; -0.40412 -0.147009 … 0.1839 0.197964], [2.15087, 1.47129, 1.14085, 0.911765, 0.850504, 0.620149, 0.545588, 0.412213, 0.305983, 0.148787], [-0.597235 -1.0 … -0.678767 -0.59054; -0.385741 0.0 … -1.0 -0.361263; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0])

In our tests (see paper/), this decomposition turns out to be superior to SVD for DQMC.

The package provides convenient access to several LAPACK algorithms for calculating singular value decompositions (SVDs):

• gesdd, gesdd!: Divide and conquer
• gesvd, gesvd!: Regular
• gesvj, gesvj!: Jacobi (based on JacobiSVD.jl)

Furthermore, you can access a type-generic, pure Julia implementation,

• genericsvd, genericsvd! (based on GenericSVD.jl)

However, to keep the dependencies of the package minimal, only the first two are available by default and loading of the Jacobi or type-generic SVD is opt-in. We provide convenience functions StableDQMC.addJacobiSVD() and StableDQMC.addGenericSVD() to facilitate this process. See below for a quick demonstration.

julia> using StableDQMC

julia> gesvj
ERROR: UndefVarError: gesvj not defined

julia> StableDQMC.addJacobiSVD()
  Updating registry at C:\Users\carsten\.julia\registries\General
  Updating git-repo https://github.com/JuliaRegistries/General.git
  Updating git-repo https://github.com/RalphAS/JacobiSVD.jl
 Resolving package versions...
  Updating C:\Users\carsten\Desktop\stabledqmctest\Project.toml
  [2ca068c6] + JacobiSVD v0.0.0 #master (https://github.com/RalphAS/JacobiSVD.jl)
  Updating C:\Users\carsten\Desktop\stabledqmctest\Manifest.toml
  [2ca068c6] + JacobiSVD v0.0.0 #master (https://github.com/RalphAS/JacobiSVD.jl)
┌ Warning: Package StableDQMC does not have JacobiSVD in its dependencies:
│ - If you have StableDQMC checked out for development and have
│   added JacobiSVD as a dependency but haven't updated your primary
│   environment's manifest file, try Pkg.resolve().
│ - Otherwise you may need to report an issue with StableDQMC
└ Loading JacobiSVD into StableDQMC from project dependency, future warnings for StableDQMC are suppressed.
[ Info: Recompiling stale cache file C:\Users\carsten.julia\compiled\v1.1\JacobiSVD\Frhox.ji for JacobiSVD [2ca068c6-2156-5cf0-8317-c67edf277a2c]

julia> gesvj # gesvj and gesvj! are available now
gesvj (generic function with 1 method)

Provided that you have JacobiSVD.jl (or GenericSVD.jl) installed, you can get the LAPACK access functions gesvj, gesvj! (or genericsvd, genericsvd!) simply by import JacobiSVD (import GenericSVD).

julia> using StableDQMC

julia> gesvj
ERROR: UndefVarError: gesvj not defined

julia> import JacobiSVD # using might lead to name conflicts

julia> gesvj # gesvj and gesvj! are available now
gesvj (generic function with 1 method)